ABSTRACT. We consider a provider of electric vehicle charging stations that operates a network of charging stations and use time varying pricing to maximize profit and reduce the impact on the electric grid. We propose a bilevel model with a single leader and multiple disjoint followers. The customers (followers) make decisions independently from each other. The provider (leader) sets the price of charging for each station at each time slot, and ensures there is enough energy to charge. The charging choice of each customer is represented by a combination of a preference list of (station, time) pairs and a reserve price. The proposed model thus accounts for the heterogeneity of customers with respect to price sensitivity and charging preferences. We define a single-level reformulation based on a reformulation approach from the literature on product line optimization, and we report computational results that highlight the efficiency of the new reformulation and the potential impact of our approach for reducing peaks on the electricity grid.

ABSTRACT. Introduction. Electric vehicle (EV) adoption depends on timely charging infrastructure revealing a chicken-and-egg dilemma. A formulation is given by the Charging Facility Location-and-Operation Problem as a bilevel program with continuous linking variables. These variables hinder the existence or attainability of the solution (Moore and Bard, 1990) and tailored resolution methods often exclude them (Bolusani and Ralphs, 2022).

Methods. This work proposes a two-phase multi-cut Benders decomposition (McDaniel and Devine, 1977) to solve an equivalent MILP of the bilevel program (Kleinert et al., 2021). More precisely, the proposed approach deal with a master problem accounting for primal variables while the subproblem provides bilevel feasibility cuts.

Results. Computational experiments show that two-phase variants consistently outperform alternatives with lower medians, upper quartiles, and reduced variability. The effect is most evident on larger networks, where gaps shrink significantly (e.g., from 14.86% to 7.14%).

Conclusion. This work solves the charging facility location-and-operation problem using a two-phase Benders decomposition, which consistently outperforms strong duality and relaxation methods. Results show systematically lower optimality gaps, reduced dispersion across instances, efficient master problem control through cut clean-up, and robust performance under varying granularity and route expansion.

ABSTRACT. With the transition to a fully renewable energy grid arises the need for a green source of stability and baseload support, which classical renewable generation such as wind and solar cannot offer due to their uncertain and highly-variable generation. In this paper, we study whether green hydrogen can close this gap as a source of supplemental generation and storage. We design a two-stage mixed-integer stochastic optimization model that accounts for uncertainties in renewable generation. Our model considers the investment in renewable plants and hydrogen storage, as well as the operational decisions for running the hydrogen storage systems. For the data considered, we observe that a fully renewable network driven by green hydrogen has a greater potential to succeed when wind generation is high. In fact, the main investment priorities revealed by the model are in wind generation and in liquid hydrogen storage. This long-term storage is more valuable for taking full advantage of hydrogen than shorter-term intraday hydrogen gas storage. In addition, we note that the main driver for the potential and profitability of green hydrogen lies in the electricity demand and prices, as opposed to those for gas. Our model and the investment solutions proposed are robust with respect to changes in the investment costs. All in all, our results show that there is potential for green hydrogen as a source of baseload support in the transition to a fully renewable-powered energy grid.

ABSTRACT. We investigate a Lagrangian particle discretization for a risk estimation problem within the framework of optimal transport. Given finitely many univariate probability densities corresponding to risk factors, as well as a certain danger function, one must find the riskiest joint law, in the sense of a certain spectral risk measure. The problem in question has been shown by Ennaji, Mérigot, Nenna, and Pass in [1] to be equivalent to standard multimarginal optimal transport problem with one additional marginal. To discretize the latter, we look for an approximate solution as a uniform point cloud of size N, and optimize the positions of the Dirac masses. The constraints on the marginals are dealt with via quadratic Wasserstein penalization terms, each of which is a semi-discrete optimal transport cost in dimension one. We prove that the discretized problem converges to the original one as the number of points goes to infinity, with a rate determined by the asymptotic uniform quantization errors of the marginals. Numerical experiments illustrate and support our theoretical findings.

Reference [1] H. Ennaji, Q. Mérigot, L. Nenna, B. Pass, Robust risk management via multi-marginal optimal transport, Journal of Optimization Theory and Applications, 202(2), 554-581, 2024.

ABSTRACT. Submission for the session "New trends in Optimal Transport and Applications", preferrably on November 18th.

Abstract: Finding regular transport maps between measures is an important task in generative modelling and a useful tool to transfer functional inequalities. The most well-known result in this field is Caffarelli’s contraction theorem, which shows that the optimal transport map from a Gaussian to a uniformly log-concave measure is globally Lipschitz. Note that for our purposes optimality of the transport map does not play a role. This is why several works investigate other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we use the control interpretation of the driving transport vector field inducing the transport map and a coupling strategy to obtain Lipschitz bounds for this map for a big class of what we call asymptotically log-concave measures. This talk is based on a joint work with Louis-Pierre Chaintron and Giovanni Conforti.

ABSTRACT. In this talk I will introduce a branched transport problem with weakly imposed boundary conditions. This problem was first derived as a reduced model for pattern formation in type-I superconductors in Conti, Goldman, Otto, Serfaty (2018). For minima of the reduced model with weak boundary conditions, it was conjectured by Conti, Otto and Serfaty that the dimension of the boundary measure is non-integer. The conjecture was linked to local scaling laws in De Philippis, Goldman, Ruffini (2025). I will present some recent advances in solving this conjecture. This talked is based on some works with Michael Goldman, Melanie Koser and Felix Otto.

ABSTRACT. As power systems integrate increasing shares of renewables, distributed energy resources, and complex market mechanisms, ensuring secure and efficient operation becomes ever more challenging. Conventional approaches such as dynamic security assessment and AC Optimal Power Flow (AC-OPF) are computationally demanding, which has motivated the use of machine learning (ML) to reduce computational cost and enhance situational awareness. However, several barriers remain: the lack of high-quality training data can limit model performance; power system physics must be respected to ensure feasibility; and the black-box nature of ML methods hinders their adoption in critical infrastructure. In this talk, I will present recent advances addressing these challenges. First, I will highlight the role of high-quality datasets and show how to generate samples that meaningfully capture system security boundaries, reflecting both static and dynamic criteria. Second, I will present physics-informed graph neural networks as promising tools for efficient N -k contingency screening. Finally, I will discuss how neural network verification can be used to rigorously enforce constraints, and introduce verification-informed ML techniques for AC-OPF that explicitly minimize worst-case violations during training. Together, these approaches provide certified guarantees of constraint satisfaction, ensure physically realizable operating points, and enable substantial computational savings, offering a pathway toward trustworthy ML for secure power system operation.

ABSTRACT. The optimal strategy for integrating large-scale EV fleets with regard to the DSO constraints in a manner that maximizes the local use of renewable energy sources is an NP-hard problem. Moreover, due to stochastic weather and human behaviors, this optimization should be done under uncertainty and potentially online. Reinforcement learning is a tool of choice under these conditions. As part of this work, bandits algorithms have been investigated. A first part of the work focussed on stochastic bandits based on Thompson sampling [1]. Following this, a comparative analysis was performed with adversarial bandits which were expected to provide theoretical bounds, at the cost of performance degradation that we aimed to quantify [2]. In particular, the “Follow-The-Perturbed-Leader” (FTPL) combined with heavy-tailed distributions (Fréchet, Pareto) was considered. A last part of the work focussed on a comparative analysis of bandit algorithms with Markov-decision process-based reinforcement learning in the form of the proximal policy optimization (PPO) algorithm. This work compared the performance of both algorithm types used independently, as well as the performance of an heterogeneous system including these different policies [3].

[1] E. Le Guern-Dall’o, R. Féraud, G. Camilleri, P. Maillé, F. Petit, R. Zorgati, H. Ben Ahmed, A. Blavette, “Muti-agent multi-armed bandits: application to EVs smart charging with grid constraints”, IEEE PowerTech conference, Kiel, Germany, 2025. [2] E. Giabicani, E. Le Guern-Dall’o, R. Féraud, G. Camilleri, P. Maillé, H. Ben Ahmed, A. Mepuis, J.-J. Prince Agbodjian, R. Zorgati, A. Blavette, “Decentralized and Robust EV Fleet Charge Scheduling Using Follow-The-Perturbed-Leader Under Multiple Uncertain- ties”, submitted to the PSCC 2026 conference, Limassol, Cyprus. [3] X. Ratte, Master thesis, in preparation, CentraleSupélec, 2025.

ABSTRACT. Demand-side management has emerged as a key strategy to enhance the flexibility and reliability of modern power systems, particularly in the context of the increasing penetration of renewable energy sources. This is achieved by leveraging the flexibility of a wide range of distributed energy resources. The aggregated control of such devices can provide valuable services to the grid (e.g. load shaping, frequency regulation, or congestion management), thus contributing to system stability and economic efficiency. In demand-side management applications, the number of controllable devices is typically very large, which naturally motivates the use of mean-field control (MFC). Optimal transport (OT) provides an intuitive approach for modeling the evolution of the aggregate distributions considered in the MFC. We develop a tractable optimization framework for the real-time coordination of heterogeneous loads (e.g. electric vehicles and water heaters), combining ideas from OT and MFC. We first extend the framework of Moment Constrained Optimal Transport for Control (MCOT-C) to a heterogeneous setting. We then propose a model predictive control approach where the agents’ data is progressively discovered during the day. The proposed approach is validated through numerical experiments on real datasets for electric vehicles and water heaters, demonstrating the effectiveness of this method in imposing global restrictions while preserving agent-level dynamics. The results show an emerging phenomenon with each system adapting to the (lack of) flexibility of the other. This proves to be very beneficial compared to a separate optimization of the different populations.

