ABSTRACT. Transferring predictive power from past data to similar future situations is a central objective in statistical learning and stochastic optimization. In the past years, distributionally robust optimization has become popular by providing a mathematically elegant proxy for this problem by relying on optimal transport between distributions. We will cover recent advances on this approach but also its limitation in deep learning and related perspectives.

ABSTRACT. The expansion planning (EP) problem in energy systems optimization aims to determine long-term investment decisions while ensuring the feasibility of short-term system operations. Long-term decisions concern investments in generation and storage capacities, as well as capacity decommissioning and lifetime extension, whereas short-term operational decisions involve generation and storage scheduling subject to supply--demand balance, technical, and network constraints~\cite{lara2018}.

We consider Benders decomposition to exploit this natural separation between investment and operational decisions. However, accurately representing short-term operations over a long planning horizon leads to large operational problems, particularly in the presence of time-coupling constraints. To address this issue, we rely on a temporal decomposition that separates the operational problem into smaller subproblems~\cite{jacobson2024}. At each Benders iteration, these subproblems can generate several candidate cuts for the current master-problem solution, raising the question of which cuts should be added to the master problem.

In this talk, we present this temporal decomposition approach and then focus on cut selection, investigating how different selection policies affect the performance of the Benders decomposition algorithm. We review several strategies from the literature~\cite{hosseini2024} and discuss theoretical properties of Benders cuts. Finally, we provide a numerical analysis on capacity expansion instances derived from the ERAA dataset to compare the performance of these selection strategies.

ABSTRACT. We study the Nuclear Outage Planning Problem (NOPP). This problem consists of finding a minimum-cost refuelling outage plan for a fleet of nuclear power plants over a multi-year horizon. Refuelling outages are one of the most complicated phases of a plant's operating cycle, requiring thousands of activities to be scheduled within a tight window. Furthermore, these outages are prone to extensions of uncertain duration. An outage that overruns shifts the start of the following production phase, and alters the unit's fuel depletion trajectory. This in turn may raise the cost of meeting demand, delay the outages of other units that share the same resources, or in the worst case leave the fleet without a feasible plan.

We propose a two-stage stochastic mixed-integer formulation to decide when to schedule the outages. The first stage selects, for each unit, a path through a transition graph encoding all feasible multi-cycle operational plans. The second stage adapts production dispatch, fuel level and production mode (full power, modulation, or a mandated decreasing power profile) to the realised outage durations in each scenario used to capture uncertainty. Our numerical experiments use the full 58-unit French fleet, with fleet parameters and historical outage durations supplied by EDF and demand taken from RTE's eCO2mix database.

The value of the stochastic formulation comes from infeasibility rather than cost. We observe that the generation cost is almost invariant across feasible schedules because the merit order and the reserve margin fix each technology's contribution to demand before any optimisation takes place. What changes instead is whether a feasible schedule exists at all. Feasible schedules of the deterministic instance are found to be infeasible as soon as any duration uncertainty is introduced. This is due to the limit on how many units can be in outage simultaneously being breached.

The mechanism used for sampling scenarios determines the frequency and severity of infeasibility. Comonotonic sampling, in which either all outages run as planned, or all overrun by the same percentile, produces infeasibility at the majority of sites but only under extreme realisations. Independent sampling produces infeasibility in the majority of realisations, but only at a few sites. Under both mechanisms the stochastic plan violates the site limits substantially less often than the deterministic one.

A limitation of the present numerical experiments is the planning horizon. The computational complexity grows linearly with the scenario count and quadratically with the horizon length. Over a single year most units undergo only one outage, so the propagation of an overrun into a unit's subsequent production cycle is not observed. Capturing this requires a horizon of at least two cycles per unit, and therefore decomposition.

ABSTRACT. SESSION: "Applications of optimization to power systems and networks"

We consider the problem of setting tolls on a subset of the arcs of a congested network so as to maximize the revenue of a network authority, anticipating that users route according to a Wardrop equilibrium. This problem is naturally cast as a bilevel program in which the leader sets tolls while the followers respond by choosing an equilibrium routing.

We focus on the important special case of separable affine arc cost functions. Under this assumption, Wardrop equilibria admit Beckmann's convex characterization, from which we derive three new single-level reformulations of the bilevel problem, exploiting both Beckmann's formulation and its dual: an arc-based formulation, which is exact, and two path-based formulations, solved by column-and-constraint generation. Beyond the reformulations, we further derive from Beckmann's potential an upper bound on the optimal revenue that can be used to certify optimality gaps uniformly across all formulations.

An extensive computational study on instances derived from the Sioux Falls and Eastern Massachusetts networks compares our formulations with the recent mixed-integer nonlinear formulation of Beck, Labbé and Schmidt [1]. The arc-based formulation solves substantially more instances to proven optimality (within a time limit of 1800s) than the formulation of [1] and, unlike it, returns a feasible solution on every instance, remaining effective on the largest 74-node network, on which [1] solves few instances, if any. The magnitude of the upper bounds on the tolls proves to be a major driver of tractability: the genuinely hard core of the benchmark emerges once the regulatory cap on tolls is removed. The two path-based formulations, used as heuristics, return a solution within 1% of the optimum on every instance whose optimal value is known, often in seconds where exact formulations reach the time limit.

[1] Y. Beck, M. Labbé, M. Schmidt, A toll-setting problem with robust Wardrop equilibrium conditions under budgeted uncertainty, Optimization Online, 2024.

ABSTRACT. Bike-sharing systems (BSS) are among the most widely adopted new shared mobility services. Because they rely on one-way trips, changes in demand cause imbalances, with bikes depleted in some locations and overflowing in others. Traditionally, this challenge is addressed through costly truck-based rebalancing operations. In high-traffic zones or during peak time periods, precisely when reliability is most critical, operators face a fundamental challenge between service and inventory management, which in station-based BSS, increases further with capacity restrictions. To improve the reliability of the service, BSS operators have introduced reservation options that guarantee bike availability at specific stations and times, which comes at a trade-off in service levels. Reserving bikes too aggressively reduces system utilization while exposing the operator to the risk of not fulfilling future commitments.

In this study, we address a choice-based dynamic pricing problem for demand management in BSS networks with a service-level-maximizing objective. We propose a novel multistage stochastic program (MSSP) in which the operator offers a menu of alternatives for each origin station, including destination stations, deferred departures, and prices. The users' choices are integrated through a multinomial choice model, where they either choose a trip alternative from the offered menu or opt out. We also include exogenous pre-booked trips; hence, the operator must decide based on residual capacities.

We propose a rolling horizon algorithm to reduce the size of the MSSP by combining a stochastic decision window and expected value approximation of the remaining horizon in a single optimization model, enabling menu decisions that adapt to the observed network state. Furthermore, we develop a station-wise Lagrangian decomposition for scalability. The proposed model and solution algorithms are evaluated on a synthetic network with instances diversified in size and demand patterns.

ABSTRACT. The single-plant hydro unit commtment problem (1-HUC) involves determining the turbine flow rate at each time step over a discretized horizon in order to maximize production value. One of the challenges lies in the power function - which is neither linear nor concave- requiring approximation. Currently, EDF discretizes this function into a finite set of operating points (PDF), significantly limiting the range of obtainable solutions. This internship investigates the alternative of piecewise linear (PWL) approximation, which allows for continuous flow rates within each segment.

We proposed five mixed-integer linear programming models for this approximation. They encode the same linearization and differ only in the method used to select the active segment: one binary variable per segment, a logarithmic encoding of these variables, one variable per concave piece, an incremental formulation requiring integrality only for the segments that initiate a piece, and Gurobi's native piecewise linear constraint. The best-performing model was then compared against the PDF model using an equivalent approximation budget -- that is, matching the number of operating points to the number of segments in the linearization. As existing problem instances could not be directly applied to our models, we also developed a generator for realistic and varied instances.

Experiments conducted on 5,200 configurations show that the incremental formulation outperforms the others. For an equivalent budget, the PWL approach yields better solutions than the PDF approach while also being faster. Piecewise linear approximation therefore represents a viable alternative to operating-point discretization for the 1-HUC problem.

ABSTRACT. The Hydro Unit Commitment (HUC) problem is a difficult sub-component of the Unit Commitment problem (UCP) in electricity production planning. We consider a reservoir valley in which several hydroelectric power plants are interconnected. The objective of the HUC is to optimize the planning of the hydroelectric units in order to satisfy the demand while maximizing the profit and respecting the operational constraints. Its combinatorial complexity as well as its strict volume constraints make it particularly difficult and real large-scale instances remain hard to solve in a satisfactory way. These problems have so far mostly been modeled using linear optimization with integer variables, which remains the dominant approach in the literature. We focus here on the combinatorial structure induced by the discrete operating points and the target volume constraints of the reservoirs. When the initial and final volumes of a reservoir are fixed, they determine a target volume to be turbined by each plant over the horizon. Since plants operate on a discrete set of cumulative flows, reaching this target constrains the possible combinations of operating points over time. We propose a new extended formulation based on these target volumes, using cardinality variables to describe how many periods are spent at each operating point. We first investigate a preprocessing approach that enumerates all combinations compatible with the given target. We then propose a decomposition scheme that avoids full enumeration by alternating between two mixed-integer programs. One relaxes operating variables to continuous values while keeping these cardinalityvariables integer, generating candidate combinations with an upper bound on their value. The other fixes a candidate combination and checks whether it can be realized by an integer-feasible schedule. The combination is then excluded from subsequent iterations via a no-good cut, until optimality is proven. These approaches proved to be very effective and led to significant improvements in computation times on our set of instances.

ABSTRACT. In this paper, we consider polyhedral symmetries, defined as permutations of variables that preserve feasibility independently of the objective function. Unlike classical symmetry-handling techniques, we develop a Branch&Cut framework with a separation in two-phases that leverages these symmetries to structure families of valid inequalities and enhance their separation. The first phase generates representative valid inequalities based solely on the constraint set, while the second phase separates their symmetry orbits during the Branch-and-Cut process. For column-wise symmetric polyhedral symmetry groups, we show that the orbit-separation problem reduces to a maximum matching problem which can be solved in polynomial time. We apply the proposed framework to the Multiple Choice knapsack problem, the core combinatorial structure of the Hydro Unit Commitment problems at EDF. Computational results show that exploiting polyhedral symmetries significantly strengthens the separation process and improves overall solution performance.

ABSTRACT. Orbital rendezvous requires efficient maneuvers while ensuring collision avoidance throughout the trajectory. We consider this problem in the context of ONERA's FlyLab mission, involving two nanosatellites in low Earth orbit with different propulsion systems and limited communication capabilities. These constraints lead to an open-loop relative guidance problem, which we formulate as an optimal control problem aiming to minimize the fuel consumption of the chaser satellite and to reach a final configuration while satisfying several operational constraints as well as a minimum safety distance constraint. In this talk, we present the modeling of the problem and introduce a simplified state-constrained formulation that allows us to analyze the structure and derive some qualitative properties using the Pontryagin Maximum Principle (PMP).

ABSTRACT. In the framework of optimal control, the Pontryagin Maximum Principle (PMP) is a well established and universal tool used to solve optimal control problems. However, PMP gives only necessary conditions of optimality and a trajectory satisfying them need not to be optimal in general. Moreover, even if some arc of such a trajectory is known to be optimal, PMP alone does not determine where the trajectory ceases to be an optimizer. For these reasons, second- order optimality conditions were introduced. I this talk, I will discuss some recent optimality conditions of second order that we obtained [1], particularly for piecewise regular extremal arcs, that is concatenation of extremal arcs satisfying the strong Legendre condition. Finally, I will discuss some simple examples to illustrate how our optimality conditions work in practice.

References [1] Andrei A. Agrachev and Michele Motta. Second order optimality conditions for piecewise regular extremals in Optimal Control. 2026. arXiv: 2607.10434 [math.OC]. url: https: //arxiv.org/abs/2607.10434.

ABSTRACT. Invited session: Geometric Control and Optimization

When considering an optimal control problem, we say that two distinct extremals branch if, up to time translations, they coincide on a subinterval. Our goal is to describe branching properties of optimal trajectories for a class of optimal control problems with cost J(u) = ∫︁|u|^2 dt, where u denotes the control. We consider nonlinear control systems in R^n, or more generally on a manifold M, with dynamics that are linear in the controls. By introducing the endpoint map it is possible to rephrase the optimal control problem as a constrained minimization problem on L^2 and deduce, as an application of the Lagrange multipliers rule, the existence of two classes of candidate optimal trajectories: normal and abnormal extremals. Normal extremals are smooth and can be lifted to solutions of a Hamiltonian system on T^∗M. Abnormal extremals do not depend on the particular choice of the cost and may be nonsmooth. However, their presence must be taken into account since there exist optimal trajectories that are strictly abnormal (i.e. they are not also normal).

In [1], it was proven that normal extremals can branch, and a characterization of the branching times was provided. Our study involves a relevant class of smooth abnormal extremals for systems with two-dimensional controls, called regular [2]. These extremals, analogously to normal ones, can be lifted to solutions of a Hamiltonian system on a submanifold R of T^∗M. This allows to adapt the arguments of [1], obtaining a characterization of their branching times which is formally analogous to the one given in [1] for normal extremals. We also discuss two cases in which the branching curves are strictly abnormal; in one of them, we consider a reference path that is both normal and regular abnormal, that splits at the same time into two distinct families of branching curves that are, respectively, normal and regular abnormal, proving that the regular abnormal ones are also strictly abnormal.