ABSTRACT. We propose a neural network architecture to address high--dimensional optimal control problems with linear and nonlinear dynamics. An important application of this method is to solve path planning problems of multi-agent vehicles in real time. The new method extends our previous SympOCNet framework by introducing a time-dependent symplectic network into the architecture. In addition, we propose a more general latent representation, which greatly improves model expressivity based on the universal approximation theorem. We demonstrate the efficiency of our approach for path planning problems with obstacle and collision avoidance, including systems with Newtonian dynamics and non-convex environments, up to dimension 512.

ABSTRACT. We present a finite-dimensional global numerical approximation for a class of extended mean field control problems. Our algorithm learns the value function on the whole Wasserstein domain, as opposed to a fixed initial condition. We leverage the approximation of the mean field problem by a finite-player cooperative optimisation problem, due to the propagation of chaos, together with the usage of finite-dimensional solvers. This avoids the need to directly approximate functions on an infinite-dimensional domain, and allows for more efficient memory usage and faster computation.

ABSTRACT. Motivated by game-theoretic models of crowd motion dynamics, this paper analyzes a broad class of distributed games with jump diffusions within the recently developed alpha-potential game framework. We demonstrate that analyzing the alpha-Nash equilibria reduces to solving a finite-dimensional control problem. Beyond the viscosity and verification characterizations for the general games, we explicitly and in detail examine how spatial population distributions and interaction rules influence the structure of alpha-Nash equilibria in these distributed settings, and in particular for crowd motion games. Our theoretical results are supported by numerical implementations using policy gradient-based algorithms, further demonstrating the computational advantages of the alpha-potential game framework in computing Nash equilibria for general dynamic games.

ABSTRACT. Deep Neural Networks are powerful tools for solving machine learning problems, but their training often involves dense and costly parameter updates. In this work, we use a novel Max-Plus neural architecture in which classical addition and multiplication are replaced with maximum and summation operations respectively. Our interest in these architectures stems from their ability to naturally induce sparsity in both forward and backward passes, making them appealing alternatives to dense networks. Indeed, only neurons that contribute to the maximum affect the loss. The (max,+) and (min,+) algebras have been applied in previous works, achieving competitive performance in various tasks. For instance, max-plus operators have been used for filter selection and model pruning, and the Linear-Min-Max-Plus architecture has been shown to be a universal approximator for continuous function. These results indicate that moving toward structured sparse architectures does not necessarily sacrifice expressivity.

However, standard backpropagation fails to exploit this sparsity, leading to unnecessary computations. To address this, we propose a sparse subgradient algorithm that explicitly exploits this algebraic sparsity. By tailoring the optimization procedure to the non-smooth nature of Max-Plus models, our method achieves more efficient updates while retaining theoretical guarantees. This highlights a principled path toward bridging algebraic structure and scalable learning. One of the fundamental tools we use to manage sparse updates is short computational trees, that allow an efficient use of computing resources in high dimensions. From the theoretical result, we also propose a novel initialization for min-max-plus networks, showing encouraging performance in preliminary numerical experiments.

ABSTRACT. In a wide range of scientific and engineering applications, such as economics, operations research, optimal transport, and energy systems, it is essential to approximate unknown convex functions based on limited data. Traditional convex regression methods, including convex programs, approximate kernel-based techniques, as well as piecewise-linear fitting approaches, often have poor scalability in high-dimensional spaces, may recover convex functions only with some errors, and may not have enough flexibility to recover the complicated convex behavior seen in real datasets. Input Convex Neural Networks (ICNNs) provide a promising alternative by encoding convexity directly into the architecture of neural networks. By enforcing weight constraints and adopting convexity-preserving activation functions, ICNNs guarantee convexity with respect to the input. Moreover, ICNN maintains the expressiveness and scalability of deep learning, making it an attractive choice for developing data-driven convex modeling in complex and high-dimensional spaces.

In this work, we introduce Input Smooth Convex Neural Networks (ISCNNs), a variant of ICNNs whose outputs are provably Lipschitz continuous. We focus on a previously underexplored property of ICNNs: the Lipschitz continuity of their output. Smooth convex functions are especially important in contexts where second-order information affects optimization performance. We investigate how the choice of activation function impacts the second-order smoothness of the ICNN output. In particular, we demonstrate that the widely used SoftPlus activation function produces significantly smoother convex functions compared to popular non-smooth alternatives such as ReLU.

ABSTRACT. One of the best-known fundamental results in neural network theory is that they are universal approximators: for any continuous function, any compact domain, and any precision epsilon, there exists a neural network within distance at most epsilon from the function over this compact domain. Mathematically, uniform approximation states that the class of functions realized by neural networks is dense in the set of continuous functions over compact domains. This holds under minimal assumptions (a single hidden layer and a non-polynomial activation function suffice).

We show the existence of a fixed recurrent network capable of approximating any computable function with arbitrary precision, provided that an encoding of the function is given in the initial input. While uniform approximation over a compact domain is a well-known property of neural networks, we go further by proving that our network ensures effective uniform approximation—simultaneously ensuring:

Uniform approximation in the sup-norm sense, guaranteeing precision across the compact domain [0,1]^d.

Uniformity in the sense of computability theory (also referred to as effectivity or universality), meaning the same network works for all computable functions.

Our result is obtained constructively, using original arguments. Moreover, our construction bridges computation theory with neural network approximation, providing new insights into the fundamental connections between circuit complexity and function representation.

We also show that the associated decision problems are complete—even under strict architectural constraints such as fixed, small network width and depth.

Furthermore, this connection extends beyond computability to complexity theory. The obtained network is efficient: if a function is computable or approximable in polynomial time in the Turing machine model, then the network requires only a polynomial number of recurrences or iterations to achieve the same level of approximation, and conversely. Moreover, the recurrent network can be assumed to be very narrow, strengthening the link between our results and existing models of very deep learning, where uniform approximation properties have already been established.

This work (preliminary version presented in MFCS 2025) is motivated by better understanding the foundations of complexity and computability theory of problems in optimisation. We will present some implications about the complexity of approximation for several optimisation problems.

ABSTRACT. In this article, we study inertial algorithms for numerically solving monotone inclusions involving the sum of a maximally monotone and a cocoercive operator. In particular, we analyze the convergence of inertial and relaxed versions of the nonlinear forward–backward with momentum (NFBM). We propose an inertial version of NFBM including a relaxation step and a second version considering a doubleinertial step with additional momentum. By applying NFBM to specific monotone inclusions, we derive inertial and relaxed versions of algorithms such as forward–backward, forward-half-reflected-backward (FHRB), Chambolle–Pock, Condat–Vu, among others, thereby recovering and extending previous results from the literature for solving monotone inclusions involving maximally monotone, cocoercive, monotone and Lipschitz, and linear bounded operators. We also present numerical experiments on image restoration, comparing the proposed inertial and relaxed algorithms. In particular, we compare the inertial and relaxed FHRB with its non-inertial and momentum versions. Additionally, we compare the numerical convergence for larger step-sizes versus relaxation parameters and introduce a restart strategy that incorporates larger step-sizes and inertial steps to further enhance numerical convergence

ABSTRACT. In this talk we present our new solver, MadIPM. MadIPM implements a regularized interior-point method (IPM) for large-scale linear programming. The algorithm solves a sequence of symmetric quasi-definite (SQD) systems, which becomes increasingly ill-conditioned as we are reaching convergence. These SQD systems can be decomposed efficiently using a signed Cholesky decomposition, requiring only static pivoting. As such, the linear systems can be factorized very efficiently in parallel. MadIPM leverages the sparse linear solver NVIDIA cuDSS to solve the linear systems at each iteration of the interior-point algorithm. To minimize the data exchange between the host and the GPU, MadIPM has been designed to run entirely on the GPU. We show that our new solver scales very well on modern GPUs. We compare MadIPM with a commercial solver (running in parallel using 16 threads) on large-scale LP instances from the MIPLIB benchmark. Our results show that our method is competitive with the commercial solver, with some significant speed-up on very large-scale LPs.

ABSTRACT. We study nonsmooth convex minimization through a continuous-time dynamical system that can be seen as a high-resolution ODE of Nesterov Accelerated Gradient (NAG) adapted to the nonsmooth case. We apply a time-varying Moreau envelope smoothing to a proper convex lower semicontinuous objective function and introduce a controlled time-rescaling of the gradient, coupled with a Hessian-driven damping term, leading to our proposed inertial dynamic. It should be noted that our dynamical system comes naturally from the high-resolution ODE framework. We provide a well-posedness result for this dynamical system, and construct a Lyapunov energy function capturing the combined effects of inertia, damping, and smoothing. For an appropriate polynomial scaling, the energy dissipates and yields decay of the objective function and gradient, stabilization of velocities, and weak convergence of trajectories to minimizers under mild assumptions. Conceptually, the system is a nonsmooth high-resolution model of Nesterov's method that clarifies how time-varying smoothing and Hessian-driven damping jointly govern acceleration and stability. The framework links and extends lines of work on high-resolution ODEs for NAG, Hessian-driven damping, and Moreau-envelope/proximal dynamics.