Finally, we will discuss examples of extremals with an arbitrarily large number of branchings, possibly accumulating at finite time.

References [1] T. Mietton and L. Rizzi. Branching geodesics in sub-Riemannian geometry. Geom. Funct. Anal., 30(4):1139–1151, 2020 [2] W. Liu and H. J. Sussman. Shortest paths for sub-Riemannian metrics on rank-two distributions. Mem. Amer. Math. Soc., 118(564):x+104, 1995

ABSTRACT. In this work, we investigate the resilience against adversarials of quadratically regularised medians in Wasserstein spaces. After highlighting the sensitivity of classical Wasserstein barycenters measured in terms of their breakdown point, we show that two models of regularised median enjoy a strong form of resilience, in which the total displacement induced by an adversary cannot exceed the ratio of its weight, divided by the regularisation parameter. Our results are illustrated by numerical simulations, relying on a variety of methods and schemes.

ABSTRACT. Optimal transport is widely used in machine learning for learning distributions, enforcing distributional constraints [3], and modeling risk aversion [2]. In one dimension, transport costs can be computed in closed form by sorting values, enabling efficient differentiation and making them practical components of learning pipelines [4, 5]. These costs are usually approximated through sampling. We study parameterized functions defined by transport costs and show that, as the sample size grows, the graphs of their subdifferentials converge [1]. In particular, this result guarantees that standard subgradient methods consistently approach population-level critical points. We illustrate our results in several settings, including risk-averse optimization, fairness, and sliced Wasserstein costs.

ABSTRACT. Day-ahead renewable nomination requires committing energy before both generation and settlement prices are known. Wasserstein distributionally robust optimization (DRO) provides a natural framework for hedging against uncertainty, especially when historical data are scarce. However, when both the power generation and settlement prices are unknown, the objective function becomes nonconvex in the uncertainty. This makes the standard tractable convex reformulation of Wasserstein DRO inapplicable directly. A common technique to restore tractability consists in linearizing the loss at the cost of excessive conservatism, which we can address via an appropriate McCormick relaxation. An alternative approach is to directly solve the original nonconvex problem. By exploiting the low-dimensional structure induced by a box support and a weighted $\ell_1$ transportation cost, we prove that each sample-wise separation problem attains its optimum on finitely many uncertainty states, yielding an exact finite reformulation. Synthetic experiments show that distributional robustness matters when limited or highly variable data misestimate marginal imbalance incentives. Experiments on actual French market data confirm that distributionally robust methods can tame exposure to extreme imbalance episodes, at the cost of lower average profit.

ABSTRACT. Since the early 2000s, real algebraic geometry found a new application in the moment-SoS hierarchy, a framework that leverages Putinar’s Positivstellensatz to solve generalized moment problems (GMP). This methodology was first applied to polynomial optimization, before finding new applications in optimal control, volume computation, option pricing, stability analysis, study of stochastic processes, solving PDEs and super resolution, to name a few. The moment-SoS hierarchy relies on positivity certificates, that consist in polynomial sums of squares of bounded degree. Finding the maximum degree bound required to certify positivity of a given polynomial has been an important research topic, to which a recent contribution, namely the Effective Putinar Positivstellensatz, provided a promising answer. Based on this, it is possible to compute the convergence rate of the moment-SoS hierarchy, depending on the considered GMP instance. In this talk, I will explain this new Effective Putinar Positivstellensatz, and illustrate its application to computing the convergence rate of some key applications of the moment-SoS hierarchy.

ABSTRACT. Policy gradient methods are known to be highly sensitive to the choice of policy parameterization. In particular, the widely used softmax parameterization can induce ill-conditioned optimization landscapes and lead to exponentially slow convergence. Although this can be mitigated by preconditioning, this solution is often computationally expensive. Instead, we propose replacing the softmax with an alternative family of policy parameterizations based on the generalized f-softargmax. We further advocate coupling this parameterization with a regularizer induced by the same f-divergence, which improves the optimization landscape and ensures that the resulting regularized objective satisfies a Polyak--Łojasiewicz inequality. Leveraging this structure, we establish the first explicit non-asymptotic last-iterate convergence guarantees for stochastic policy gradient methods for finite MDPs without any form of preconditioning. We also derive sample-complexity bounds for the unregularized problem and show that f-PG, with Tsallis divergences achieves polynomial sample complexity in contrast to the exponential complexity incurred by the standard softmax parameterization.

ABSTRACT. We provide a novel analysis of the stochastic proximal point (SPP) algorithm under remarkably weak assumptions. Namely, we provide explicit convergence rates without requiring the objective function to satisfy a regularity condition, such as Lipschitz continuity, which is a very standard assumption in the analysis of this algorithm \cite{ref3, ref4}, or smoothness, less commonly used in the SPP literature. Existing results \cite{ref} under our working assumptions provide convergence guarantees, but typically do not provide explicit convergence rates, and the present work fills this gap. Our analysis is based on a non-standard choice of metric for monitoring the progress of the algorithm: rather than the usual optimality gap in the convex setting, we consider a regularized version of the optimality gap. We extend our analysis to a stochastic forward-backward algorithm, where both the proximal backward step and the (sub-)gradient forward step are stochastic (in the fashion of \cite{ref, ref2}). Finally, we emphasize that our analysis does not rely on a strong variance assumption, such as a uniform bound on the gradient variances; instead, we require the variance to be finite only at a solution of the problem.

ABSTRACT. We analyze the constant step size subgradient method on nonsmooth, nonconvex functions. We identify geometric assumptions on the objective function under which i) its domain admits a partition (stratification) into smooth manifolds (strata) on which the function is smooth; ii) a global projection formula for Clarke subgradients holds; and iii) quantitative curvature bounds hold on each stratum.

Under these conditions, we prove that the iterates of the subgradient method locally shadow a Riemannian gradient descent on nearby strata, which we use to measure stationarity. Introducing a selection rule for the active stratum, we assemble local descent inequalities across successive strata into explicit convergence rates, expressed through a conservative set-valued field associated to the stratification.

Finally, we show that our assumptions are automatically satisfied for semialgebraic functions, yielding the first known convergence rates in this setting.

ABSTRACT. We consider the problem of routing robotaxis, i.e., autonomous vehicles for taxi transportation, within in a city. We assume that the city is divided into regions and that demand is organized in origin-destination markets. We model the operations of robotaxis in this network under steady-state conditions and we aim to determine optimal dispatching and repositioning strategies. We assume that vehicle-passenger matching is endogenous to the state of the system, i.e., in each region, the number of idle vehicles affects traveler's waiting time and thus service quality. We adopt the stock-based matching function from the transit systems literature which is based on the nearest idle vehicle geometry. We embed the stock-based matching function within a network flow model to represent multi-region vehicle dispatching operations. This leads to a nonconvex mathematical optimization formulation and we conduct a theoretical analysis. To solve this mathematical optimization problem, we develop a spatial branch-and-bound algorithm that exploits several convex relaxations. In particular, we identify a relaxation of the matching function that is second-order cone representable. We report preliminary results on problem instances based on realistic urban transportation networks that illustrate the impact of endogenous en-routes times onto system performance and service quality.

ABSTRACT. In the fresh-food supply chain—specifically at the Rungis food market, multiple suppliers each sell a different type of goods (meat; fish; and fruits and vegetables). Each must independently perform deliveries to their own set of clients, and there is heavy overlap between these sets of clients, most of them being located in close proximity to each other. This independent routing leads to two types of inefficiencies: partially filled trucks and multiple visits to the same client. To alleviate this, Califrais aims to introduce a new actor to the system, called the ``pooling operator,'' that can take over some of the deliveries of the different suppliers and pool them, leading to better efficiency than what the suppliers can achieve on their own.

We propose a new form of (vertical) collaborative routing model, where the pooling operator provides a ``pooling service'' that the suppliers can use to reduce their own routing costs. In our model, the pooling operator prices the service in advance and chooses daily how to operate the system. To make sure the different suppliers want to participate, the pooling operator must ensure that operating costs are reduced for suppliers who participate in the pooling scheme, while trying to maximize its own profit.

Our main contribution lies in using a pooling model where the allocation of profits is done in advance rather than through the use of a posteriori reallocation of the profits, which is the more common approach in the literature. We formulate the model, provide bounds on the usefulness of pooling in simple cases, and state properties that will help us solve the model. Finally, we provide initial numerical results that demonstrate the viability of the proposed pooling system, based on a real-world data set from suppliers operating at the Rungis food market.

ABSTRACT. Ride-sharing, also known as carpooling, is an emblematic form of intermediate transport. Achieving a satisfactory level of service remains challenging, particularly because drivers must be encouraged to share trips they have already planned. Local initiatives have emerged to support the development of such services, often through monetary incentives. Motivated by discussions with Ecov, a French mobility operator committed to solidarity-based transport, this work aims to provide methodological tools for public operators designing such incentives. Although casual carpooling and dynamic pricing in large stochastic ride-sharing markets have been studied, limited attention has been paid to settings combining pre-planned driver trips, incentive-dependent acceptance, and a limited operator budget. We therefore consider a streamlined model with a single route and one origin-destination pair. Passengers and drivers arrive according to inhomogeneous Poisson processes with fixed and exogenous rates. Passengers are willing to wait for a limited time, while each arriving driver accepts a waiting passenger with a probability that increases with the incentive offered by the operator.

We propose several heuristics for determining incentive levels, mainly based on approximate deterministic formulations of the problem. The simplest formulation, although potentially less efficient, leads to two-dimensional non-convex optimization problems that can nevertheless be solved efficiently. A discrete-event simulator is currently being developed to compare fixed and dynamic pricing strategies under the same stochastic conditions. The objective is to determine when dynamic pricing can increase the number of passengers served and when its effectiveness is limited by stochastic demand and supply, driver acceptance behaviour, and budget constraints.

ABSTRACT. Optimizing interconnected hydro-thermal systems raises two structural difficulties, in particular with multi-reservoir prospective studies led by RTE. First, dealing with the uncertainties requires a multi-stage stochastic optimization formulation. Second, spatial coupling through interconnections prevents each node from being treated independently. Classical algorithms such as Stochastic Dynamic Programming (SDP) rapidly face the curse of dimensionality when the numbers of reservoirs to optimize rise. Price decomposition schemes such as Dual Approximate Dynamic Programming (DADP) decouple the problem spatially, resulting in a non-smooth dual problem. Existing works solve it either with subgradient ascent, whose convergence requires diminishing step-size rules and is slow in practice, or with L-BFGS, whose guarantees assume differentiability and therefore do not apply here. Moreover, its convergence analysis relies on convexity, which in practice is obtained by assuming linear reservoir dynamics, a modeling approximation we aim to relax. In order to tackle these issues, a new framework is presented, based on the Alternating Direction Method of Multipliers (ADMM). This framework enables convergence in linear and non-linear cases. Unlike the DADP framework, which implements deterministic price decomposition processes, we aim to implement stochastic dual multipliers, which are assumed to follow a Markovian process. The dynamic of the dual multiplier is learned through the process. This framework is compared to exact reference obtained with SDP and approximated result obtained with DADP. This work paves the way to optimizing multi-reservoir systems in a disaggregated fashion, which represents a significant achievement for transmission system operators such as RTE.

ABSTRACT. Seasonal hydropower operation requires release decisions that account for uncertain future inflows and the opportunity cost of stored water. We assess the Modelling and Optimisation of Stochastic Seasonal Hydropower Storage (MOSSHOOS) framework for the Manantali reservoir using 660 monthly energy-inflow observations from 1961--2015. The framework combines multiscale inflow representation, blocked resampling and scenario reduction with stochastic dual dynamic programming, and uses a seasonal square-root diffusion and Hamilton--Jacobi--Bellman formulation as a continuous-time interpretation of marginal water value. Scenario reduction improves the energy score by 47.6% relative to independent monthly sampling, but the marginal distribution is rejected by a Kolmogorov--Smirnov test and the lower-tail p10 error remains 103.9%, indicating inadequate dry-tail calibration. Consistent with that limitation, 50 paired Monte Carlo evaluations show no detectable cost or deficit advantage of the stochastic policy over its deterministic-equivalent counterpart at the archived operating configuration. As an independent benchmark, the repaired open-source plan4res seasonal storage valuation chain yields a dominant marginal value of 81.66~USD/MWh, within 3% of the MOSSHOOS mean of 79.25~USD/MWh. These results support a coherent stochastic inflow-to-water-value methodology while showing that hydrological tail representation, rather than economic scaling, is the principal limitation to address before operational application.

ABSTRACT. Enedis, as the French main DSO, can be severely affected by storms and other extreme weather conditions. In particular, the medium-voltage (MV) network located in wooded areas is prone to be affected by vegetation-related incidents, which may lead to long outages and significant numbers of customers without supply. This paper presents an Operations Research framework developed to transform a national resilience objective into an executable multi-year investment programme.

For each MV feeder, all possible electrical networks are generated, including existing electrical assets and feasible underground routes along roads. These networks are combined into weighted graphs. Edge costs represent construction conditions, junctions, substation adaptations, asset removal, and exposure to vegetation-related risks. Shortest-path based algorithms are used to generate network configurations that preserve connectivity while exploring different trade-offs between investment cost and resilience. Each configuration is then assessed in terms of cost and expected reliability.

Since feeder-scale configurations cannot be implemented as single construction operations, a graph-based decomposition identifies connected groups of assets that should be removed and uses shortest-path computations to associate the required new sections and substation adaptations. This produces operationally independent projects that preserve the required reliability when other projects are postponed. Each project is characterized by geographical scope, construction actions, investment cost, avoided risk, and reliability benefit.