ABSTRACT. Wasserstein distributionally robust optimization (WDRO) offers an attractive framework for model fitting in machine learning as it systematically accounts for data uncertainty. While the Wasserstein distance is often associated with the curse of dimensionality, recent works have established generalization bounds with dimension-free sample rates for WDRO, similar to those in empirical risk minimization. However, exact generalization bounds—which fully capture WDRO's performance—remain unexplored in general settings. In this work, we derive an exact generalization bound for WDRO with dimension-free sample rates. Our analysis relies on a careful study of the nonsmooth structure of the robust objective, leading to a unified and broadly applicable framework.

ABSTRACT. The standard quadratic optimization problem (StQP) consists of minimizing a quadratic form over the standard simplex. If the quadratic form is neither convex nor concave, the StQP is NP-hard. This problem has many interesting applications ranging from portfolio optimization to machine learning. Sometimes, the data matrix is uncertain but some information about its distribution can be inferred, e.g. the first two moments or else a reference distribution (typically, the empirical distribution after sampling). In distributionally robust optimization, the goal is to minimize over all possible distributions in an ambiguity set defined based upon above mentioned characteristics. We explore the distributionally robust StQP and an ambiguity set based upon maximal Wasserstein distance to the sampled empirical distribution.

ABSTRACT. Distributionally Robust Optimization (DRO) is a robust framework based on optimal transport theory for solving optimization problems under uncertainty. Recent research provides tractable formulations based on the Wasserstein distance between probability distributions. For the 1-Wasserstein distance, strong duality results hold under convexity and continuity assumptions. Those approaches have proven their interest in deriving distributionally robust counterparts for various problems, such as regressions or supervised learning. We investigate an approach for solving optimal control problems under uncertainty by including distributionally robust formulations. We derive a system of ordinary differential equations leading to a two-point boundary value problem, thanks to Pontryagin's Maximum Principle, which determines the optimal control from the initial conditions of the costates of the problem.

ABSTRACT. A common practice for representing flexible populations of assets in Unit Commitment (UC) problems, which aim at optimizing electricity production while ensuring energy supply and demand balancing [7], is the aggregation of similar units in order to decrease the computational difficulty of the resolution [5]. However, in many cases the naive aggregation, consisting in summing individual constraints, is not exact and can lead to infeasibilities on the individual level. Interestingly, in the specific case of Electric Vehicles (EVs), individual constraint sets can be seen as generalized polymatroids, or g-polymatroids, a special class of polytopes introduced by Frank and Tardos [3] and also studied by Danilov and Koshevoy in the general setting of discrete convexity [2]. In particular, g-polymatroids ensure exact aggregation when summed as their Minkowski sum remains a g-polymatroid with the correct characteristics, and they can allow efficient optimization of linear objectives by the greedy algorithm. This observation has been made independently by Mukhi, Loho, and Abate [6]. We show how to integrate EV constraint sets in the UC problem by leveraging the g-polymatroid structure through price decomposition methods [1] and submodular function minimization techniques [4]. We numerically illustrate this approach on a realistic UC model of the French electricity production.

References [1] Pierre Carpentier and Guy Cohen. Décomposition-coordination en optimisation déterministe et stochastique. Vol. 81. Mathématiques et Applications. Berlin, Heidelberg: Springer Berlin Heidelberg, 2017. doi: 10.1007/ 978-3-662-55428-9. [2] 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. [3] András Frank and Éva Tardos. “Generalized polymatroids and submodular flows”. In: Mathematical Programming 42.1-3 (1988), pp. 489–563. doi: 10.1007/BF01589418. [4] Satoru Fujishige and Shigueo Isotani. “A Submodular Function Minimization Algorithm Based on the Minimum-Norm Base”. In: Pacific Journal of Optimization 4 (2009). [5] 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. [6] Karan Mukhi, Georg Loho, and Alessandro Abate. Exact Characterization of Aggregate Flexibility via Generalized Polymatroids. 2025. doi: 10.48550/arXiv.2503.23458. [7] Milad Tahanan et al. “Large-scale Unit Commitment under uncertainty”. In: 4OR 13.2 (2015), pp. 115–171. doi: 10.1007/s10288-014-0279-y.

ABSTRACT. The recent development of the electric vehicle sharing concept is due to its environmental and social benefits and to its potential to promote the use of electric vehicles. However, the success and prosperity of an electric vehicle sharing system (EVSS) heavily depend on the quality of planning of the system design and operations. This work focuses on the design of a one-way station-based EVSS. We propose a bi-objective two-stage stochastic programming model to simultaneously optimize the economic performance of the system and its quality of service under uncertainties of customer trip demand. Our model makes use of a risk measure, the conditional value-at-risk, to control the risk of obtaining a poor customer service level in the worst-case scenarios. Moreover, to enhance the practical relevance of the model, we integrate a detailed representation of tactical decisions on fleet deployment, as well as operational decisions related to vehicle relocation and battery recharging into the modelling of the EVSS design problem. This integrated approach results in the formulation of a large-size mixed-integer linear program. In order to solve it efficiently, we develop an original approximate Benders decomposition-based method in which explicit information from the scenario sub-problems is included into the master problem formulation under the form of aggregate second-stage variables and constraints.Numerical experiments on randomly generated instances highlight the performance of the developed procedure as compared to a state-of-the-art MILP solver. Finally, using a case study, we investigate the trade-off between economic profitability and quality of service.

ABSTRACT. In this work, we consider a station-based one-way vehicle sharing system, like, e.g., V\'elib, whose rebalancing is exclusively performed by the users. This rebalancing is achieved by inciting the users to sometimes modify their destination stations, so as to reroute the flow of vehicles towards stations where demand is high. Dynamic prices attached to the stations are used for the incitation: the amount to pay for a ride is the price attached to the destination station. A user arriving at an origin station may not perform a ride only for two reasons: either there is no available vehicle at the station when the user arrives, or the current prices attached to the stations make the user prefer to walk.

We assume a discrete choice model to describe the users' decisions. Based on this framework, we address the problem of maximizing the proportion of users who eventually take a ride, when prices can be updated according to the state of the system. We propose two heuristics. The first one is a simple rule where the price depends only on the current state of the station. The second one relies on strong duality in the theory of optimal transport. We also propose upper bounds based on mixed integer linear programming, close to those proposed by Waserhole and Jost. The quality of our heuristics is assessed via simulation on synthetic data, with the help of these upper bounds, and with respect to ``static'' strategies, where the prices cannot change over time. Experiments are currently conducted. In addition, special cases where exact optimal solutions can be determined are also identified.

ABSTRACT. This presentation addresses the numerical resolution of an aggregative stochastic optimal control problem in which agents are subject to a common source of uncertainty. The agents' dynamics may be nonlinear, rendering the problem typically nonconvex. Our problem is motivated by the practical framework for optimal and coordinated management of a large number of nonlinear energy dynamical systems under uncertainty.

A direct application of Rockafellar's expansion lemma [Rockafellar, 2019] allows a reformulation of our problem with separability across the agents and scenarios. This structure is particularly well-suited for a numerical resolution using Douglas-Rachford Splitting (DRS) algorithm, a well-known method for minimizing the sum of two convex functions, with proven weak convergence to a stationary point. In the nonconvex setting, when one function is smooth with a Lipschitz-continuous gradient and the other is proper and lower semicontinuous, convergence in finite dimensions to a point satisfying the first-order necessary optimality condition has been established in [Themelis & Patrinos, 2020]. We adapt the proof of this result to our infinite-dimensional setting, where the second function is assumed to be convex. Under suitable conditions on the dynamics and running cost, we establish the existence of a cluster point for the sequence generated by the DRS scheme when applied to our problem.

ABSTRACT. The increasing penetration of renewable energy requires greater use of storage resources to manage system intermittency. As a result, there is growing interest in evaluating the opportunity cost of stored energy, or usage values, which can be derived by solving a multi-stage stochastic optimization problem. Stochasticity arises from net demand (the aggregation of demand and non-dispatchable generation), the availability of dispatchable generation, and inflows when the storage facilities considered are hydroelectric dams.

We aim to compute these usage values for each market zone of the interconnected European electricity system, in the context of prospective studies currently conducted by RTE using the Antares simulation tool. The energy system is mathematically modelled as a directed graph, where nodes represent market zones and arcs represent interconnection links between these zones. In large energy systems, spatial complexity (thirty nodes in the system, each with at most one aggregated storage unit) compounds temporal complexity (a one-year horizon modelled with two timescales: weekly subproblems with hourly time steps).