Projects are then selected and scheduled over several years using a budget-constrained knapsack model that maximizes resilience and reliability benefits within annual regional budgets. Project dependencies and shared construction costs are accounted for to produce a consistent multi-year investment plan.

Applied to mainland France, this framework generated thousands of projects, with scenarios reducing vegetation-related exposure by more than 70 \%, removing approximately 140,000 km of overhead lines, and improving estimated network reliability by 5–10 \%. The approach combines network design, Steiner-tree approximation, project decomposition, and multi-period knapsack optimization into an end-to-end decision-support method linking strategic resilience targets to implementable construction programmes.

ABSTRACT. Decarbonising industrial heat has become a pivotal challenge for policymakers and industrial stakeholders alike. However, investment in low-carbon heat technologies is still timid. This is hindered, among other factors, by risk aversion from economic agents having to take investment decisions, the irreversible nature of these investments and the high sunk costs they would involve if future outcomes were unfavourable. As a result, firms may delay or avoid investments, waiting for greater certainty in energy prices, regulatory frameworks, or market signals. In the context of the development of innovative, decarbonised heat production technologies, Small Modular Reactors (SMRs) are gaining attention as potential solutions for providing low-carbon, medium-to-high temperature heat to industrial sites. This paper explores the potential role of Small Modular Reactors (SMRs) in supplying decarbonised heat and electricity to industrial sites. In particular, we explore the possible emergence of bilateral contracts between SMR developers and industrial firms, which may offer a viable pathway to overcome investment hurdles and align supply with demand.

To investigate this, we develop a coupled investment model within a Real Options Analysis framework, capturing the strategic decision-making processes for both an industrial firm and a SMR developer. We combine Dynamic Programming with Game Theory to solve for the strategic behavior of both firms. The model considers multiple investment choices for the industrial firm, incorporating the irreversibility of capital expenditures and the potential for bilateral contracting in heat and electricity supply with the SMR firm. We solve the model by backward induction, adapting the Least-Square Monte Carlo method to a two-agent framework in a risk averse fashion. Risk aversion is modelled using the Expected conditional risk measures.

Our results suggest that both firms delay their investment decisions when given the option to wait before investing: The industrial firm keeps its initial gas boiler for a longer period, while the SMR firm holds off investment and construction of the SMR. Furthermore, risk aversion exacerbates the postponing of investments in non-nuclear investments. Finally, when contracting between firms is allowed, an increased risk aversion motivates both firms' to sign heat supply contracts in order to hedge against energy price risks.

ABSTRACT. Across transportation, communication, and graph-based physical systems, topology determines how traffic and information flow and how physical components interact, thereby shaping systemlevel behavior and robustness. Consequently, adding only a small number of edges can substantially alter graph-level outcomes. However, identifying effective edge additions is challenging: the candidate space grows quadratically with graph size, while the objective is often accessible only through an expensive and non-differentiable black-box solver or simulator. Existing learningbased approaches mainly formulate edge selection as sequential reinforcement learning, often requiring long training and scale-specific optimization. In this work, we propose Search-Supervised Transformer Edge Addition (SS-TEA), a unified Transformer framework for budgeted graph edge addition under black-box feedback. SS-TEA captures global node dependencies through full self-attention and uses a masked query–key decoder to score feasible nonedges, constructing a complete fixed-budget edge set from a single model forward pass. To train the Transformer without evaluator gradients, high-quality edge sets discovered during search are converted into supervision for the edge-selection policy. Extensive experiments demonstrate that SS-TEA achieves strong performance with a small training budget and transfers directly from small to large graphs without retraining. The same framework applies to both exposing solver vulnerabilities and improving network robustness, demonstrating its cross-scale transferability and broad cross-task applicability.

ABSTRACT. Large-scale infrastructures, such as communication networks, energy systems, and computing platforms, increasingly rely on joint investments by independent stakeholders. Such arrangements enable stakeholders to share deployment and maintenance costs while remaining competitors in the subsequent use of the infrastructure. This creates a coupling between infrastructure investment, resource allocation, and coalition formation: the benefit of forming a coalition depends on both the infrastructure it can jointly finance and the resulting resource allocation. We develop a market-based framework to jointly model these interactions. Stakeholders form co-investment coalitions and pool their budgets to finance infrastructure capacity, which is subsequently allocated among coalition members through a Fisher-market mechanism. Unlike classical Fisher markets with fixed supply, infrastructure capacity is endogenous and determined by the coalition’s aggregate investment and production cost. We further allow for allocation externalities, where a stakeholder’s utility may depend on both its own allocation and those of other stakeholders, capturing cooperation in infrastructure investment alongside competition in resource utilization. For a fixed coalition structure, we introduce a Partition-Induced Fisher Market Equilibrium (PIFME), which jointly determines infrastructure production, resource allocation, and coalitionspecific prices. We characterize this equilibrium through a variational inequality (VI), where the multipliers associated with capacity constraints recover the equilibrium prices. To study the stability of co-investment structures, we associate each coalition structure with its market equilibrium and formulate a partition-function-form game. Stability is analyzed through the recursive core, which accounts for coalition deviations and the subsequent reorganization of the remaining stakeholders

ABSTRACT. We consider an optimal trading problem under a market impact model with endogenous market resistance generated by a sophisticated trader who (partially) detects metaorders and trades against them to exploit price overreactions induced by the order flow. The model features a concave transient impact driven by a power-law propagator with a resistance term responding to the trader's rate via a fixed-point equation involving a general resistance function. We derive a (non)linear stochastic Fredholm equation as the first-order optimality condition satisfied by optimal trading strategies. Existence and uniqueness of the optimal control are established when the resistance function is linear, and an existence result is obtained when it is strictly convex using coercivity and weak lower semicontinuity of the associated profit-and-loss functional. We also propose an iterative scheme to solve the nonlinear stochastic Fredholm equation and prove an exponential convergence rate. Numerical experiments confirm this behavior and illustrate optimal round-trip strategies under "buy" signals with various decay profiles and different market resistance specifications.

ABSTRACT. We provide quantitative estimates on the long-time behavior of the opti- mal particle system associated with a mean-field control problem. Our analysis covers both displacement and flat convex settings, and extends to flat semiconvex regimes in which the concave part is absorbed by heat dissipation. Combining the modulated free energy method with functional inequalities on N -particle configurations, we compare arbitrary particle flows to their mean-field counterpart, obtaining a uniform-in-time propagation of chaos, as well as a turnpike estimate around the mean-field ergodic state up to an approximation error.

ABSTRACT. We introduce two primal-dual algorithms for nonconvex-nonconcave nonsmooth saddle point problems (NC-SPP) characterized by the weak Minty Variational Inequality (MVI). The first, Nonconvex-Nonconcave Primal-Dual Hybrid Gradient (NC-PDHG), extends the Primal-Dual Hybrid Gradient (PDHG) method. The second, Nonconvex-Nonconcave Stochastic Primal-Dual Hybrid Gradient (NC-SPDHG), incorporates a randomly extrapolated primal-dual coordinate descent approach, extending the Stochastic PDHG (SPDHG) algorithm.

To our knowledge, designing a coordinate-based algorithm for NC-SPP is unprecedented, and proving its convergence posed significant difficulties. This challenge motivated us to utilize PEPit, a Python-based tool for computer-assisted worst-case analysis of first-order optimization methods. By integrating PEPit with automated Lyapunov function techniques, we successfully derived the NC-SPDHG algorithm.

Both methods converge under a mild condition on the weak MVI parameter, using constant, problem-adapted step sizes. Experiments on sigmoid regression with squared loss and perceptron regression confirm our theory and outperform state-of-the-art methods, where linear convergence is observed. Additionally, we conduct a convex-concave least-squares experiment to show that NC-SPDHG performs competitively with SAGA, a leading algorithm in the smooth convex setting.

ABSTRACT. Screening is a technique introduced in [El Ghaoui et al., 2012], which leverages optimality conditions of a non-smooth convex optimization problem to retrieve sparsity in features or samples. Most existing works are using non smooth objectives to guarantee null variables (via the loss for sample elimination [Ogawa et al., 2013] or on the penalty for feature sparsity [Ndiaye, 2018] or both [Shibagaki et al., 2016]). Safe screening based on duality gap bounds allows one to test during the training phase for relevance of optimization variables. In the context of reproducing kernel Hilbert spaces (RKHS), where the number of optimization variables is equal the number of samples, screening is a good way to relax the optimization problem on the go and reduce the problem’s dimensions while being solved. However actual safe screening method that ensure nullity of optimization variables heavily rely on non smoothness. Given the same tools but with a smooth objective, we introduce here a way to screen variables that are not null but rather lie in a small interval near zero at the optimum.

ABSTRACT. We analyze a zeroth-order particle algorithm for the global optimization of a non-convex function, focusing on a variant of Consensus-Based Optimization (CBO) with small but fixed noise intensity. Unlike most previous studies restricted to finite horizons, we investigate its long-time behavior with fixed parameters. In the mean-field limit, a quantitative Laplace principle shows exponential convergence to a neighborhood of the minimizer x^*. For finitely many particles, a analysis based on \textit{propagation of chaos} yields explicit error bounds: individual particles achieve long-time consistency near x^*. We also analyze the performance of the ``global best'' particle. The proof technique combines a quantitative Laplace principle with block-wise control of Wasserstein distances, avoiding the exponential blow-up typical of Grönwall-based estimates.

ABSTRACT. Differentiating through parametric optimization problems is central to bilevel programming and meta-learning, often accomplished using approximate implicit differentiation. The implicit function theorem requires inverting a partial Jacobian of the optimality condition, which fails when there are many solutions. Nonetheless, in such cases it is possible to relax invertibility to a strictly weaker uniform range condition, under which it is shown that the minimal-norm solution mapping admits generalized derivatives by using a limiting Tikhonov regularization argument and conservative set-valued field theory. With additional control on the eigenvalues of the generalized Hessians, a pseudoinverse formula is justified. This is established for a class of smooth convex objectives and extended to nonsmooth composite problems. These assumptions are verified for Least-Squares, Huber regression and LASSO. The resulting extension of nonsmooth implicit differentiation to ill-posed settings is examined experimentally on data poisoning and data hypercleaning problems.

ABSTRACT. We establish a superposition principle for perturbed sweeping processes governed by a moving prox-regular set. The classical Superposition Principle connects the Eulerian description of dynamics through the continuity equation with their Lagrangian description in terms of characteristic trajectories. Our result extends this framework to constrained dynamics and, in the unconstrained case, extends the superposition principle for differential inclusions of Cavagnari, Marigonda, and Piccoli to non-autonomous dynamics that may be discontinuous in time.

More precisely, starting from a probability measure on path space concentrated on Eulerian admissible sweeping trajectories, we show that its time marginals form a narrowly Lipschitz continuous curve of probability measures supported on the moving constraint set and solving a constrained continuity equation driven by an admissible Borel velocity field. Conversely, every solution of this continuity equation admits a superposition representation concentrated on Eulerian admissible sweeping trajectories. This yields a rigorous correspondence between the Lagrangian and Eulerian descriptions of the dynamics.

We also prove a discrete version of the principle, showing that admissible sweeping trajectories can be approximated in Wasserstein distance by empirical measures supported on finitely many characteristics. As an application, we consider an optimal control problem governed by the perturbed sweeping process and prove, under a convexity assumption on the running cost, that its Kantorovich and Eulerian formulations have the same value and both admit minimizers. This equivalence is in the spirit of the multi-agent control framework of Cavagnari, Lisini, Orrieri, and Savaré.

ABSTRACT. Many optimization and game models evolve in heterogeneous random environments. A natural question is whether, at large space and time scales, microscopic randomness averages out and gives rise to a deterministic effective dynamics. For Hamilton-Jacobi equations, this is the problem of stochastic homogenization, initiated by Lions, Papanicolaou and Varadhan.

I will discuss a game-theoretic approach to this question. Via the representation of Hamilton-Jacobi equations as values of zero-sum differential games, homogenization can be viewed as a long-run problem for a game played in a random medium. Our main result establishes homogenization for a class of first-order equations with nonconvex and noncoercive Hamiltonians in environments with finite-range dependence, and provides quantitative convergence estimates. The proof works directly with the game and combines geometric control of trajectories, concentration and approximate subadditivity. It is inspired by the metric approach of Armstrong and Cardaliaguet (2015) and by the work of Feldman and Souganidis (2017); the probabilistic arguments are close in spirit to techniques from first-passage percolation.

ABSTRACT. Many learning and decision-making problems are solved in an environment that reacts to the decision itself: the data distribution is not fixed, but is induced by the deployed decision. This is the setting of stochastic optimization with decision-dependent distributions, also known as performative prediction, where a natural solution concept is an \emph{equilibrium}, i.e. a decision which is optimal for the distribution it generates~\cite{DX}.

In a real Hilbert space $\mathcal{H}$, this leads to the closed-loop monotone inclusion \[ \text{find } \bar{x} \in \mathcal{H} : \quad 0 \in A(\bar{x}) + \mathsf{B}_{\mathsf{m}_{\bar{x}}}(\bar{x}), \] where $A$ is maximally monotone, $\mathsf{B}_{\mathsf{m}}(x) = \mathbb{E}_{\xi \sim \mathsf{m}}[B(x,\xi)]$ is cocoercive, and $(\mathsf{m}_x)_{x \in \mathcal{H}}$ is a family of probability measures indexed by the decision $x$. Taking $A = \partial g$ and $B(x,\xi) = \nabla_x f(x,\xi)$ recovers regularized stochastic optimization. This framework was introduced in~\cite{EFA}, where first- and second-order dynamics are analysed through a reformulation as a Lipschitz perturbation of a monotone inclusion.

Building upon this framework, we introduce second-order \emph{splitting} dynamics~\cite{SEF} driven by a distributionally evaluated forward-backward operator. This approach is particularly efficient as it requires only forward evaluations of $\mathsf{B}_{\mathsf{m}}$ and resolvents of $A$. Under a uniform monotonicity assumption, the equilibrium exists and is unique. Furthermore, a vanishing viscous damping yields strong convergence of the trajectories along with fast rates for the velocities, whereas a constant Polyak-type damping ensures global exponential convergence when the regularizing operator is strongly monotone.

ABSTRACT. We study the numerical optimization and bifurcation analysis of an Airborne Wind Energy (AWE) system. Using kites to harness wind energy has been intensively studied in the last decade, see e.g. the survey by M. Diehl et al. In the framework of the KEEP project (Kite Electrical Energy Production), funded by PGMO and CNRS and gathering researchers from ENSTA Bretagne and Université Côte d'Azur, our goal is to maximize the electrical power generated by a kite attached to a lever arm; flying the kite along a crosswind lemniscate drives the arm to generate electricity. We build a point-mass mechanical model where a fictitious aerodynamic force steers the kite along a prescribed lemniscate-shaped cone. The resulting differential equation is expressed as a second-order ODE with two degrees of freedom (4-dimensional state space). The power maximization problem on the periodic regime is solved via a single-shooting formulation that jointly optimizes the design parameters, initial state, and period subject to periodic boundary conditions, bypassing slow asymptotic time-averaging. Finally, we use BifurcationKit.jl to perform continuation on the resulting limit cycle with respect to wind speed, revealing the existence of a fold bifurcation at low wind speeds as well as confirming the local uniqueness of the limit cycle at normal operating conditions. This is an example of joint use of optimization and bifurcation theory for the numerical analysis of a dynamical system.

ABSTRACT. In this paper, we investigate dynamic meta-parameter control for the covariance matrix adaptation evolution strategy (CMA-ES) through deep reinforcement learning (DRL). We formalize the online adaptation of meta-parameters as a Markov decision process (MDP), where each iteration corresponds to one decision step. Based on this formalization, we integrate a controller designed with proximal policy optimization (PPO) into CMA-ES that adapts the learning rates of the mean and the damping parameter of cumulative step-size adaptation (CSA) during the optimization run. Importantly, the controller preserves all known invariance properties of CMA-ES. We train the PPO model only on the Rastrigin function. The learned controller enables CMA-ES with default population size to solve a variety of multimodal functions: on the Rastrigin function, the PPO-CMA-ES is four times faster than IPOP-CMA-ES, and it is also highly competitive on a variety of other multimodal functions, while maintaining reasonable performance on unimodal functions.

ABSTRACT. Parameter control aims at dynamically adapting the behavior of an optimization algorithm to the different stages of the search. While dynamic parameter choices can provide substantial benefits over static configurations, deriving effective multi-parameter control policies remains challenging, especially when the policy must both perform well empirically and remain interpretable enough to support theoretical analysis. In this talk, we present an investigation of deep reinforcement learning (deep RL) as a potential tool for discovering such policies for evolutionary algorithms. We consider the $(1+(\lambda,\lambda))$-Genetic Algorithm on the OneMax benchmark, a well-studied setting in which dynamic parameter control is known to yield a provable asymptotic advantage. We formulate the control of its four main parameters: the mutation and crossover population sizes, the mutation-rate coefficient, and the crossover-bias coefficient, as a Dynamic Algorithm Configuration (DAC) problem. Using Double Deep Q-Networks (DDQN), we show that deep RL can discover highly effective multi-parameter policies, despite the non-trivial structure of the resulting decision problem. A simple policy derived from the learned behavior consistently outperforms previously proposed theoretical and algorithmically tuned policies on problem sizes up to $n=40\,000$. More importantly, we address a key limitation of deep-RL-based control: although neural policies can be highly effective, they are difficult to interpret and analyze theoretically. We therefore develop a two-stage distillation procedure that transforms learned policies into explicit symbolic control rules. The resulting white-box policies expose interpretable relationships between the algorithm's parameters and the optimization state while retaining strong empirical performance. We further automatically fine-tune the symbolic policy using SMAC3, obtaining additional improvements over the initial symbolic representation. Together, these results demonstrate a data-driven route from black-box policy discovery to interpretable algorithm design. Beyond the considered benchmark, this approach suggests that deep RL can serve not only to optimize evolutionary algorithms, but also to generate concrete mathematical hypotheses for the analysis of adaptive optimization methods.

ABSTRACT. Vehicle Routing Problems (VRPs) are a fundamental class of combinatorial optimization problems aimed to find high-quality routes for vehicles under operational constraints. Despite decades of study in operations research, VRPs remain computationally challenging due to their NP-hardness, with additional difficulty introduced by the diverse constraint structures that arise across different variants. Hybrid Genetic Search (HGS) has emerged as a state-of-the-art metaheuristic framework for many VRP variants. However, its performance still depends heavily on manually designed search operators and empirical parameter tuning, limiting its ability to adapt automatically to complex and diverse problem settings. To address the limitation, we propose EvoHGS, a large language model (LLM)-guided evolutionary framework that automatically improves HGS across the VRP family. EvoHGS integrates LLM-based heuristic generation with evolutionary search, allowing candidate HGS operators to be generated, evaluated, selected, and refined through an iterative optimization process. To further align generation with downstream search performance, EvoHGS uses feed- back collected during evolution to finetune the LLM via reinforcement learning. This enables the model to progressively produce operators that are better adapted to the search dynamics of HGS across VRP variants. Experiments on multiple VRP variants show that EvoHGS consistently outperforms the original HGS baseline at the scale used for evolution, remains competitive when the evolved operators are applied to larger instances, and demonstrates promising zero-shot generalization to unseen VRP variants.

ABSTRACT. The effectiveness of optimization algorithms depends drastically on the problem at hand, and often it can change during the optimization process. This implies that choosing different algorithms for different stages of search might outperform any single algorithm that is used from the beginning until the end. However, finding a good strategy of switching between algorithms from a given portfolio for a specific problem is a non-trivial task.

In this work, we provide the first foundations for \emph{online algorithm selection} (OAS) and show an example where its superiority over single-algorithm approaches is well-pronounced. Namely, we consider a setting when we have a portfolio of two algorithms, the $(1 + 1)$~EA and the $(1 + (\lambda, \lambda))$~GA, and we solve \textsc{OneMax} benchmark by starting with the $(1 + 1)$~EA and switching to the $(1 + (\lambda, \lambda))$~GA when close to the optimum. While both algorithms from the portfolio have $\Omega(n \sqrt{\log n})$ expected runtime (or by sub-polylogarithmic factors better), the OAS can find the optimum in $O(n\log\log n)$ expected time, which is asymptotically faster. We then provide a broad empirical study which confirms that our observation that these two algorithms demonstrate different effectiveness at different distances from the optimum. It also indicates that the proposed switching mechanism is not specific to the \textsc{OneMax} benchmark, but rather general.

We also present the current work-in-progress, where we explore more complicated mechanisms of switching between algorithms such as reinforcement learning and bandit-based approaches. We show that these approaches can be used to learn the best switching strategy for a given problem, and we provide preliminary results on their performance.

ABSTRACT. Baldwinian and Lamarckian evolution have existed for a long time in evolutionary algorithms (EAs) without ever dominating the academic literature or practical applications. In this work, we use modern empirical and theoretical methods to revisit Lamarckian and Baldwinian evolution and rigorously compare them with the generic Darwinian evolution. On the empirical side, we run a comprehensive suite of experiments on graphs from six different datasets from the recent GraphBench benchmark on Maximum Independent Set and Maximum Cut problems. Our results show that Baldwinian and Lamarckian evolution consistently outperform Darwinian evolution, confirming the great potential of local search augmented evolutionary algorithms. Notably, in the great majority of cases, all EAs outperform recent deep learning baselines and approach the performance of highly specialised heuristic and exact solvers. We furthermore report a high-performing set of generalist parameters for all studied evolution types that we hope will be of use to practitioners in future. On the theoretical side, we extend the existing DeceptiveLeadingBlocks benchmark to arbitrary block length k. For all constant k, we then prove asymptotically tight runtime bounds for the (1 + 1) EA in the three evolution types on this benchmark. For Baldwinian evolution, these are independent of k, whereas for the other two evolution types, the runtimes steeply increase with growing value of k.

ABSTRACT. We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a model of scale-free networks combining geometry with heterogeneous vertex degrees. Geometric routing is a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length \(\Theta(\log \log n)\), it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation.

We prove that, for power-law weight exponent \(\tau \in (2,3)\) and geometric decay parameter \(\alpha > \tau-1\), geometric routing succeeds with constant probability and finds ultra-short paths of length \(\Theta(\log \log n)\), matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network's geometry alone implicitly guides the path to the target through its high-weight core.

ABSTRACT. We study the complexity of several usual problems when we consider the question of the unicity of the searched entity. More precisely, we intend to compare the complexity of the existence of a solution to the complexity of the unicity of such a solution. Some problems are polynomial for both versions. For some other problems, under appropriate hypotheses relative to the complexity classes, the two versions do not belong to the same class. For instance, the usual Satisfiability problem belongs to NP while the Unique Satisfiability version is US-complete (where ``US'' stands for the ``Unique Solution'' class). The aim of the talk is to compare the complexities of such problems when taking the unicity property into account.

ABSTRACT. PDE-constrained optimization requires repeated solutions of state and adjoint equations and can become computationally prohibitive in many-query, high-dimensional, or uncertainty-aware settings. We propose a rank-adaptive tensorized neural operator specifically designed for surrogate-based PDE-constrained optimization. The parametric solution map is learned by a Fourier-type neural operator whose high-dimensional spectral weights are represented in Tensor-Train form. Rather than fixing the tensor ranks a priori, the proposed framework treats them as discrete complexity variables and adapts them using joint state, derivative, and compression indicators. The training objective combines solution accuracy, physics consistency, optimization-relevant gradient information, and an explicit low-rank complexity penalty. The resulting surrogate is embedded within a gradient-based optimization loop and evaluated on parametric elliptic/Darcy-type control problems. Comparisons with a full Fourier neural operator and fixed-rank tensorized models assess relative state error, reduced-gradient accuracy, objective optimality gap, PDE feasibility, computational cost, and robustness under parameter distribution shift. The framework connects PDE-constrained optimization, numerical analysis, low-rank tensor approximation, and scientific machine learning, targeting compact surrogates whose derivatives remain sufficiently accurate for reliable optimization.

ABSTRACT. Optimal control problems with discrete-valued controls are typically tackled via a first-discretize-then-optimize strategy, in which the original infinite-dimensional problem is approximated by a large-scale finite-dimensional mixed-integer nonlinear program (MINLP) and handed to a general-purpose solver. A fundamental drawback of this approach is that conventional discretization schemes shrink the feasible set, so that dual bounds on the discretized problem are not valid in general for the original infinite-dimensional formulation. Moreover, such bounds do not carry over when the mesh is refined. Although one can in principle correct for this by estimating the discretization error, the resulting procedure is both technically involved and tends to produce only weak bounds.

To overcome these limitations and to support a branch-and-bound-and-refine scheme for global optimization in function space, we introduce a first-dualize-then-discretize paradigm. In this framework, a convex relaxation of the original problem is dualized directly in function space; the resulting dual problem can then be discretized with any conforming scheme, and the dual objective value of the discretized dual problem constitutes a valid lower bound for the original infinite-dimensional problem. We demonstrate the approach on two classes of problems featuring total variation or minimum dwell-time constraints, for which we also provide extended formulations in function space.

ABSTRACT. We consider a linear parabolic partial differential equation in which the control appears as a distributed source term and uncertainty acts in a structured way on the right-hand side. The objective is find a control such that some outcome of the random state occurs with maximal probability. The outcome can be enforced pointwise, for instance when requiring the state to remain bounded at every point in time and space. This type of model offers a compromise between the robust "lookback" problem and the problem of maximizing on average. Probability functions are attractive because of their interpretability, but are generally nonsmooth and nonconvex. In the case of Gaussian-like noise, the properties of the underlying control-to-state operator allow us to obtain a fully explicit formula for the Clarke subdifferential of the probability function. The formula depends on certain solutions to an adjoint equation with measures appearing in source terms. We also consider optimality conditions depending on solutions to a related adjoint equation in the case of the lookback problem. These formulae are used to compute solutions in a numerical example.

ABSTRACT. We establish that nonconvex definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, conic constraint systems, and non-unique primal and dual solutions admit an adjoint state formula under a mere qualification condition. The adjoint construction yields a selection of a conservative field for the value function, providing a computable first-order object without requiring differentiation of the solution mapping. Through examples, we show that even in smooth problems, the formal adjoint construction fails without conservativity or definability, illustrating the relevance of these concepts to grasp theoretical aspects of the method. This work provides a tool that can be directly combined with existing primal-dual solvers for a wide range of parametric optimization problems.