This work addresses three main sources of complexity: temporal, spatial, and stochastic. We begin by examining the interaction between temporal and stochastic structures through a detailed study of information structures in a two-timescale setting for a single-node system, supported by numerical experiments. Based on this, we extend the framework to the full multinode case by incorporating a spatio-temporal decomposition scheme. To efficiently compute usage values, we apply Dual Approximate Dynamic Programming (DADP), which enables tractable decomposition across both time and space. This approach yields nodal usage values that depend solely on the local state of each node, independently of the others. To assess the quality of this approximation, we conduct numerical studies on a three-node system and compare the usage values obtained via DADP with those computed using traditional methods such as Stochastic Dynamic Programming (SDP) and Stochastic Dual Dynamic Programming (SDDP). Finally, we conduct numerical studies on a more realistic system composed of thirty nodes (modelling part of Europe) and show that DADP scales well.

ABSTRACT. In this presentation, we will describe our recent results on the convergence of the Augmented Lagrangian Algorithm with inexact proximal method, as well as its application to the numerical approximation of a solution to a mean-field control problem.

The Augmented Lagrangian is an algorithm for the minimization of a convex cost function which can be written as the sum of two convex functions f and g. Motivated by problems in which the direct minimization of f+g is difficult but that of the sum of f with a regularization of g is simpler, the Augmented Lagrangian algorithm proceeds by iteratively minimizing the latter sum, modifying the regularization of g at each iteration. This regularization is obtained through a dual principle, so that our regularized problem is the dual problem associated to an augmented dual cost which penalizes the distance to the dual parameter μ. A more complete definition can be found for instance in [1].

Our contribution consists here in studying the behavior of this algorithm when we cannot find an exact solution of the regularized problems. More precisely, we use the Frank--Wolfe Algorithm for the resolution of these problems. We show a sublinear convergence speed for this method, in terms of the number of calls to the oracle for the Frank--Wolfe Algorithm.

In a second part, we show that this algorithm can be used to find a numerical solution of a mean-field control problem. Our main idea consists in putting the contribution of the running cost and the coupling constraint in the function f. The oracle then consists in a numerical scheme for the coupled system of PDEs Hamilton--Jacobi--Bellman and Fokker--Plank.

References

[1] Jorge Nocedal and Stephen J. Wright, Numerical optimization, Springer, 1999

ABSTRACT. We consider linear Chance-Constrained Stochastic Problems (CCSPs) with finite support, a class of optimization problems known for its challenging non-convex structure. To improve the performance of branch-and-cut solution methods, we introduce a new class of valid inequalities, termed multi-disjunctive inequalities. These inequalities are constructed based on a disjunctive property inherent to the mathematical formulation of CCSPs. We present a heuristic separation procedure for these inequalities and provide extensive numerical results. The computational study demonstrates that our proposed method significantly strengthens the formulation and improves solver performance compared to existing families of valid inequalities from the literature, such as quantile and mixing inequalities.

ABSTRACT. In the railway industry, as well as in other branches of transportation, the construction of yearly schedules for drivers is a key step of the planning process. Due to the high complexity of the problem, the SNCF applies a classical two-step approach. In a first step—the Crew Scheduling—, driving tasks (assumed to be the same every week) are combined together to form daily duties. In a second step—the Cyclic Crew Rostering—, these duties are organized so as to form cyclic rosters, one per team of drivers. The sequential nature of this approach results in sub-optimal solutions. Additionally, this approach does not account for uncertainties, which is known to lead to increased costs due to train delays. In this work, we propose to modify this approach by handling the two steps simultaneously and by including stochastic constraints, so as to tackle both suboptimality and train delays. This “integrated” approach relies on column generation and models exactly the impact of train delays on crew rosters when the delay distributions are known. A contribution is the modeling of the pricing sub-problem as a Resource Constrained Shortest Path Problem, within the setting of Parmentier. In particular, distributions are kept as a resource and the stochastic order is used to discard paths in a label correcting path enumeration algorithm. In the case where there is no train delays, the integrated approach has already been able to provide a solution to a regional freight instance of 280 trains, with a cost 8% lower that the one achievable with the current sequential approach. In the case with train delays, the integrated approach showed significant reduction in the impact of these delays on the previous solution, while having a cost lower than with the current sequential approach.

ABSTRACT. We introduce a proximal-type optimization algorithm designed for minimizing nonsmooth and nonconvex composite functions. The composite structure comprises a weakly-concave outer function and a Lipschitz continuous inner mapping. By linearizing the outer function, the problem is approximated by a nonlinear, potentially nonconvex master program. The algorithm's iterates are defined as stationary points of this master program, guided by an improvement function that integrates both the objective and constraint into a single mathematical object. We establish convergence to critical points and present promising numerical results on chance-constrained optimization problems, demonstrating the practical effectiveness of the proposed approach.

ABSTRACT. We study affine algebraic varieties parametrized by sine and cosine functions, generalizing algebraic Lissajous figures in the plane. We show that, up to a combinatorial factor, the degree of these varieties equals the volume of a polytope. We deduce defining equations from rank constraints on a matrix with polynomial entries. We discuss applications of Lissajous varieties in dynamical systems, in particular the Kuramoto model. This leads us to study connections with convex optimization and Lissajous discriminants. Joint work with Francesco Maria Mascarin.

ABSTRACT. Medical emergency departments are complex systems in which patients must be treated according to priority rules based on the severity of their condition. We develop a model of emergency departments using Petri nets with priorities, described by nonmonotone piecewise linear dynamical systems. The collection of stationary solutions of such systems forms a "phase diagram”, in which each phase corresponds to a subset of bottleneck resources (like senior doctors, interns, nurses, consultation rooms, etc.). Since the number of phases is generally exponential in the number of resources, developing automated methods is essential to tackle realistic models. We develop a general method to compute congestion diagrams. A key ingredient is a polynomial time algorithm to test whether a given "policy" (configuration of bottleneck tasks) is achievable by a choice of resources. This is done by reduction to a feasibility problem for an unusual class of lexicographic polyhedra. Furthermore, we show that each policy uniquely determines the system’s throughput. We apply our approach to a case study, analyzing a simplified model of an emergency department from Assistance Publique - Hôpitaux de Paris

ABSTRACT. Given a multivariate polynomial with integer coefficients, we present rational certificates of nonnegativity based on sum-of-squares decompositions, without making any the assumptions on the input polynomial. We introduce the stereographic transformation, which allows us to transform any polynomial to a coercive polynomial. We also employ perturbation methods to ensure that a coercive polynomial has a zero-dimensional gradient ideal. All these transformations preserve the nonnegativity of the input polynomial and reduce the problem to our previous work (hal-04909464). The complexity of our algorithm is single exponential in the number of variables. This is a joint work with Matías Bender, Khazhgali Kozhasov and Elias Tsigaridas.

ABSTRACT. The Gilbert--Steiner problem is a generalization of both the Steiner tree problem and optimal mass transportation problems, allowing the use of additional (branching) points in a transport plan. This problem was first introduced for measures with finite support in a paper by Gilbert in 1967~\cite{gilbert1967minimum}. The name ``Steiner'' refers to the branched structure of Steiner trees. A book by Bernot, Caselles, and Morel~\cite{bernot2008optimal} provides a modern perspective on the problem.

The most widely studied model assumes that the cost of transporting mass $m$ along a segment is proportional to $m^p$ and linear in the segment length, where $0 \leq p < 1$. This cost function, $c(x) = x^p$, satisfies several natural conditions such as monotonicity, concavity, and $c(0) = 0$.

We determine all pairs $(p, d)$ for which the Gilbert--Steiner problem in $d$-dimensional Euclidean space admits only branching points of degree~3. Namely, this occurs if and only if $d = 2$ or $p < 1/2$. This classification was obtained in~\cite{cherkashin2025branching2}, building on results from~\cite{volz2013gilbert} and~\cite{cherkashin2025branching}.

An \emph{irrigation setup} is a subproblem in which one of the measures is restricted to be a Dirac measure. As an application of the main result, one can construct a universal solution to the irrigation Gilbert--Steiner problem for $d = 2$ or $p < 1/2$. Universality here means that this solution contains every possible combinatorial structure with the same $p$ and $d$ as a substructure.

ABSTRACT. In 1992, Bollobás and Meir showed that for any n points in the k-dimensional unit cube [0,1]^k, k >= 2, one can always find a tour x_1, ..., x_n through these n points with |x_1 - x_2|^k + |x_2 - x_3|^k + ... + |x_n - x_1|^k <= c_k * k^(k/2), where |x - y| is the Euclidean distance between x and y, and c_k is an absolute constant depending only on k. Remarkably, this bound does not depend on n, the number of points. They further conjectured that the best possible constant for every k >= 2 is c_k = 2 and showed that it cannot be taken lower than that. The conjecture is only known to be true for k = 2 due to Newman. Recently, Balogh, Clemen and Dumitrescu revised the conjecture for k = 3 by showing that c_3 >= 4 * (2/3)^(3/2) > 2.17. They also gave the best currently known bounds c_k <= (2/3) * 6.709^k and, for large k, c_k <= 2.91^k * (1 + o_k(1)).