ABSTRACT. In this work, we introduce a static notion of differentiability for functionals defined on the space of positive measures, based on the Wasserstein-Fisher-Rao geometry, which accounts simultaneously for mass transport and mass creation or destruction. We establish general differentiability criteria and derive explicit formulas for relevant classes of functionals, including expectations, cylindrical functions, and interaction energies. A key feature of the resulting calculus is that differentiability along optimal Wasserstein-Fisher-Rao perturbations yields robust first-order expansions along more general, non-optimal perturbations.

On the space of probability measures, our framework naturally connects with both Wasserstein differentiability and flat differentiability, providing a common geometric perspective on these two classical notions.

Finally, we show how this differential calculus can be used in the optimal control of interacting particle systems with time-dependent weights, where both particle positions and weights are dynamical variables.

ABSTRACT. In this talk, we study free-terminal-time optimal control problems for non-local continuity equations over the Wasserstein space \(\sP_2(\R^d)\). Although Pontryagin’s maximum principle and dynamic programming provide complementary descriptions of optimal trajectories, connecting them through sensitivity relations is delicate in this setting. In particular, the value function is generally nonsmooth, and subgradients computed at different probability measures belong to different L^2 spaces.

Under a Petrov-type condition on the terminal target, we obtain a quantitative steering estimate and prove local Lipschitz continuity of the value function around an optimal trajectory. We then adapt the classical auxiliary-problem approach to the Wasserstein framework and establish the existence of a Pontryagin coextremal satisfying both full time–measure and partial measure sensitivity relations.

ABSTRACT. We study the optimization landscape induced by the true loss for shallow polynomial neural networks (PNNs) with h ∈ N neurons in the hidden layer, one-dimensional input and output layers, and a monomial activation function of degree d ∈ N, trained against a non-constant affine linear target function. Our first main result provides, for arbitrary activation degree d, a sharp existence and non- existence criterion for global minimizers together with necessary structural conditions. We show that the infimum of the loss is always zero and is attainable with at least d active and visible hidden neurons, that is, hidden neurons with non-zero inner and outer weights, having pairwise distinct pivots. In contrast, if h < d, then the infimum cannot be attained, and every minimizing sequence of parameters necessarily diverges to infinity. Our second main result provides a complete classification of all critical points of the loss function for cubic activation. We show that the loss landscape admits no local maximizers, that critical points cannot have exactly two distinct pivots, and that global minimizers require at least three distinct pivots. Critical points with no active hidden neurons correspond only to saddle points. Consequently, non-global local minimizers and non-trivial saddle points arise only in networks where all pivots coincide. Moreover, non-global local minimizers require all hidden neurons to be active and visible, with exactly one hidden neuron having a slope sign matching that of the target function. We further show that each hidden neuron of a critical point that is not a global minimizer has either an input- dependent or zero contribution, but no nonzero input-independent contribution, to its corresponding realization function.

ABSTRACT. We investigate the explanability properties of the recently proposed linear-min-max neural networks. At initialization, they can be interpreted as k-medoids with the infinity norm as a distance. Then, they are trained using subgradient descent to better fit the data. The model has been shown to be a universal approximator. Yet, we can trace the decision process because a single most activated neuron is responsible for the value of the output. Using this property, we designed a pixel fragility measure that determines whether changes to a single pixel may be responsible to a change in the classification output. Experiments on the PneumoniaMnist dataset show that this explanation for the output of the neural network compares favorably to SHAP and Integrated Gradient.

ABSTRACT. Many real-world decision problems require solving, again and again, combinatorial optimization instances drawn from a common distribution. Instead of solving each instance independently, a recent line (COAML) of structured learning methods, with many empirical achievements, exploits the statistical regularity by learning policies that pair a statistical model with a tractable combinatorial oracle. We here analyze smoothed policies: adding controlled random perturbations to the direction used by the linear oracle yields a differentiable surrogate risk and improves generalization. Our main contribution is a generalization bound that decomposes the excess risk into i) perturbation bias, ii) statistical estimation error, and iii) optimization error. The perturbation bias is controlled by the fan-crossing probability, a new geometric quantity measuring the likelihood that a perturbation changes the oracle solution. The statistical model is then eventually replaced by a denoising diffusion model, allowing to use probabilities over continuous sets despite the discrete setting of combinatorial problems.

Based on ongoing work with Yohann De Castro (ICJ Lyon) and Axel Parmentier (CERMICS, ENPC) and on https://arxiv.org/abs/2407.17200 (to be accepted at SIOPT)

ABSTRACT. In the context of global decarbonization, optimizing industrial electricity consumption is a critical lever for reducing greenhouse gas emissions. Non-Intrusive Load Monitoring (NILM) estimates individual load consumption from an aggregate meter, offering a cost-effective means to gain this visibility. Although deep learning has advanced residential NILM, industrial deployment remains difficult because machine signatures overlap, operating cycles are long and heterogeneous, variable-speed loads are common, and operating states are strongly imbalanced. This difficulty is compounded by the scarcity of publicly available industrial datasets, which in turn limits the availability of standardized benchmarks for fairly comparing methods across machine types. We propose an industrial benchmark of representative deep learning methods for NILM, including convolutional, recurrent, multi-task, attention-based, Transformer, and generative architectures. We compare seventeen implementations on five industrial machines from the HIPE dataset, spanning binary, multi-state, and continuously variable behavior. We organize our analysis around three axes: performance (measured via Total Energy Correctly Assigned, TECA), learning speed, and sampling resolution. Results show strong machine dependence: BiLSTM gives the highest mean TECA overall (0.543), while BiGRU leads on Chip Press (0.735) and Attention CNN on the High Temperature Oven (0.726). A separate runtime experiment shows that a sequence-to-sequence CNN trains roughly 13x faster than BiGRU and 21x faster than Energformer under our measured setting. A sampling-resolution sweep from 5 s to 1 h across ten machines shows 5 s gives the highest mean TECA (0.570), though the optimum remains machine-specific. These results support architecture and sensing choices conditioned on machine dynamics rather than a universal model or sampling interval.

ABSTRACT. France aims to deploy 45 GW of offshore wind capacity by 2050. The French Transmission System Operator (TSO) is responsible for both building and maintaining the offshore substations that connect these wind farms to the grid, and must pay substantial penalties to producers whenever a substation is partially shut down. Because maintenance costs account for a large share of a substation's lifetime cost, estimating them accurately is essential for investment decisions, such as substation design, that must be made well before the substation enters service. This requires optimizing maintenance operations and evaluating their cost.

We model the maintenance of such a substation as a decomposable Markov Decision Process (D-MDP), in which each piece of equipment (transformer, converter, cooling system, etc.) is a subsystem with its own degradation state. While the dynamics decompose across components, maintenance decisions do not: feasible maintenance schedules must satisfy shared operational constraints, such as technician availability and HR rules, while penalties depend on maintenance and degradation across the whole substation. We therefore represent a maintenance schedule as a joint combinatorial action with a non-additive cost. This creates two sources of complexity: the joint state space grows exponentially with the number of components, while the number of feasible joint actions is itself too large to enumerate explicitly.

We introduce a fluid linear programming approximation of the MDP that exploits the local structure of the dynamics: each component's evolution depends only on whether it is maintained, while the joint action retains the operational constraints and joint cost. The resulting relaxation avoids the full joint state space but still contains a very large number of action variables, which we handle using column generation. We then derive tighter relaxations by tracking the states and actions of groups of components. Numerical experiments study the trade-off between computational effort and accuracy and demonstrate the performance of the approach. The methodology applies more generally to D-MDPs with decomposable dynamics but joint combinatorial actions and costs.

ABSTRACT. Gas-insulated substations (GIS) are critical components of power transmission systems and require recurrent sulfur hexafluoride refilling to preserve dielectric integrity. Scheduling maintenance interventions independently at their predicted execution dates can lead to fragmented plans, repeated crew dispatches, and high coordination costs. We propose a profit-maximization framework for consolidating predictive maintenance tasks across GIS compartments. The approach combines physics-informed pressure degradation models with an integer linear programming formulation, allowing tasks to be shifted within pressure-based safety windows subject to daily capacity limits. The objective balances dispatch-cost savings from task consolidation against asymmetric time-shift penalties, with delays penalized more heavily than advances.

The computational study compares three strategies: M1, a cluster-minimization strategy developed in our previous work; M2, the proposed profit-maximization model; and M3, a fixed-day greedy heuristic. A sliding-window procedure extends the optimization over long planning horizons while preserving feasible task spacing. Results show that M1 reduces the number of maintenance groups by 30 - 60% but incurs substantial time-shift penalties, whereas M2 achieves the lowest total cost, reducing it by 50 - 70% relative to M1 and by 12 - 22% relative to M3. M3 offers competitive cost performance with lower computational complexity but requires more maintenance groups. Sensitivity analysis identifies setup-dominated, penalty-constrained, and saturation regimes, highlighting the trade-off between task consolidation, scheduling

ABSTRACT. Buildings account for 15--20\% of global CO$_2$ emissions, with heating and cooling alone responsible for roughly 8\%. This work studies how combining photovoltaic (PV) production with geothermal storage can reduce this footprint: PV output is used first to cover a building's immediate electricity needs, and any surplus is stored as heat several tens of meters underground, through a network of buried pipes, to be recovered months later -- typically in winter -- to preheat the building and cut electricity demand for heating. This principle is being tested at full scale on four residential buildings equipped by Energy4Climate (E4C) on the Saclay plateau, where replacing a large share of the gas supply with an optimized electrical management strategy could cut CO$_2$ emissions by up to 78\%. Operating such a system well is nonetheless a genuinely hard control problem: at every instant, the building must arbitrate between drawing on PV, discharging thermal storage, and buying grid electricity, under uncertainty on sunshine, occupant demand, and prices -- a decision that must also be reconciled with the far slower, months-long dynamics of the ground storage itself, whose state today constrains what will be available the following winter. The demonstrator will later be extended with electric-vehicle charging stations, adding both further flexibility and further uncertainty, as vehicles double as consumption points and as mobile, decentralized storage whose availability depends on user behavior.

A first contribution of this work is a time-stepped simulator of such hybrid PV, geothermal, and gas building energy systems. Both the network topology (which components are present and how they are connected) and the operating scenario (weather, prices, carbon intensity) are fully user-defined and independent of one another, so the same tool can evaluate a candidate control policy -- a simple heuristic or an externally supplied, time-varying strategy -- across many different building configurations and many realistic, possibly multi-year, conditions imported directly from public meteorological and grid-operator data. It reports the resulting energy consumption, cost, carbon footprint, and comfort, already providing reliable and realistic metrics to compare candidate strategies.

The second contribution is ongoing. We are currently developing simplified and tractable versions of the model of the operation problem described above, together with efficient solution methods. These methods will then be evaluated in the simulator to assess their expected performance in practice. This model will serve as a basis to determine how much precision the predictions and the decisions actually require.

ABSTRACT. Flow-based market coupling clears the European day-ahead electricity market subject to a zonal approximation of transmission constraints constructed ex ante. Because this approximation is imperfect and zonal net positions do not uniquely determine nodal injections, the resulting dispatch may violate physical limits and require costly ex post remedial actions during congestion-management stage. We ask whether nodal security can instead be incorporated into flow-based capacity calculation while leaving the standard zonal clearing problem and its pricing rule unchanged.

We formulate capacity calculation stage as a bilevel inverse-optimization problem. The upper level chooses admissible flow-based parameters, which define the zonal approximation of transmission constraints, and a nodally secure dispatch. The lower level represents conventional zonal market coupling. Since both levels maximize the same welfare function, the formulation seeks the best secure outcome that an admissible flow-based feasibility domain can induce. Our decomposition exploits a structural partition of the lower-level Karush--Kuhn--Tucker (KKT) conditions. Resource-side conditions link dispatch quantities to zonal prices independently of the flow-based parameters and strengthen a hybrid zonal--nodal master problem. The master thus selects a secure target consistent with zonal merit order. An inverse-optimization subproblem then calibrates the flow-based parameters, with regularization toward conventional capacity-calculation estimates, and reconstructs the omitted network-side conditions. Compared with a conventional decomposition using a purely nodal master, this construction excludes targets that no zonal price vector can support, removing a major source of inverse-subproblem infeasibility.

Feasibility of the inverse subproblem completes the KKT system and certifies that the target is globally optimal for the unchanged zonal clearing problem under the calibrated parameters. Global solution of the master additionally certifies global optimality for the bilevel problem. We characterize supporting zonal prices, give a rank condition for uniqueness, and identify structural causes of reconstruction infeasibility.

Experiments on a stylized network and the IEEE 118-bus system show that calibrated domains can eliminate physical violations at market clearing and reduce total operating cost when ex post congestion-management costs are included. The framework thus provides both a capacity-calculation method and explicit certificates that nodally secure outcomes can be implemented through zonal market coupling.

ABSTRACT. Road freight transport is a critical target of European decarbonization policy: heavy-duty trucks in particular are subject to a regulatory trajectory mandating a 90% reduction by 2040 relative to 2019 levels of CO2 emissions. Battery electric trucks (BETs) are a leading decarbonization pathway for the sector yet fleet operators lack decision tools that jointly account for the physical feasibility of electrification, given real vehicle usage, and its economic and environmental consequences.

This work develops a data-driven optimization framework, operated by a European manufacturer, for the electrification of heavy-duty truck fleets, applied to real client fleets. Raw telematics pings are processed through a pipeline that reconstructs missions, along which potential charging opportunities are identified, from which battery state of charge trajectories are simulated for a catalog of candidate BETs' models. Building on these simulations, we formulate mixed integer linear programs assigning BET models to vehicles.