We reduce the gap between the lower and upper bounds on c_k from exponential to linear by showing that c_k <= min(6(k+1), 2e(k+2)). Moreover, we prove that for large k the conjecture holds asymptotically, i.e. show that c_k = 2 + o_k(1). We obtain similar results for related problems on Hamiltonian paths and perfect matchings in the unit cube. To achieve this, we substantially improve the ball packing argument used in previous works and exploit connections with optimal (with respect to the minimum angle) spherical codes.

ABSTRACT. Steiner tree problem is a problem of connecting the given compact set by the shortest way. I will consider the problem in Euclidean space.

By indecomposable Steiner tree we name the solution of the Steiner tree problem without vertices of degree $2$. I will talk about examples of indecomposable Steiner trees with infinite number of branching points (vertices of degree 3).

I will provide different examples of Steiner trees with infinite number of branching points: small (connecting countable number of the terminal points), middle (connecting an uncountable set of zero Hausdorff dimension) and large (connecting totally disconnected set of positive Hausdorff dimension).

Based on joint works with D. Cherkashin, E.Paolini and E. Stepanov.

ABSTRACT. In this presentation, we revisit the problem of minimizing the epidemic final size in the SIR model through social distancing of bounded intensity. In the existing literature, this problem has been considered imposing a priori interval structure on the time period when interventions are enforced. We show that when considering the more general class of controls with an L1 constraint on the confinement effort that reduces the infection rate, the support of the optimal control is still a single time interval. This shows that, for this problem, there is no benefit in splitting interventions on several disjoint time periods. However, if the infection rate is known in advance to vary over time in a piecewise constant manner, then we show that the optimal solution could consist of splitting the interventions in at most two disjoint time periods. Finally, we provide a couple of numerical examples illustrating the structure of the optimal control derived in both cases: constant and piecewise constant transmission rate.

ABSTRACT. Dans cette présentation, nous étudions le contrôle optimal d’un système dynamique à trois espèces, modélisé par les équations de Lotka–Volterra, dans le but de réduire l’infection d’un microbiote complexe. Ce type de modèle s’inspire de travaux récents sur la dynamique intestinale, notamment Stein et al. (2013), qui ont utilisé des modèles écologiques pour analyser la stabilité et les interactions du microbiote intestinal à partir de séries temporelles. Le contrôle étudié correspond à un régime de dosage antibiotique ou probiotique, et notre approche s’appuie sur une analyse géométrique des trajectoires optimales. En appliquant le principe du maximum de Pontryagin et les méthodes de classification des contrôles développées dans Bonnard & Rouot (2024), nous paramétrons les trajectoires sous forme de géodésiques et identifions les ensembles limites α et ω, constitués de points d’équilibre isolés correspondant à des rayons anormaux. L’étude montre que ces géodésiques anormales forment un feuilletage d’une surface quadratique et permet de classifier la dynamique complète des trajectoires à l’aide d’un système d’invariants calculé de manière explicite et symbolique. Cette méthodologie offre un outil puissant pour analyser et optimiser les régimes de traitement microbiens afin de réduire les infections, tout en donnant une vision géométrique claire de la structure des trajectoires optimales dans ces systèmes dynamiques.

References [1] B. Bonnard, J. Rouot, Feedback Classification and Optimal Control with Applications to the Controlled Lotka-Volterra Model, Optimization 74 no. 5 (2024), pp. 1173–1196. [2] R.R. Stein, V. Bucci, N.C. Toussaint, C.G. Buffie, G. Rätsch, E.G. Pamer, et al., Ecological modelling from time-series inference: insight into dynamics and stability of intestinal microbiota, PLos Comp. Biology, 9 no. 12 (2013).

ABSTRACT. This presentation focuses on hybrid optimal control, the goal is to develop a new regularization technique. For this purpose, we consider a Mayer optimal control problem governed by a dynamics defined regionally. Using the concept of Fillipov solutions, it is shown that this problem is equivalent to a new optimal control problem, with additional controls. Next, we introduce a family of auxiliary optimal control problems by regularizing the nonsmooth interfaces and we add a boundary layer as well as a penalization term to the cost function to account for auxiliary controls. Our main result is that solutions to these (smooth) problems converge (up to a subsequence) to a solution of the original one.

This presentation if part or the session "Optimal Control and its Applications".

ABSTRACT. We consider a manufacturing facility producing multiple finished products on a single capacitated machine. The production activities in this facility require energy under the form of electricity. We assume that this electricity is provided by a grid-connected decentralized energy system comprising renewable energy conversion devices (e.g., photovoltaic panels) and an energy storage system (e.g., lithium-ion batteries). We seek to simultaneously plan the industrial production and the energy supply of this facility over a short-term horizon comprising a few days and divided into a discrete set of periods. Specifically, this plan should determine, for each period, the timing of the startup operations to be carried out on the machine and the size of the production lots for the items to be manufactured, as well as the amount of energy to be purchased from the national grid and the quantities to be charged into or discharged from the batteries.

The joint optimization of the short-term production plan and of the energy supply plan is made particularly challenging by the uncertainties on the demand for the finished products and on the renewable power generation. In this work, we consider a multi-stage decision process in which the actual value of the uncertain parameters is revealed progressively over time and some planning decisions can be adjusted after the realization of these parameters. This leads us to develop a multi-stage stochastic integer programming approach for the problem.

We introduce a dynamic-programming type formulation of our multi-stage stochastic integer problem taking the form of a set of nested single-stage sub-problems linked together by cost-to-go functions. The sub-problem corresponding to stage 0 is a mixed-binary linear program while the sub-problems corresponding to the later stages are linear programs.

We propose a novel solution approach based on the hybridization of a branch-and-cut algorithm with a stochastic dual dynamic programming algorithm. Through computational experiments on a diverse set of randomly generated instances, we show that our hybrid algorithm outperforms benchmark methods in both solution quality and speed, particularly for medium to large instances involving a large number of decision stages.

ABSTRACT. For a given integer linear program (ILP), a symmetry is a variable permutation which trans- forms any solution to another solution with same cost. Symmetries arising in ILP can impair the solution process, in particular when symmetric solutions lead to an excessively large Branch and Bound (B&B) search tree. Various techniques, so called symmetry-breaking techniques, are available to handle symmetries in ILP. In this presentation, we focus on so-called ßtructured" symmetries, where all column permutations of the solution matrix are symmetries. Many prac- tical problems related to energy management feature such structured symmetry. We first review existing techniques to handle structured symmetries. We also propose new techniques, bases on variable fixing during B&B. We also introduce the concept of “sub-symmetries", which are symmetries arising in subproblems of the considered ILP. This extends the concept of symmetry and enables to capture more symmetries of the considered ILP. Sub-symmetry breaking tech- niques are also proposed. Numerical experiments on the Unit Commitment Problem show the efficiency of the proposed techniques.

ABSTRACT. System periodicity is exploited in long-term planning to reduce the size of the model by reducing the time horizon to a limited set of representative periods. System dynamic is then captured by a fine time discretization within each period, but this temporal representation prevents to model interdependencies between the periods, including storage conservation. We consider modelling non-periodic storage in integrated models for long-term expansion planning of power systems, from micro to continental grids, using temporal decomposition and block coordination between investment and operation.

ABSTRACT. In the last couple of decades, metric spaces have found themselves at the forefront of optimization and analysis research. A wide range of problems with important applications in physics, chemistry, and biology require the generality of metric spaces as a basis for their formulation. Lueg et al. provided a geometrical interpretation to the space of phylogenetic trees, allowing for the problem of interpolating between two such trees, constructed on different genetic data, to be formulated as Fréchet mean problems on a metric space. The space of protein structures has been given a \lparan{}pseudo-\rparan{}metric structure, and protein optimal design problems can be formulated with respect to this structure. This work fits into the broader topic of knot theory, where metric-geometric aspects have blended successfully with optimization notions. In the study of our universe, computational shape optimization problems have often been required. Such optimization problems for subsets of Finsler manifolds have been solved.

This work expands on this recent interest by developing a framework for differentiability on a quasi-metric space suitable to Newton-type methods. This theoretical framework can be employed in constructing root-finding and optimization algorithms. Numerous results relating to the traditional Newton's methods can be adapted to the more general setting of this framework. We thus develop a calculus theory and an existence theory, thus providing a version of an inverse function theorem applicable to functions defined on a metric space.

On the algorithmic side, our work develops a Newton-type, superlinearly convergent algorithm for root finding problems on quasi-metric spaces. Quasi-metric spaces do not provide a canonical notion of invertible linear maps, and as such this concept has to be narrowly defined for our purposes. Riemannian optimization provides a blueprint for defining our algebraic constructs, in the Riemannian exponential and logarithm. The main analytical technique giving rise to superlinear convergence of Newton-type methods consists of fixed point iterations of operators derived from Newton differentiable mappings.

The existence of solutions to optimization problems in metric spaces has attracted the interest of the variational analysis community. Newton-type methods have been successfully used in order to obtain inverse function theorems. The works of Smale and Kantorovich provide classic examples of these ideas. The work of Kantorovich has been successfully applied to nonsmooth problems, defined on Euclidean spaces by Cibulka et al.. Continuing in the realm of nonsmooth problems defined on linear spaces, Páles has used Ekeland's principle for problems on Banach spaces. A Kantorovich-type result, from, for equations defined on quasi-metric spaces stands as the crowning achievement of this work.

ABSTRACT. In this talk we study optimal control problems with state constraints in Mayer form over the Wasserstein space $\sP_2(M)$ of a not-necessarily compact Riemannian manifold $M$. We focus on the Hamilton-Jacobi (HJ) equation on $\sP_2(M)$ using the viscosity theory, proving a general comparison principle for the HJ equation, and thus establishing the uniqueness of solutions.

Finally, using classical tools from optimal control theory, we prove that the value function of the state-constrained optimal control problem is a solution of a certain, but known, HJ equation. An additional result -and a cornerstone of this work- is that, under suitable curvature assumptions on the manifold $M$, we show that the squared Wasserstein distance is directionally differentiable.

ABSTRACT. While gradient descent method on Euclidean spaces is deeply studied, similar results on nonlinear metric spaces are still underdeveloped. Due to the lack of linearity, the usual definition of subgradient can not be directly adapted. In this work, directional derivatives along geodesics of functions defined on a p-uniformly convex metric space (G, d) are introduced. The necessary condition for being a local minimizer of f turns out to have nonnegative directional subderivative along all geodesics. If f is strongly convex, then this condition is also sufficient. Based on optimality conditions, algorithms such as basic descent method and steepest descent method are designed and analyzed. Meanwhile, a linear convergence result of the steepest descent method under Polyak-Łojasiewicz property or strong convexity is proved. This work is intended to serve as a foundational result for further research on this topic.

ABSTRACT. Invited session: New trends in Optimal Transport and Applications

Abstract: In this talk we discuss how to perform Principal Component Analysis (PCA) of a dataset whose elements are probability distributions, compared using the Wasserstein distance. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. To achieve this, we leverage the Riemannian interpretation of the Wasserstein metric. We first address the case of a collection of centered Gaussian distributions, and show how to lift the computations in the space of invertible linear maps using Bures-Wasserstein geometry. For the more general setting of absolutely continuous probability measures, we leverage Otto-Wasserstein geometry and neural networks to parameterize geodesics in Wasserstein space. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.

Preferred date: November 18

ABSTRACT. Unbalanced optimal transport (UOT) is a natural extension of optimal transport (OT) allowing comparison between measures of different masses. It arises naturally in machine learning by offering a robustness against outliers. The aim of this work is to provide convergence rates of the regularized transport plans and potentials towards their original solution when both measures are weighted sums of Dirac masses.

Invited sessions : New trends in Optimal Transport and Applications

Comment : I prefer the date of November 19.

ABSTRACT. In this talk we will propose a notion of geodesic extrapolation in the Wasserstein space and more generally in a metric space. This notion appears to be a convex optimization problem. We then discuss an extension of this problem: the Generalized Wasserstein Barycenter. It is a Wasserstein barycenter with multiple positive and negative weights. Finally we will discuss a convex relaxation of this problem and some cases when the relaxation appears to be tight.

ABSTRACT. Consider the problem of constrained stochastic multi-objective optimization. We are inter- ested in minimizing a set of objectives subject to inequality constraints, all of which are twice differentiable and have the functional form of an expectation of a stochastic quantity of interest. Since we rarely have access to analytic expressions for the expectations of either the objectives or the constraints, we try to find the minima through stochastic approximation. However, complications arise due to the uncertain nature of the stochastic quantities of interest. Not only is it complicated to project onto the constraint set, but the evolution of the primal and dual problems are now coupled, leading to oscillations that can inhibit convergence to a saddle point without setting an arbitrary upper limit [1]. Moreover, even after arriving at a stationary solution which is asymptotically viable, it is still relatively unexplored how to assess its admissibility. We propose a saddle point algorithm to solve constrained stochastic multi-objective opti- mization problems. The first ingredient to our approach is the stochastic multi-gradient, which we use as a common direction of descent for all objectives [2]. The second is a dual step which dynamically adjusts the value of the Lagrange multipliers for the problem. We go on to prove the convergence of our approach almost surely to the Pareto set. References [1] Harold Joseph Kushner and Dean S Clark. Stochastic approximation methods for constrained and unconstrained systems. Vol. 26. Springer Science Business Media, 2012. [2] F. Poirion Q. Mercier and J.-A. Désidéri. “SMGDA: an uncertainty based multi- objec- tive optimization approach. Illustration to an airplane composite material”. In: Procedia engineering 199 (2017), pp. 1199–1203

ABSTRACT. We study dynamic programming in optimal stochastic control and its generalizations in finite discrete time. The extended problem format accommodates nonlinear and set-valued dynamical systems often encountered in applications. We consider generalized forms of Bellman equations whose solutions (Bellman functions) allow for computationally attractive optimality conditions for the control problem. Our focus is on problem structures where the arguments of the Bellman functions take values in low-dimensional spaces. The dimensionality is critical for the complexity of computational techniques, including those based on state space discretization or cutting planes, such as the Stochastic Dual Dynamic Programming (SDDP) algorithm, which has become popular in real-life applications.

Our first main result characterizes optimal solutions and optimum values in terms of solutions to the generalized Bellman equations. Existence of solutions to the Bellman equations is established under general conditions that do not require compactness. In the convex case, the existence condition can be seen as a far-reaching generalization of the "no-arbitrage condition" in mathematical finance. We allow for general randomness but show that, in the Markovian case, the dimensionality of the Bellman equations reduces with respect to randomness just like in more traditional control formulations. Besides the theoretical results, we propose a version of the SDDP algorithm applicable to the extended control format.

ABSTRACT. We study continuity properties of solutions to Bellman equations (Bellman functions) in discrete-time stochastic optimization and control. Continuity of Bellman functions is important e.g. in the sensitivity analysis of the Bellman equations with respect to changes of the underlying probability measure. Continuity is instrumental also for the convergence of various numerical schemes such as traditional state space discretizations as well as cutting plance approximations that employ convexity to avoid discretizations which tend to be very sensitive to the dimension of the state space. Sufficient conditions are given for lower semicontinuity, continuity and Lipschitz continuity, respectively, of the Bellman functions with respect to their effective domains. Requiring continuity only with respect to the effective domains allows for extended real-valued Bellman functions and more general problem formulations than earlier results on the topic. In particular, our results allow random constraints which are common in most industrial applications of stochastic optimization and control.

ABSTRACT. We introduce a decision-focused scenario generation framework for contextual two-stage stochastic linear programs that bypasses explicit conditional distribution modeling. A neural generator maps a context x to a fixed-size set of scenarios {ξs(x)}S s=1. For each generated collection we compute a first-stage decision by solving a single log-barrier regularized deterministic equivalent whose KKT system yields closed-form, efficiently computable derivatives via implicit differentiation. The network is trained end-to-end to minimize the true (unregularized) downstream cost evaluated on observed data, avoiding auxiliary value-function surrogates, bi-level heuristics, or differentiation through generic LP solvers. Unlike single- scenario methods, our approach natively learns multi-scenario representations; unlike distribution-learning pipelines, it scales without requiring density estimation in high dimension. We detail the barrier formulation, the analytic gradient structure with respect to second-stage data, and the resulting computational trade-offs. Preliminary experiments on contextual synthetic instances illustrate that the method can rival current state-of-the-art methods, even when trained on small amounts of training data.

ABSTRACT. The Kelly mechanism is a proportional allocation auction widely adopted in decentralized resource allocation systems to share an infinitely divisible resource among competing agents. We analyze the sequential game it induces when agents have $\alpha$-fair utilities and behave strategically. Our main result proves that synchronous best-response updates drive bids to the unique Nash equilibrium at a linear rate for $\alpha\in \{0,1,2\}$. Extensive simulations reveal that best-response dynamics reach equilibrium significantly faster than previously proposed no-regret learning algorithms.

ABSTRACT. Rational behaviour of agents is a classical assumption in game theory. However, as seen from various experiments, rationality is limited when human agents make decisions. Prospect theory models the non rational behaviour of human agents arising from a subjective perception of risk. While Prospect theory has been used in economics, it has not been widely applied in game theory. We apply Prospect theory in games with aggregative structure. Such games arise, for example, in electricity markets where strategic end users are exposed to risk. We show that the irrationality of the agents is seen to modify the structure of the set of Nash equilibria. Further, we obtain conditions under which equilibria are preserved or vanished. Taking a system-wide perspective, we show how the impact of irrational behaviour on the equilibria can be controlled by learning optimal values of a system price parameter. Thus, even in the presence of irrationality, a coordinator can drive system behaviour in a desired direction.