We then perform an a posteriori assessment, comparing the initial all-diesel fleet against each optimized electrified configuration along three criteria: total cost of ownership (TCO), life cycle greenhouse gas emissions and criticality of materials use through an antimony equivalent indicator.

This framework grounds fleet electrification decisions in real operational data, with cost and environmental criteria to be integrated directly into the optimization objective in future work.

ABSTRACT. This paper introduces an analytical approximation of the static traffic assignment (STA) problem that provides an explicit mapping from network characteristics and origin destination (OD) demand to link flows and travel times. The approach combines two key elements: (i) a projection of path flows onto a lower-dimensional set of auxiliary flow variables, and (ii) a local linearisation of the link performance functions around a converged traffic assignment. Under a fixed path support, these ingredients lead to a closed-form affine representation of link flows and OD travel times, together with their analytical sensitivities with respect to network and demand variables, avoiding the repeated iterative assignment procedures required by conventional STA solvers.

The analytical model is validated against the Method of Successive Averages (MSA) on the Sioux Falls benchmark and on the real-scale road network of Lyon, France, comprising 8,140 links, 4,128 nodes and up to 33,258 OD pairs. The results show a high agreement with the numerical reference, with an $R^2$ of approximately 0.99 for OD travel-time prediction.

Beyond accelerating the assignment itself, the main interest of the proposed framework lies in its integration into larger optimization problems. By replacing the iterative lower-level traffic assignment with an explicit analytical mapping, the model provides a tractable surrogate for bilevel and nested optimization problems in which traffic equilibrium is repeatedly evaluated. This opens the way to gradient-based formulations of inverse and network-design problems, including OD demand calibration, network parameter estimation and graph reduction. The proposed framework therefore provides a computationally efficient bridge between traffic assignment and large-scale mathematical optimization, with particular relevance to backcasting-oriented transport planning and other decision-making problems involving repeated traffic assignment evaluations.

ABSTRACT. We develop a probabilistic model for free-floating car-sharing systems. Our goal is to characterize the macroscopic behavior of the system and provide insights for optimizing vehicle distribution. In these systems, shared vehicles occupy the same parking spaces as private cars. The availability of parking spaces across different zones of a city depends on private cars, which are far more numerous than free-floating vehicles. The system dynamics are described by a closed network of queues in which private and free-floating cars move among the same nodes, representing city zones. Nodes have finite capacity, and saturation is handled through a blocking and rerouting policy. The model accounts for spatial heterogeneity in both user demand and availability of parking spaces. We show that these dynamics preserve the product-form structure of the invariant distribution. We identify, in this setting, phase transitions between an overloaded regime where all nodes are saturated and underloaded regimes with saturated and non-saturated nodes. Scaling limits and stochastic averaging methods are used to analyze the behavior of the system when the capacity of the nodes is large. The analysis is performed when the average number of private cars per zone increases linearly with capacity, while the number of free-floating cars remains of smaller order.

ABSTRACT. Online learning usually assumes that the action selected by a learner is also the action that is executed. In safety-critical systems, protection mechanisms may instead invalidate part of a decision during execution, so that the realized action, loss, and feedback differ from the initial plan. We study this \emph{execution mismatch} in decentralized electric-vehicle (EV) charging. Each EV selects a combinatorial charging schedule within a stochastic availability window, while transformer-protection events may interrupt charging and force causal rescheduling to future feasible slots.

We introduce \emph{Causal Rescheduling Regret} (CR-Regret), which compares the executed trajectory with the best fixed causal deployment policy in hindsight. Our main result is a structural transfer theorem from adversarial combinatorial semi-bandits to this execution-aware regret notion. If $r_t\in\{0,1\}^d$ denotes the interruption mask and $F_T=\sum_{t=1}^{T}\|r_t\|_1$, then for any admissible online initial-selection algorithm, \[ \mathrm{CR\text{-}Regret}(T) \leq \mathrm{Regret}(T)+\mathbb{E}[F_T]. \] Hence, guarantees for the underlying combinatorial bandit learner transfer to the rescheduling problem with only an additive execution-mismatch term, independently of the state-space size of the induced recourse MDP.

Combining this framework with SleepingCatBandit for stochastic decision sets and adversarial semi-bandit losses~\cite{neuvalko} yields \emph{FTPL-Repair}, with \[ \widetilde{O}((mdT)^{2/3})+O(F_T) \] CR-Regret, which is sublinear whenever $F_T=o(T)$.

We evaluate the method in controlled experiments and in a fleet-scale study using four years of photovoltaic-generation and electricity-consumption data. With 480 EVs, causal rescheduling maintains 100\% demand satisfaction. A safety-oriented FTPL-Repair variant reduces congestion by more than 99\% relative to immediate charging, with fewer than two disconnections per EV, while a lower-price variant remains competitive with a centralized forecast-based optimizer. These results illustrate how execution-time safety interventions can be incorporated into adversarial online optimization while preserving regret guarantees.

ABSTRACT. Bundle methods are a versatile framework for nonsmooth convex optimization, particularly for structured problems arising from decomposition and Lagrangian relaxation (Frangioni, 2002). However, their effectiveness relies on how implementations exploit problem structure, manage accumulated information, and interface with master-problem solvers (Frangioni & Gallo, 1999; Bacaud et al., 2001).

We present BundleSolver 2.0, a general-purpose implementation of generalized bundle methods for nonsmooth convex optimization within the SMS++ framework. Although the solver does not require a decomposable objective, SMS++ naturally supports structured problems and motivates a specialized treatment of objectives expressed as sums of components. In this setting, components that are sufficiently "easy" can be represented exactly in the master problem (Frangioni & Gorgone, 2014), while each remaining component is approximated independently through a cutting-plane model constructed from first-order oracle information.

A central design principle of BundleSolver 2.0 is the separation between the bundle algorithm, the representation of the master problem, and its numerical solution. A dedicated master-problem component encapsulates the cutting-plane models and stabilization terms, while numerical optimization is delegated to solvers through the modeling infrastructure provided by SMS++ \cite{SMSpp}. This architecture supports both primal and dual master formulations within a unified framework and accommodates proximal, level, and doubly stabilized bundle variants.

We discuss key implementation choices, including incremental master-problem updates (VanAckooij & Frangioni, 2018), linearization management and aggregation, sparse component-variable dependencies, and efficient handling of problem modifications \cite{BundleSolver20}. The goal is to combine flexibility and maintainability with computational efficiency, while supporting experimentation with alternative master formulations and stabilization strategies.

Finally, we present preliminary numerical results on selected applications, illustrating the computational effectiveness of the proposed implementation and the impact of different design choices on the behavior and performance of the solver.

ABSTRACT. We propose a new perspective on bundle methods that, rather than relying on the classical combination of a cutting-plane lower model and a stabilization mechanism (Kelley, 1960; Lemarechal & Mifflin, 1978; Frangioni, 2002), models the objective simultaneously from below and above. In this framework, a lower model \(f_B\) and an upper model \(f_P\) jointly define the bi-objective problem \(\min_x \bigl(f_B(x),f_P(x)\bigr)\), which provides the master-problem formulation of the method.

We show that, through appropriate choices of the two models, classical stabilization techniques - including proximal, level, and trust-region methods (Lemarechal & Mifflin, 1978; Lemarechal et al., 1995; Schramm & Zowe, 1988) - can be interpreted as different rules for selecting points on the same Pareto frontier generated by \((f_B,f_P)\). In the polyhedral setting, this viewpoint connects naturally with the framework of Bertsekas and Yu (2011), where outer and inner linearizations play roles analogous to our lower and upper models, respectively.

A merit function is then introduced to measure both model agreement and approximate optimality. Rather than prescribing the serious/null decision and the aggregation of cutting planes externally, we determine them by directly minimizing this merit function. In a two-point aggregation scheme, this leads to an explicit update: the center decision emerges as a binary outcome of merit minimization, while the optimal cut-aggregation parameter is obtained in closed form. Numerical experiments indicate that the resulting Upper–Lower Model Bundle Method is competitive with the classical proximal bundle method and that the associated model gap provides an effective proxy for true accuracy.

We also prove convergence under the actual merit-minimizing update rule and establish an \(O(1/\varepsilon^2)\) complexity bound for the associated optimality certificate. We then show that, by combining bundle refinement with a relative-accuracy serious/null step mechanism and a proximal-distance argument, the framework recovers the optimal \(O((LD/\varepsilon)^2)\) nonsmooth first-order complexity bound for true objective accuracy (Nemirovski & Yudin, 1983; Diaz & Grimmer, 2023).

ABSTRACT. Primal-dual hybrid gradient (PDHG) and its variants have proven to be efficient for solving large-scale linear programs, thanks to the parallelism enabled by GPUs. Since integer flows can be modeled as linear programs, this begs the question whether primal-dual methods are suitable for minimum cost flow problems. We assume multiple similar instances need to be solved at the same time, as is common when training differentiable optimization layers, and investigate the competitiveness of PDHG against specialized flow solvers. Prior work focused on using PDHG for solving large multicommodity flow problems, but to the best of our knowledge, general minimum cost flow problems have not been tackled before. We will use the problems posed in the 13th DIMACS challenge on network flows as our benchmarks.

Since sparse matrix-vector products represent the main bottlenecks of PDHG, we aim to accelerate these products by taking advantage of two special properties of the node-arc incidence matrix: there are exactly two entries per column, and arcs can always be sorted according to their origin node, putting half of the coefficients in a straightforward staircase structure. Handling multiple similar instances at the same time allows us to use batching: Sparse matrix-vector products become matrix-matrix products, which improves throughput. Our preliminary results suggest that taking advantage of special properties of the incidence matrix alone does not bring significant benefits, but they reveal other ways of speeding up existing PDHG implementations.

ABSTRACT. Segment Routing is a recent network technology that helps optimizing network throughput by providing finer control over the routing paths. Instead of routing directly from a source to a target, each demand is assigned a list of waypoints called segment path, and between consecutive waypoints, the packets are routed according to traditional shortest path routing protocols. Segment Routing offers greater flexibility by allowing packets to deviate from the shortest paths, with minimal additional implementation costs for network administrators.

The associated computational problem is Min Segment Routing: Given a network on n vertices with weighted and capacitated links, d traffic demands (vertex pairs), and a number k, the task is to find, for each demand, at most k waypoints such that routing the demands according to these segment paths minimizes congestion in the network. Congestion is usually measured by the maximum link utilization (MLU), that is, the largest ratio of load to capacity over the links of the network.

The problem is known to be weakly NP-hard. Approaches based on linear programming or constraint programming have been employed in previous works to solve it heuristically or speed-up solvers. We complement these practical approaches with a rigorous analysis of the computational complexity of Min Segment Routing. In particular, we study the problem within the theoretical framework of parameterized complexity: we investigate if special structures of real-world communication networks could be exploited algorithmically.

ABSTRACT. Machine learning applications may involve data distributed across multiple sources rather than collected at a central location. In-network learning addresses this setting by deploying machine learning and data processing functions within the network, closer to the data sources. Deciding where functions are placed, how they are chained from sources to aggregators, and how the resulting flows are routed leads to a combinatorial optimization problem in which chaining, assignment, and routing decisions are coupled. These decisions directly affect network energy consumption, motivating the design of energy-efficient function placement and routing strategies. Building on the Data Pipeline System framework of [1], we formulate this joint chaining, assignment, and routing problem as a Quadratically Constrained Quadratic Program that minimizes energy consumption, subject to node and link capacity constraints and end-to-end latency requirements. We prove NP-hardness by reduction from the Uncapacitated Facility Location Problem. We derive families of valid inequalities and strengthening constraints to tighten the formulation. We also investigate alternative formulations and decomposition approaches to improve scalability on larger instances. Computational experiments on SNDlib topologies, with varying numbers of data sources, quantify the impact of latency constraints on optimal energy consumption and assess the computational performance of the proposed approaches. This work extends the model presented at INOC 2026 [2].

ABSTRACT. When considering substantial Unit Commitment (UC) problems, a common practice for reducing the problem size is that of aggregating similar units in order to reduce the computational complexity of the problem [1]. However, naive aggregation methods generally result in infeasibilities on the individual level. Nonetheless, it has been shown that Electric Vehicle (EV) exibility sets are in fact generalized polymatroids [2, 3], or g-polymatroids, a class of polytopes introduced by Frank [4] and further studied in the context of discrete convexity by Danilov and Koshevoy [5]. As g-polymatroids are stable by Minkowski sum and allow optimization of linear objectives via the greedy algorithm, this observation ensures an exact aggregation of EVs and an efficient resolution. We explore the other components of the UC problem as reported by the European Resource Adequacy Assessment (ERAA) and show that other storage and flexibility means, such as batteries or heat networks, also have g-polymatroid constraints. We also evoke more difficult models, such as detailed hydropower, which do not exhibit the g-polymatroid structure (at least not in their most natural formulation).

References [1] Jelle Meus, Kris Poncelet, and Erik Delarue. "Applicability of a Clustered Unit Commitment Model in Power System Modeling". In: IEEE Transactions on Power Systems 33.2 (2018), pp. 2195-2204. doi: 10.1109/TPWRS.2017.2736441. [2] Karan Mukhi, Georg Loho, and Alessandro Abate. "Exact Characterization of Aggregate Flexibility via Generalized Polymatroids". In: IEEE Transactions on Smart Grid 17.3 (2026), pp. 2131-2142. doi: 10.1109/TSG.2026.3659513. [3] Hélène Arvis et al. "Integrating Aggregated Electric Vehicle Flexibilities in Unit Commitment Models using Submodular Optimization". In: 2026 European Control Conference (ECC). ISSN: 2996-8895. 2026, pp. 760-766. [4] A. Frank. "Generalized Polymatroids". In: Finite and Infinite Sets. Elsevier, 1984, pp. 285-294. doi: 10.1016/B978-0-444-86893-0.50021-8. [5] Vladimir I Danilov and Gleb A Koshevoy. "Discrete convexity and unimodularity-I". In: Advances in Mathematics 189.2 (2004), pp. 301-324. doi:10.1016/j.aim.2003.11.010.

ABSTRACT. We investigate and compare four draw procedures for incomplete round-robin tournaments with p seeding pots and q teams per pot, where each team faces exactly k different opponents from each pot. First, we consider a method where matchups are drawn before the match schedule is built. We show, using a graph-theoretical argument, that, if q = s(k+1) with s ≥ 3, the scheduling issue cannot be ignored when drawing the matchups, by exhibiting a draw outcome that cannot be scheduled within pk match days. Second, we study an alternative method where one first builds a schedule template satisfying all the pot constraints before randomly populating it with the pq teams. We study the overall minimum number of breaks achievable by the schedule template, and in the case of the UEFA Champions League (UCL), we build a template that minimizes the number of breaks and optimizes various schedule fairness and attractiveness criteria. For both methods, a randomized variant is also considered. Using Monte Carlo simulations, we assess opponent strength fairness for the four methods in the case of the 2024–25 UCL by comparing the distributions of average opponent strength that they produce for all the teams. We find that all four procedures are fair to all teams—no team is significantly aggrieved—and the first procedure yields more uniform matchup probabilities than the second one. Finally, we introduce a luck index that objectively ranks teams from the luckiest to the unluckiest during the actual draw.

ABSTRACT. Linear programming, mean-payoff games, and parity games raise long-standing questions in algorithmic complexity. Tropical geometry connects these problems: mean-payoff games are equivalent to feasibility problems for tropical linear inequalities (Akian, Gaubert and Guterman, 2012). In particular, a strongly polynomial combinatorial simplex algorithm for linear programming would yield a strongly polynomial algorithm for mean-payoff games (Allamigeon, Benchimol, Gaubert and Joswig, 2014).

These connections lead us to study the underlying combinatorial structures using oriented matroids, which encode the determinant signs on which combinatorial pivoting rules depend. In this talk, we compare the families of oriented matroids arising from classical, tropical, and parity point configurations in general position.

For every fixed rank r ≥ 2, we prove that the tropical and parity families of realizable oriented matroids on n labeled elements have respective sizes n^((r−1)^2 n + o(n)) and n^((r−1)n + o(n)) as n → ∞. Although tropically realizable oriented matroids form a subclass of classically realizable ones, both families have size n^((r−1)^2 n + o(n)), whereas parity-realizable oriented matroids form a family of size n^((r−1)n + o(n)) that grows strictly more slowly when r ≥ 3. These results give a quantitative comparison of the combinatorial expressivity of linear programming, mean-payoff games, and parity games.

Beyond the oriented matroid results, we show that Santos’s proposed estimate (nr)^(Θ(nr)) for the number of regular subdivisions of Δ^(n−1) × Δ^(r−1) cannot hold uniformly in n and r.

ABSTRACT. We study the direct use of $\ell_0$ regularization in sparse multiple kernel learning (SMKL) problems. We introduce an explicit $\ell_0$ norm on the kernel combination coefficients, leading to min--max mixed-integer optimization formulations in which kernel selection is handled in the outer problem and a standard SVM is solved in the inner problem; we solve these formulations to within a prescribed tolerance of global optimality. For feature-wise rank-one linear base kernels, we show that the number of selected kernels provides an explicit upper bound on the rank of the combined kernel. Under this construction, the $\ell_0$ constraint directly controls the number of selected kernels and admits an interpretation in terms of feature selection. Furthermore, to disentangle the sparsity-inducing and kernel-scale-controlling effects of $\ell_1$ regularization in standard MKL, we introduce an $\ell_0$--$\ell_p$ MKL framework that uses explicit $\ell_0$ regularization to control the selected support and continuous $\ell_p$ regularization to control the magnitude and scale of the kernel weights. We compare our $\ell_0$-MKL formulations with conventional $\ell_1$-based MKL~\cite{sonnenburg2006large,rakotomamonjy2008simplemkl} and a recent heuristic approach~\cite{bertsimas2025sparse} for $\ell_0$-based MKL. Extensive experiments on synthetic and UCI benchmark datasets demonstrate that the proposed $\ell_0$-regularized SMKL approach selects fewer kernels than conventional $\ell_1$-MKL while maintaining comparable test accuracy and generally achieving higher predictive accuracy than the alternating best-response heuristic. These results complement previous observations in MKL where less sparse solutions are often observed to perform better.

ABSTRACT. K-means clustering is a nonconvex optimization problem whose many local minima and computational cost pose challenges for big datasets. We present a multimodal perspective based on three complementary ways of influencing the search: objective formulation, data selection and representation, and solution landscape exploration. Our studies show that combining data selection and landscape exploration improves solution quality and computational efficiency over corresponding unimodal methods. Our newly proposed methods exploit repeated optimization on small random subsets, while preliminary results suggest further gains from collective intelligence methods. Integrating objective reformulation offers a promising direction for extending this multimodal approach.

ABSTRACT. Demand-management policies aim to shift behavior toward sustainable choices, such as goods or transport modes. Personalized incentive policies tailor subsidies to agents' needs, but they also enable strategic manipulation: an agent may imitate a less responsible profile to obtain larger incentives. When this is profitable remains poorly understood.

We model the repeated interaction as a Stackelberg game. Prior work often represents manipulation as strategic misreporting: a follower supplies fake payoffs or best responses, and the leader adapts its commitment accordingly. We instead place preference elicitation inside the game: the regulator infers a profile from a history of random-utility choices and applies its optimized incentive. This separates the agent's true, imitated, and regulator-inferred profiles. To imitate another profile, the agent must repeatedly choose as that profile would rather than follow its true preferences, thereby incurring a *Cost of Manipulation* (CoM).

We decompose the advantage from manipulation into a *Gain from Manipulation* (CoM) (obtained from shifting the distribution of future incentives towards higher values) minus the CoM. Information-theoretic arguments upper-bound the GoD in terms of the regulator's ability to distinguish profiles from past choices. Comparing this bound with the CoD yields a threshold on the inter

ABSTRACT. GPU schedulers must allocate limited capacity among jobs with different deadlines and uncertain runtimes. Earlier work has studied deadline-aware GPU scheduling [1] and robust use of runtime predictions in cluster scheduling [2]. We examine how estimation error changes the performance of simple priority rules.

We simulate 80 jobs competing for four identical GPUs. Four non-preemptive rules use the same backfill procedure: first-come-first-served, earliest-deadline-first (EDF), estimated slack, and buffered slack. For a waiting job j at time t, estimated slack is d_j - t - r_j, where d_j is its deadline and r_j its estimated runtime. Buffered slack instead uses s_j(t) = d_j - t - r_j exp(1.28 sigma_j), where sigma_j specifies the simulated level of runtime-estimate error. Jobs with the smallest slack receive priority. We compare the rules on paired synthetic job streams while varying arrival rate, deadline tightness, and multiplicative estimation error.

Across 100 trials per setting, no rule dominates. Under heavy load with exact estimates, buffered slack missed 54.80 deadlines per trial versus 51.96 for EDF. Under medium load, tight deadlines, and high estimate noise, buffered slack missed 38.54 versus 40.79 for EDF. A second set of 100 trials with a different random seed reproduced the latter reversal.

These results illustrate how runtime-estimate uncertainty can change the performance of a priority rule. The findings are limited to synthetic jobs and a specified error model; evaluation on measured GPU workloads remains necessary.

References: [1] W. Gao et al., “Chronus: A Novel Deadline-aware Scheduler for Deep Learning Training Jobs,” ACM SoCC, 2021. doi:10.1145/3472883.3486978.

[2] A. Tumanov et al., “JamaisVu: Robust Scheduling with Auto-Estimated Job Runtimes,” Carnegie Mellon University, 2016. Technical report..

ABSTRACT. We study finite-horizon stochastic optimal control problems and approximate the resulting time-discrete formulation by a direct policy-learning problem over neural-network feedback maps. We prove a quantitative convergence estimate, in an averaged sense, for the error between the time-discrete value and the value induced by an approximately optimized neural policy. The bound separates the approximation of near-optimal feedback policies, the localization of stochastic trajectories on compact sets, and the optimization tolerance in training. The analysis does not require transition-density assumptions and covers possibly degenerate diffusions and deterministic controlled dynamics in a unified framework. Numerical experiments are provided for a degenerate stochastic radial target problem, a Hamilton--Jacobi--Bellman benchmark, and a gas storage problem, illustrating the approach and separating the main error sources: time discretization, restriction to piecewise-constant policies, neural-network approximation, and Monte Carlo evaluation.

ABSTRACT. session: "Neural Network approximations and related techniques for high-dimensional PDEs and Control Problems"

In this talk, we discuss how to produce a numerical interval which contains (or approximately contains) the price of an European option under the Uncertain Volatility Model (UVM). The pricing function, in such models, is solution to a fully non-linear Hamilton-Jacobi-Bellman equation.

We first present a numerical method based on stochastic policy gradients for approximating the option price. By construction, this approach provides a lower bound on the true price. For industrial applications, however, this can be unsatisfactory. We therefore construct an upper bound that can also be approximated numerically to provide a computable certificate on the pricing error. The construction is based on PDE residuals and can be interpreted as an a posteriori error estimate.

ABSTRACT. We consider a data-driven formulation of the classical discrete-time stochastic control problem. Our approach exploits the natural structure of many such problems, in which significant portions of the system are uncontrolled. Employing the dynamic programming principle and the mean-field interpretation of single-hidden layer neural networks, we formulate the control problem as a series of infinite-dimensional minimisation problems. When regularised carefully, we provide practically verifiable assumptions for non-asymptotic bounds on the generalisation error achieved by the minimisers to this problem, thus ensuring stability in overparametrised settings, for controls learned using finitely many observations. We explore connections to the traditional noisy stochastic gradient descent algorithm, and subsequently show promising numerical results for some classic control problems. Joint work with Boris Baros and Samuel N. Cohen.

ABSTRACT. We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by parameterizing solutions via Hilbert-Galerkin Neural Operators (HGNOs). We prove the first Universal Approximation Theorems (UATs) which are sufficiently powerful to address these problems, based on novel topologies for Hessian terms and corresponding novel continuity assumptions on the fully nonlinear operator. These topologies are non-sequential and non-metrizable, making the problem delicate. In particular, we prove UATs for functions on Hilbert spaces, together with their Fréchet derivatives up to second order, and for unbounded operators applied to the first derivative, ensuring that HGNOs are able to approximate all the PDE terms. For control problems, we further prove UATs for optimal feedback controls in terms of our approximating value function HGNO.

We develop numerical training methods, which we call Deep Hilbert-Galerkin and Hilbert Actor-Critic (reinforcement learning) Methods, for these problems by minimizing the $L^2_\mu(H)$-norm of the residual of the PDE on the whole Hilbert space, not just a projected PDE to finite dimensions. This is the first paper to propose such an approach. The models considered arise in many applied sciences, such as functional differential equations in physics and Kolmogorov and HJB PDEs related to controlled PDEs, SPDEs, path-dependent systems, partially observed stochastic systems, and mean-field SDEs. We numerically solve examples of Kolmogorov and HJB PDEs related to the optimal control of deterministic and stochastic heat and Burgers' equations, demonstrating the promise of our deep learning-based approach.

ABSTRACT. Positive-definite matrices and operators provide a natural language for covariance structure, diffusion operators, kernels, and probability distributions, but standard geometric constructions are usually presented either in finite dimensions or under assumptions that become restrictive in infinite-dimensional Hilbert spaces. We develop a unified family of Mahalanobis-weighted α–Procrustes geometries that connects Bures–Wasserstein and Log-Euclidean/Log-Hilbert–Schmidt construc- tions across these settings. In finite dimensions, the generalized Bures–Wasserstein distance is recovered at α = 1 , while the generalized Log-Euclidean geometry arises as α → 0. In infinite dimensions, we work on positive definite unitized Hilbert–Schmidt operators and introduce a regularized Mahalanobis norm based on (M + ρI)−1, which is bounded even when M−1 is not. This yields infinite- dimensional generalized Bures–Wasserstein, generalized Wasserstein, and Log- Hilbert–Schmidt distances, together with a robust Wasserstein formulation. A spectral truncation gives a practical generalized Log-Euclidean Signature (GLES) distance. On the synthetic torus benchmark of prior work, GLES preserves the geometric ordering of the original LES while producing smoother, more stable distances; learning the Mahalanobis operator further improves separation between geometries of different intrinsic dimension. The framework therefore provides a common geometric language for optimal transport, structured distributions, and spectral representations in high-dimensional and infinite-dimensional learning.

ABSTRACT. In this talk, I present an exact global extremal problem on nonconvex equal-diagonal isospectral families of symmetric matrices. Let A be a real symmetric matrix with constant diagonal d, simple largest eigenvalue λ, and remaining eigenvalues in [a,b], where b < λ. If v is a top eigenvector, an off-diagonal entry A_ij is wrong-sign when A_ij v_i v_j < 0.