ABSTRACT. This work addresses the issue of the convergence of an $N$-player game towards a limit model involving a continuum of players, as the number of agents $N$ goes to infinity. More precisely, we investigate the convergence of Nash equilibria to equilibria of Cournot--Nash type. When the cost function of the players is the first variation of some potential function, equilibria can be characterized by a stationarity condition, satisfied in particular by the minimizers of the potential. We demonstrate such a characterization under low regularity assumptions. We give a positive answer to the convergence question in the case where players interact in a pairwise fashion; in this case we show that the original sequence of $N$-player games also admit a potential structure and prove that their corresponding potential functions converge in the sense of $\Gamma$-convergence to the potential function of the limit game. Aligned wih the strong characterization of equilibria, we can show that any convergent sequence of Nash equilibria has a Cournot--Nash equilibria as limit.

ABSTRACT. With the increasing adoption of Electric Vehicles (EVs), the charging demand at public charging stations with a fast charging mode is expected to grow, leading to congestion, such as long waiting time. Such congestion can influence user’s behavior, creating interaction between them. From the point of view of a charging station operator, it is crucial to anticipate such congestion in order to design appropriate strategies (e.g., sizing, pricing, etc.). In this context, we study a mixed-strategies and non-atomic game, where the players are the EV users, their strategies are the choice of a Charging Station Pool (CSP), and their individual cost include the waiting time at the chosen CSP. It is assumed that the players (i.e., the EV users) impact on congestion is heterogeneous and continuously distributed. We show that the Nash equilibrium is connected to the solution of a social welfare maximization, meaning that the game admits a potential. As a consequence, we demonstrate the existence and, under monotonicity assumption (leading to the convexity of the potential), the uniqueness of the Nash equilibrium. Under additional assumptions, the solution of the social planner, and hence the Nash equilibrium, induces a Voronoï-type partition of the domain, meaning that the Nash equilibrium is in pure strategies. In addition, we propose a numerical scheme that approximates the Nash equilibrium, and show a toy example

ABSTRACT. In this article, we introduce and study a new class of robust mean field control problems, formulated in min-max form, where the mean field interaction is subject to some uncertainty. We establish the existence and uniqueness of a solution under appropriate assumptions, including relevant convexity-concavity conditions, and establish a related stochastic maximum principle. Additionally, we introduce a corresponding class of risk-averse mean field games, which includes variational (or potential) mean field games that arise as the first-order condition of a robust mean field control problem. We also address the existence of solutions to these mean field games and discuss different sets of conditions under which uniqueness holds true.

ABSTRACT. We investigate optimal control strategies for probability distributions governed by linear and nonlinear Fokker-Planck (FP) equations, with a particular focus on McKean-Vlasov (MV) dynamics. These equations describe the time evolution of interacting particle systems under a stochastic external force and play a central role in statistical physics and collective behaviour. By controlling these dynamics, we aim to accelerate convergence to equilibrium and stabilise metastable or unstable steady states, i.e., to control the long-time behaviour of FP equations.

We develop two approaches. Both modify the confining potential via space- and time-dependent controls. First, for the linear FP equations, we consider the case in which there is a unique equilibrium and introduce a spectral optimal control framework based on the ground-state transformation to a Schrodinger operator. A gradient-based iterative scheme that combines Pontryagin's maximum principle with Barzilai-Borwein updates computes controls that target slow modes, thus accelerating the convergence to the unique steady state. Second, for MV (nonlinear FP), we treat equations that may admit multiple equilibria, for which our goal is to accelerate convergence to a chosen steady state. We design feedback controls derived from a linear-quadratic problem built on the linearisation around this steady state and involving a Riccati operator. We rigorously prove that the proposed approach stabilises the desired equilibrium at any chosen rate.

Numerical experiments on Kuramoto and O(2) models (with magnetic field), ill-conditioned Gaussian, and double-well potentials in one and two dimensions demonstrate that our approaches accelerate convergence to equilibrium and stabilise otherwise unstable states. These results fall within PDE-constrained optimisation and stabilisation, connecting continuous optimisation, spectral methods, and controlled sampling.

ABSTRACT. Optimizing over parametric probability distributions underpins sampling tasks in modern generative modeling applications. Gibbs gradient descent (GGD) alleviates costly sampling acquisitions arising from the use of first-order optimization techniques by introducing a singleloop, two-timescale optimization algorithm to optimize through sampling. We provide upper bounds for Gibbs gradient descent in the continuous-time setting with finite and infinite number of particles. Along the way, we introduce an exponential moving average (EMA) algorithm, called EMA-GGD, which naturally extends the method into a three-timescale regime, enabling variance reduction. We illustrate our convergence bounds on several numerical examples.

ABSTRACT. Many applications in machine learning involve data represented as probability distributions. The emergence of such data requires radically novel techniques to design tractable gradient flows on probability distributions over this type of (infinite dimensional) objects. For instance, being able to flow labeled datasets is a core task for applications ranging from domain adaptation to transfer learning or dataset distillation. In this setting, we propose to represent each class by the associated conditional distribution of features, and to model the dataset as a mixture distribution supported on these classes (which are themselves probability distributions), meaning that labeled datasets can be seen as probability distributions over probability distributions. We endow this space with a metric structure from optimal transport, namely the Wasserstein over Wasserstein (WoW) distance, derive a differential structure on this space, and define WoW gradient flows. The latter enables to design dynamics over this space that decrease a given objective functional. We apply our framework to transfer learning and dataset distillation tasks, leveraging our gradient flow construction as well as novel tractable functionals that take the form of Maximum Mean Discrepancies with Sliced-Wasserstein based kernels between probability distributions.

ABSTRACT. At KPLER, we develop advanced methodologies to estimate vessel-level trade flows — including product types and quantities — at global ports. By integrating fragmented data sources (such as draft surveys, public and private reports, analyst inputs, and port infrastructure constraints), we aim to reconstruct granular trade activity with precision, enabling deeper insights into maritime logistics and commodity markets. To achieve this, KPLER employs a vessel-by-vessel approach that provides a real-time, macro-level view of commodity volumes in transit, offering unparalleled insights into maritime logistics and global commodity markets. To estimate quantities, we rely on draft change recordings, which are not always reliable and may contain data gaps. We complement this with all available product- and port-level information to enhance flow consistency. However, data sources are not always coherent, which necessitates sophisticated conflict resolution between constraints. Furthermore, vessels can exchange commodities at sea through ship-to-ship operations, leading to complex interactions between vessels and trade formations. Currently, this is handled by separating tasks: first estimating quantities, then products, and finally building trades. However, this approach poorly handles product-quantity constraints that reflect real vessel behavior. For instance, some ports have specific export/import capabilities dependent on product type, which leads to inconsistencies between estimated quantities and the chosen product at the port level. In this work, we designed a Mixed-Integer Linear Programming (MILP) model that incorporates all three tasks (quantity, product, and trade) to address the interdependence between quantity, product, and trade constraints. Additionally, the model must be computationally efficient, as it will be executed hundreds of thousands of times daily. This is particularly challenging due to the combinatorial nature of the problem.

ABSTRACT. Gas-Insulated Substations (GIS) are compact and highly reliable electrical substations widely deployed in modern power systems, where key high-voltage components are enclosed within sealed metal compartments filled with sulfur hexafluoride (SF6) gas. Despite their operational advantages, GIS maintenance poses significant challenges due to the hazardous environmental impact of SF6 leakage and and the significant cost of maintenance.

An integrated optimization framework for GIS maintenance planning is proposed, which jointly addresses the optimal scheduling of maintenance actions and the temporal grouping of maintenance tasks to enhance operational efficiency. The scheduling problem is formulated as a Mixed Integer Linear Programming (MILP) model that aims to minimize maintenance costs while constraining environmental impacts, ensuring system reliability. Complementing this, a sliding-window based Integer Linear Programming (ILP) model clusters temporally proximate maintenance tasks, consolidating interventions and reducing operational inefficiencies such as frequent site visits and redundant resource deployment. Numerical experiments on various scenarios validate the effectiveness of the proposed framework, demonstrating improved scheduling scalability, enhanced environmental compliance, and operational efficiency gains.

ABSTRACT. The districting problem involves partitioning a geographical region into smaller subregions called districts. This optimization problem has been applied across various domains, including police districting, political redistricting, and healthcare districting, among others [1], with the primary objective of optimizing resource allocation and service delivery. Depending on the specific application, districting solutions must satisfy different criteria and constraints. There are fundamental properties that are particularly critical, such as contiguity, population homogeneity, which seeks to achieve balanced population distribution across districts, and spatial compactness, which aims to minimize intra-district distances by creating geometrically compact districts.The challenge lies in simultaneously optimizing these often conflicting objectives while satisfying additional domain-specific constraints. This study investigates the healthcare districting problem, specifically focusing on primary and secondary care levels within healthcare systems. The objective is to minimize the maximum population-resource difference across all districts, with the aim of maximizing the population served or covered by the healthcare system and helping to improve accessibility to healthcare services. The proposed approach considers essential districting properties including population homogeneity, spatial compactness, and territorial contiguity. A key aspect of this work is the simultaneous analysis of both care levels, recognizing that if they are managed separately, then imbalances could occur in the level that is not the focus. One of the most important challenges in this type of problem is how to impose the contiguity property, so we analyze four different mathematical formulations to ensure this constraint. Additionally, we examine structural properties of these formulations in order to manage the computational complexity of the problem. From this analysis, we derive different strategies such as symmetry-breaking constraints and variable fixing techniques that help us reduce computational time and improve solution quality for instances that are not solved to optimality.