For a prescribed mean-d spectrum Σ with simple maximum λ, second-largest member b, and smallest member a, I consider the problem of maximizing |A_12| over the equal-diagonal isospectral fiber subject to a coordinatewise positive top eigenvector and A_12 < 0. I determine the supremum exactly. If b ≤ d, no wrong-sign pair is feasible. If b > d, the optimal value is sqrt((d-a)(b-d)(λ-d)/(λ-a-b+d)) when a+b < 2d, and d-a when a+b ≥ 2d. The supremum is not attained under the strict positivity constraint. In the first regime, this value is strictly below both classical interlacing caps, so incorporating the sign constraint gives a genuine improvement over sign-blind spectral bounds.

I will explain how complementary positive-semidefinite spectral residuals produce the upper bound and how Schur-Horn realizations together with diagonal-preserving deformations inside the isospectral fiber establish global sharpness. I will also discuss the fixed-dimensional problem, where optimizing the remaining spectral freedom yields a second exact extremal value and an exact feasibility condition. For correlation matrices, the resulting constant becomes explicit in the second-largest eigenvalue, the smallest eigenvalue, and the dimension.

ABSTRACT. Reliability-based design optimization requires accurate first- and second-order derivatives of the failure probability to construct the gradient and Hessian of the Lagrangian. For computationally expensive implicit limit-state models, finite-difference estimation is often impractical because each perturbation requires an additional reliability analysis. Moreover, first-order reliability approximations may be insufficient for strongly nonlinear limit states [1]. This work derives analytical gradient and Hessian of the failure probability within a second-order reliability framework based on the Gram-Charlier A series. All sensitivities are obtained from a single search for the most probable failure point (MPP). For implicit limit-state functions, a quasi-Newton approximation of the limit-state Hessian is recovered from the search trajectory without additional model evaluations. Numerical examples demonstrate close agreement with Monte Carlo (MC) [2] reference results and a substantial reduction in computational effort compared with second-order reliability analysis using finite-difference sensitivities.

ABSTRACT. Proton Exchange Membrane Fuel Cells (PEMFCs) are electrochemical systems that generate electrical energy through the reaction between dihydrogen at anode and dioxygen at cathode. In these systems, dihydrogen (on the anode side) and dioxygen (on the cathode side) flow through microchannels—defined by ribs—and diffuse toward their respective catalytic areas, where the electrochemical reaction occurs. Uniform gas distribution across these surfaces enhances power efficiency, optimizes catalyst utilization and/or limit ageing.

In this work, we address the topology optimization of the distribution ribs using a dehomogenization method~\cite{feppon_multiscale_2024},~\cite{dede_inverse_2020},~\cite{Wu2021-tt},~\cite{Li2024}. The behavior of the gases within the fuel cell is modeled using two isotherm stationnary compressible Stokes systems coupled through a non-linear reaction type boundary term and solved using finite element methods. The ribs are represented as a periodic arrangement of squares, deformed by a diffeomorphism that locally modulates their shape and orientation. We begin by asymptotically deriving a homogenized system, where fine geometric details are embedded in the effective coefficients. The dependence on local dilation and orientation of the deformed periodic medium is explicitly characterized. After an offline computation phase where effective coefficients are precomputed for various values of spacing and orientation parameters of the ribs, an adjoint-based optimisation procedure is applied to determine the optimal geometric arrangements of local rib spacing and orientation. Due to feasibility constraints, ribs at cathode et anode must align correctly. To manage this constraint, established methods for reconstructing ribs and fine channels require adaptation.

ABSTRACT. In this talk, a model for emergency department as a stochastic network with a two-level service is analyzed. In the model, a patient care pathway consists of a first consultation, followed by medical examinations and finally an exit consultation before discharge. The first level (the technical cubicles) contains the patients before or during the first consultation. The second level (a waiting zone) consists of patients undergoing examinations and patients waiting for or attending a second consultation. Each level has a finite capacity, so when the second level is full, patients must stay at the first level. A main feature of the model is the fact that first and second consultations are managed by the same pool of emergency physicians.

Under a scaling assumption, for different service policies, the evolution of the number of patients in each step of the pathway is investigated. It is shown that depending on the parameters, there is a phase transition for the system between two different regimes: a regime limited by the number of emergency physicians and a regime limited by the second level capacity. Stochastic calculus from the evolution equations of the 2 or 3-dimension Markov process describing the system, martingale properties and differential inclusions are the main mathematical tools to obtain the limit of the scaled process.

Using this limiting process, the various service policies are described and compared in order to determine, for each regime, the policy that best optimizes either the number of patients treated per unit of time or the average duration of a care pathway.

ABSTRACT. Hospital inventory management is challenged by budget constraints, demand variability, and the large variety of items. These items also differ in their levels of criticality, requiring management approaches that account for their impact on healthcare services, such as the VED (Vital, Essential, and Desirable) classification. Moreover, stockouts may compromise the continuity of care and patient safety. In Brazilian public hospitals, these challenges are further compounded by bureaucratic procurement processes, in which the same item may be associated with different price registration agreements, each with distinct suppliers, prices, available quantities, and priority levels. In this problem, the purchasing decision involves determining how much of each product to acquire and from which of the available price registration agreements, while considering the desired inventory level based on product demand, the criticality of each item, and the priorities among available agreements. This work proposes two mathematical programming formulations with multiple objectives to support purchasing decisions in a university hospital, integrating operational and legal aspects of the procurement process with product prioritization according to item criticality. Both models determine what and how much to acquire while considering budget limitations, agreement priorities, and penalties for understocking. The first model considers a single purchasing period, oriented toward operational decisions and emulating current purchasing practice, while the second adopts a multi-period formulation that supports more tactical and strategic planning over a time window. The approaches were evaluated using real hospital instances constructed from historical data on inventory levels, consumption, and active price registration agreements. The results indicate that the proposed models can generate replenishment decisions that maintain inventory coverage within the desired range while respecting budget and procurement constraints, with more stable inventory levels than those observed historically.

ABSTRACT. Califrais, the digital operator of the Rungis wholesale market, must nightly collect ("ramasse") ordered goods across market pavilions before delivery trucks depart. We formalize the ramasse as a parallel-machine batch scheduling problem with family setup times, aiming to minimize either maximum delivery completion time or (weighted) average delivery completion time.

We establish NP-hardness via reductions to classical scheduling problems, derive structural properties of optimal solutions, lower bounds, and both compact and extended MILP formulations. We then design and compare several solution methods: greedy heuristics with local search, a MILP rounding heuristic with proven approximation guarantees, and a column-generation algorithm on a vehicle-based formulation.

Experiments on real Califrais data show column generation achieves solutions of close quality to MILP-based methods at a lower computational cost. We conclude with perspectives on integrating ramasse scheduling with downstream delivery routing.

ABSTRACT. Estimates suggest that about 14% of the food produced are wasted between production and distribution (FAO 2019). This waste leads to an economic cost for the companies involved and an ecological cost for the society. This abstract addresses an inventory problem with operational constraints from Califrais, the digital operator of Rungis, with a consideration on food waste.

The problem studied is a multi-item uncapacitated lot-sizing problem where several constraints are considered: the minimum order quantity, the perishability of products, and the inventory bound. Each item has a unit volume, a lifespan and a minimum order quantity, and unsold units reaching their lifespan must be disposed of at a cost penalizing food waste. The items are coupled by the storage capacity of the warehouse. This problem is a harder version of the multi-item uncapacitated lot-sizing problem with inventory bound, proven strongly NP-hard in Akbalik et al. (2015), so it is NP-hard as well.

As the inventory bound constraint is the only coupling constraint, a Lagrangian relaxation heuristic is made to decompose the problem into simple single-item lot-sizing problems. We prove that the resulting single-item lot-sizing problems can be solved polynomially using dynamic programming. The subplan decomposition used to build the recurrence will be presented, as well as the dominance properties derived from the constraints. As the Lagrangian relaxation modifies production, holding and disposal costs, a cost transformation is made to obtain non-speculative disposal costs. A repair heuristic restores the storage capacity at each subgradient iteration, providing feasible solutions whose quality is assessed on instances derived from Trigeiro et al. (1989), as well as on real-case instances from Califrais. A performance comparison is given against mixed-integer linear programming formulations and sensitivity analysis of multiples parameters will be shown.

ABSTRACT. Recent mathematical runtime analyses have shown that the crowding distance used by the NSGA-II multi-objective optimizer does not well reflect how isolated, and hence worth keeping, solutions are. This has provably led to drastic difficulties when the number of objectives is three or more. Based on the insights of these works, we design a variant of the crowding distance, called truthful crowding distance, and analyze the resulting truthful NSGA-II with mathematical means. (i)~We show that it solves the many-objective versions of the OMM, COCZ, LOTZ, and OJZJ$_k$ problems in the same (polynomial) asymptotic runtimes as the NSGA-III and the SMS-EMOA, contrasting the exponential lower bounds for the classic NSGA-II. (ii)~For the bi-objective versions of these problems, our NSGA-II maintains the good performance of the classic NSGA-II, but is successful already with population sizes smaller by a constant factor. (iii)~For the bi-objective OMM problem, we observe a (minimally) better performance in approximating the Pareto front. These results indicate that our truthful crowding distance is an interesting alternative to the classic crowding distance.

ABSTRACT. Besides domination-based algorithms such as the NSGA-II, the decomposition-based MOEA/D is one of the most popular and successful optimization heuristic for multi-objective problems. It decomposes the problem into several, single-objective subproblems. By solving these in a co-evolutionary manner and archiving non-dominated solutions, it approximates or computes the Pareto front. Interestingly, so far no advantages from the co-evolutionary communication of subproblem solutions were proven. In this first mathematical runtime analysis addressing this research question, we investigate how the mutation-based MOEA/D solves the classic \textsc{OneMinMax} benchmark when solutions are exchanged among subproblems in different distances $r$. We show, for the first time, that the MOEA/D profits from an exchange of solutions, and that already very small communication radii $r$ are sufficient to maximally profit from this feature of the algorithm. More precisely, for all values $r \ge 1$ we prove a tight runtime guarantee of $\Theta(n^2)$, where $n$ is the problem size. This is asymptotically faster than the known bound of $O(n^2 \log n)$ guarantee for $r=0$, which we complement with a matching bound of $\Omega(n^2 \log n)$.

ABSTRACT. The mathematical runtime analysis of multi-objective evolutionary algorithms has made huge progress in the last years. Yet, most results regard bi-objective problems only. The few existing many-objective benchmarks are mostly obtained by defining classic bi-objective benchmarks on disjoint sets of decision variables. This leads to specific structural properties, among others, that only even numbers $m$ of objectives can occur, and that the Pareto front of such problems is an only $m/2$ dimensional object, whereas in most practical problems, $m$-objective problems lead to $m-1$ dimensional Pareto fronts.

In this talk, we propose an alternative way to construct many-objective benchmarks from bi-objective ones. It overcomes the weaknesses of the previous construction but preserves the property that the many-objective benchmark is a natural extension of the bi-objective one. Using techniques similar to \cite{WD24}, we show that the runtime guarantees for the OneMinMax, LeadingOnesTrailingZeros (LOTZ) and OneJumpZeroJump$_k$ benchmarks in the classic construction also hold for our new construction. Because the new construction has the same Pareto front dimension with fewer objectives, the size of the largest incomparable set shrinks, leading to generally tighter bounds. For the LOTZ benchmark, in particular, this tightens the guarantee from $O^*((n/m'+1)^{2m'})$ to $O^*((n/m'+1)^{m'+1})$, where $n$ is the length of the bit string and $m'$ the number of blocks. No such bound is known for the classic construction.

This work is part of the PGMO-funded project \emph{From Mathematical Runtime Analyses to Better Evolutionary Multi-Objective Optimizers} (PI: Benjamin Doerr).

ABSTRACT. Multi-objective evolutionary algorithms (MOEAs) are among the most widely and successfully applied optimizers for multi-objective problems. However, because the true Pareto front can be prohibitively large, it is necessary to bound the population size while still computing a good approximation of the optimal trade-offs. MOEAs frequently utilize grid-based box archiving, parameterized by a grid size \delta, to compute an additive \epsilon-approximation of the Pareto front. Yet, standard archiving policies that strictly retain one individual per box, such as the Global Diversity Evolutionary Multi-Objective Optimizer (GDEMO), suffer from a structural vulnerability. Driven by the bias of standard mutation toward central solutions, unguided intra-box replacement fails to preserve extreme solutions, preventing the algorithm from progressing into new boxes. This traps algorithms in exponential runtime bottlenecks on boundary-critical pseudo-Boolean problems.

To this end, we propose GDEMO-II, a refined evolutionary archiver. It introduces a global filter that rejects a new offspring only if its targeted box is strictly dominated by the combined population, and it abolishes the single-occupant restriction to explicitly retain the local boundary extremes for each objective.

Via mathematical runtime analysis, we prove an upper bound on the population size that takes into account the range of the objective functions and the grid parameter. For bi-objective problems where objective values share the same asymptotic bound n, this guarantees a linear memory complexity of \mu = \mathcal{O}(n/\delta). Furthermore, we prove that GDEMO-II completely resolves the exponential delay of the original GDEMO on the OneMinMax benchmark, computing a tight approximation in expected time \mathcal{O}(n^2 \log n / \delta). Finally, it efficiently discovers the complete Pareto front of the LOTZ benchmark in expected time \mathcal{O}(n^3 / \delta).