ABSTRACT. Multi-objective evolutionary algorithms (MOEAs) are among the most widely and successfully applied optimizers for multi-objective problems. However, to store many optimal trade-offs (the Pareto optima) at once, MOEAs are typically run with a large, static population of solution candidates, which can slow down the algorithm. We propose the dynamic NSGA-II (dNSGA-II), which is based on the popular NSGA-II and features a non-static population size. The dNSGA-II starts with a small initial population size of four and doubles it after a user-specified number $\tau$ of function evaluations, up to a maximum size of $\mu$. Via a mathematical runtime analysis, we prove that the dNSGA-II with parameters $\mu \geq 4(n + 1)$ and $\tau \geq \frac{256}{50} e n$ computes the full Pareto front of the OneMinMax benchmark of size $n$ in $O(\log(\mu) \tau + \mu \log(n))$ function evaluations, both in expectation and with high probability. For an optimal choice of $\mu$ and $\tau$, the resulting $O(n \log(n))$ runtime improves the optimal expected runtime of the classic NSGA-II by a factor of $\Theta(n)$. In addition, we show that the parameter $\tau$ can be removed when utilizing concurrent runs of the dNSGA-II. This approach leads to a mild slow-down by a factor of $O(\log(n))$ compared to an optimal choice of $\tau$ for the dNSGA-II, which is still a speed-up of $\Theta(n / \log(n))$ over the classic NSGA-II.

ABSTRACT. In discrete optimization, we seek to find "high-quality" solutions within a vast search space. Evolutionary algorithms form a family of randomized methods specifically designed to generate and refine promising candidates. Inspired by natural processes such as mutation, they explore the search space and employ heuristics to decide whether new candidates should be accepted or not. A notable approach [1], the move-acceptance hyper-heuristic (MAHH), alternates - via random coin flips - between an elitist selection (the only-improving operator, OI) and accepting any new candidate (the all-moves operator, AM), and was recently shown to escape local optima with remarkable efficiency.

In this talk [2], we propose two modifications to this algorithm that demonstrate impressive performances on a large class of benchmarks including the classic CLIFF_d and JUMP_m function classes and we provide an unified analysis of our method to this whole new class (contrary to [1]). (i) Instead of randomly choosing between the only-improving (OI) and any-move (AM) acceptance operator by flipping a coin, we take this choice via a simple two-state Markov chain. This modification alone reduces the runtime on JUMP_m functions with gap parameter $m$ from $\Omega(n^{2m-1})$ to $O(n^{m+1})$. (ii) We then replace the all-moves (AM) acceptance operator with the operator that only accepts worsenings (OnlyWorsening, OW). Such a, counter-intuitive, operator has not been used before in the literature. However, our proofs show that our only-worsening operator can greatly help in leaving local optima, reducing, e.g., the runtime on Jump functions to $O(n^3 \log n)$ independent of the gap size.

In general, we prove a remarkably good runtime of $O(n^{k+1} \log n)$ for our Markov move-acceptance hyper-heuristic on all members of a new benchmark class SEQOPT_k, which contains a large number of functions having $k$ successive local optima, and which contains the commonly studied JUMP_m and CLIFF_d functions for $k = 2$.

This work is part of the PGMO-funded project Mathematical Analysis of Complex Randomized Search Heuristics (PI: Benjamin Doerr).

ABSTRACT. We introduce linear surrogate functions for modeling inequality constraints to solve constrained blackbox optimization problems with the Augmented Lagrangian CMA-ES. Each surrogate is constructed from a binary classifier that predicts the sign of the constraint value. The classifier, and consequently the resulting algorithm, is invariant under sign preserving transformations of the constraint values and can handle binary, flat, and deceptive constraints. Somewhat surprisingly, we find that adopting a sign-based classification model of the constraints allows to solve classes of constrained problems which can not be solved with the original Augmented Lagrangian method using the true constraint value.

ABSTRACT. Randomized search heuristics (RHSs) are generally believed to be robust to noise. However, almost all mathematical analyses on how RSHs cope with a noisy access to the objective function assume that each solution is re-evaluated whenever it is compared to others. This is unfortunate, both because it wastes computational resources and because it requires the user to foresee that noise is present (as in a noise-free setting, one would never re-evaluate solutions).

In this work, we show the need for re-evaluations could be overestimated, and in fact, detrimental. For the classic benchmark problem of how the $(1+1)$ evolutionary algorithm optimizes the LeadingOnes benchmark, we show that without re-evaluations up to constant noise rates can be tolerated, much more than the $O(n^{-2} \log n)$ noise rates that can be tolerated when re-evaluating solutions.

This first runtime analysis of an evolutionary algorithm solving a single-objective noisy problem without re-evaluations could indicate that such algorithms cope with noise much better than previously thought, and without the need to foresee the presence of noise.

ABSTRACT. Stochastic Gradient Descent (SGD) is an algorithm widely used to tackle high dimensional convex (possibly smooth) optimization problems that are very common in machine learning. This algorithm has first been proposed in the pioneering work of Robbins et al. (1951) but was studied under very strong variance assumptions that are virtually never verified in practice. Since then, an extensive body of work has emerged to develop frameworks that are both realistic and provide rigorous performance guarantees for SGD. In this talk, we propose to provide a brief yet thorough account of the assumptions underlying the analysis of SGD, highlighting the contexts where each has been relevant. However, despite a very rich literature, some open questions still surround SGD. Historically, most results on SGD in the convex and smooth setting were presented in the form of bounds on the ergodic function value gap and until very recently it remained unknown if the last iterate could achieve similar performance. We will present a result that answers this question positively, a result that was independently established in Attia et al. (2025).

ABSTRACT. The analysis of Stochastic Gradient Descent (SGD) often relies on making some assumption on the variance of the stochastic gradients, which is usually not satisfied or difficult to verify in practice. This paper contributes to a recent line of works which attempt to provide guarantees without making any variance assumption, leveraging only the (strong) convexity and smoothness of the loss functions. In this context, we prove new theoretical bounds derived from the monotonicity of a simple Lyapunov energy, improving the current state-of-the-art and extending their validity to larger step-sizes. Our theoretical analysis is backed by a Performance Estimation Problem analysis, which allows us to claim that, empirically, the bias term in our bounds is tight within our framework.

ABSTRACT. While many approaches were developed for obtaining worst-case complexity bounds for first-order optimization methods in recent years, there remain theoretical gaps in cases where no such bound can be found. In such cases, it is often unclear whether no such bound exists (e.g., because the algorithm might fail to systematically converge) or simply if the current techniques do not allow finding them. In this work, we propose an approach to automate the search for cyclic trajectories generated by first-order methods. We then show that the heavy-ball (HB) method provably does not reach an accelerated convergence rate on smooth strongly convex problems. More specifically, we show that for any condition number and any choice of algorithmic parameters, either the worst-case convergence rate of HB on the class of L-smooth and μ-strongly convex quadratic functions is not accelerated (that is, slower than 1 − O(κ)), or there exists an L-smooth μ-strongly convex function and an initialization such that the method does not converge. To the best of our knowledge, this result closes a simple yet open question on one of the most used and iconic first-order optimization techniques. Our result is supported by a systematic and constructive approach to the study of cycling behaviors of first-order methods. We show the robustness of our results to perturbations of the cycle, and extend them to classes of functions that also satisfy higher-order regularity conditions.

ABSTRACT. In this recent project, our goal was to resolve a fascinating biological puzzle: why toxin-producing microbes almost always win against toxin-sensitive microbes in idealized environments such as a lab, but the situation is often reversed in the real world? Finding an answer required a combination of wet lab experiments, mathematical models, and optimal control theory. It turns out, frequent environmental disruptions such as "dilutions" (e.g., flushing of the gut, or tooth-brushing, or dry soils experiencing heavy rains) give non-antagonistic microbes a good chance to win. This is the case for a fairly broad range of model parameters and remains true whether the dilutions are periodic or randomly timed, and even if antagonistic microbes can use population sensing to produce toxin selectively. We verified the latter by solving non-local Hamilton-Jacobi-Bellman PDEs, encoding the optimal control of toxin production for antagonistic microbes, with dilutions modeled as discontinuous jumps in population trajectories. More broadly, our results also offer some caution about limitations of dynamic models with time-averaged rates.

ABSTRACT. We demonstrate a novel characterization of law-invariant and coherent risk measures, based on a generalized optimal transport problem. By generalized, we mean that the second marginal of the transport plan to be optimized is not fixed, but constrained to a given set. Extending the classical Kantorovich duality theorem, we derive a general duality formula for risk measures, of numerical interest.

ABSTRACT. Synchronous machines, a dominant technology in electric vehicle and industrial drives, exhibit strong nonlinearities due to magnetic saturation and cross-coupling between current components. These effects significantly impact efficiency, torque estimation, and control stability, but their accurate representation leads to optimization problems that are inherently non-convex. This talk presents two convex optimization methodologies that address these challenges, focusing on both modeling and control.