ABSTRACT. Model Predictive Control (MPC) is one of the most successful modern control techniques. It generates a feedback control by successively solving finite-horizon optimal control problems. A key question is whether the resulting feedback is approximately optimal in some sense. In this talk we give a positive answer to this question using dissipativity and turnpike properties. We first describe results for deterministic MPC, then present an industrial use case, and finally give a survey on recent results for stochastic MPC schemes.

ABSTRACT. This talk presents Luca Brunod Indrigo's thesis, supervised by Pascale Bendotti, Philippe Chrétienne and Bruno Escoffier, which was defended in December 2025. Scheduling is the field of optimization that aims at deciding when to process the jobs of a project so that certain constraints are respected and a performance criterion is maximized. In a real-world context, a scheduling problem typically needs to be solved at various stages of a project, between which uncertainty may disrupt project data. At each stage, new scheduling decisions are expected to match the decisions that were taken at earlier stages as best as possible. Indeed, deviating too much from previous decisions can entail additional costs, e.g., penalties from subcontractors, and should therefore be avoided. The thesis is organized in two parts. The first part deals with a special case of robustness called anchor robustness that aims at guaranteeing the starting times of some jobs. Several problems are studied that consider both the proactive aspect, i.e., building schedules with some guaranteed starting times, and the reactive aspect, i.e., rescheduling once uncertainty is realized. The second part deals with resource leveling, which in the present case is to respect a given resource level as best as possible. Polynomial solution methods, NP-hardness proofs as well as approximation results are provided for such problems. This thesis is an industrial thesis in cooperation between EDF, the main French power provider, and the LIP6 laboratory at Sorbonne-Université. The research carried out is inspired by problems found in the industrial context of EDF. More precisely, the main motivation for this work is the scheduling of large-scale maintenance tasks performed during outages of power plants. Such projects contain many jobs, involve a variety of subcontractors and must be planned sufficiently in advance, which makes them prone to uncertainty. The heavy equipments and the specialized workforce that take part in those projects are rare resources that need to be properly managed to avoid delays and additional costs. For all these reasons, anchor robustness and resource leveling are helpful as dedicated optimization tools to improve performance and reliability at each stage of planning. This aspect of practical applicability has been a central concern all along and some algorithms have indeed been implemented for life-size instances.

ABSTRACT. Factored formulations of semidefinite programs are a popular approach to handle such instances at scale, even though such reformulations worsen the problem conditioning. When the original semidefinite programs arise from phase retrieval problems, additional ill-conditioning may also be present. As a result, first-order algorithms may struggle to converge quickly towards a solution. In this talk, we consider nonconvex factored formulations of phase retrieval semidefinite programs. Using a second-order Riemannian trust-region framework, we establish fast local convergence of the method on a subclass of these problems. In practice, we observe that this fast convergence holds more broadly.

This talk is part of the session "Nonconvex landscapes and algorithms in data science" organized by Andrew McRae.

ABSTRACT. Invited session : "Nonconvex landscapes and algorithms in data science"

Orthogonal synchronization is the problem of estimating orthogonal matrices from relative pairwise measurements. This problem admits a convex semidefinite program as a relaxation, which is solvable in theory but non scalable in practice. The Burer-Monteiro factorization of this convex program gives a more scalable alternative at the cost of losing convexity. In practice however, local algorithms such as gradient descent find global optima of the non-convex problem which suggests that, in some case, the landscape is benign : all local minima are global minima.

The goal of this presentation is to understand why that is the case. More precisely, we transform the problem of « explaining why there are no spurious local minima » into an optimization problem (P), the value of which we aim to lower bound. We then discuss how to construct a family of manageable dual certificates for the dual of (P) yielding benign landscape results.

ABSTRACT. Invited session: Nonconvex landscapes and algorithms in data science

Transformer architectures have achieved remarkable success across diverse tasks, yet the theoretical foundations driving their ability to learn rich representations without labels remain an active area of study. In this talk, we explore the theoretical capabilities of attention mechanisms through the lens of two classic unsupervised problems: data clustering and Principal Component Analysis (PCA).

First, we examine how attention-based layers automatically capture underlying distributional structures. By analyzing a simplified multi-head attention layer under a Gaussian Mixture Model (GMM), we demonstrate that minimizing population risk on unlabeled data drives head parameters to align directly with true mixture centroids. We further reveal that even fixed attention layers without trainable parameters can perform in-context quantization, highlighting an innate capacity to dynamically adapt to input-specific distributions.

Second, we establish a direct, explicit connection between attention layers and PCA. When trained on Gaussian data, both softmax and linear attention mechanisms learn parameters that align with the principal eigenvectors of the covariance matrix. We extend this analysis across both finite-prompt regimes and the infinite-prompt limit—proving convergence to globally optimal spectral directions—and demonstrate that attention successfully recovers latent signal directions under spiked Wishart covariance models.

Together, these results provide a unified theoretical framework showing that attention mechanisms inherently execute fundamental unsupervised tasks like clustering and dimensional reduction, shedding light on the fundamental representation-learning power of Transformers.

ABSTRACT. \documentclass[11pt,a4paper]{article} \usepackage{graphicx} \usepackage{hyperref}

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\title{\textbf{Regularization by Common Noise for Mean Field Games and Mean Field Control}}

\author{ Fran\c{c}ois Delarue$^1$\quad \underline{Lukas Wessels}$^2$ \\[5pt] % Affiliations are put manually \small $^1$Universit\'{e} C\^ote d'Azur, France, \texttt{francois.delarue@univ-cotedazur.fr} \\ \small $^2$Universit\'{e} C\^ote d'Azur, France, \texttt{lukas.wessels@univ-cotedazur.fr} \\ }

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{{\bf Keywords}: Mean field games, mean field control, common noise, regularization by noise, backward stochastic Hamilton--Jacobi--Bellman equations \\[2em]

In this talk, we demonstrate that an infinite dimensional multiplicative common noise has regularizing properties for mean field games (MFGs) and mean field control (MFC) problems. More precisely, in the case of MFGs, we show without relying on classical monotonicity conditions that the master field U(t,x,m) and its x-derivatives are Lipschitz continuous in the measure argument. As a corollary, in the case of MFC problems, we show that the value function is Lions differentiable which allows us to construct optimal feedback controls. Our proof strategy is based on Girsanov's theorem and a regularity result for backward stochastic Hamilton-Jacobi-Bellman equations. This talk is based on ongoing joint work with F. Delarue.

\begin{thebibliography}{00}

\bibitem{ref} F. Delarue, L. Wessels, Regularization by Common Noise for Mean Field Games and Mean Field Control, \emph{Work in progress}.

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ABSTRACT. Invited session “Mean Field Models”

Classical Backward Stochastic Differential Equations (BSDEs) theory provides an elegant probabilistic representation of stochastic control problems and their associated PDEs. However, this correspondence breaks down in the mean field (McKean-Vlasov) setting, where the law of the state process enters the dynamics intrinsically, rendering the standard framework inadequate. We discuss a notion of BSDE on the Wasserstein space of probability measures, called a W-BSDE, whose solutions are intrinsically measure-dependent, and its connections with mean field control problems, mean field games, and PDEs on the Wasserstein space. In particular, this provides a probabilistic counterpart to analytic approaches based on the master equation. We finally present a well-posedness result for this class of equations. The key point is that the whole problem can be transferred to a single characteristic equation: whenever the latter admits a solution, both components of the W-BSDE can be reconstructed from it.

This talk is based on a joint work with Mao Fabrice Djete.

ABSTRACT. In this talk based on an article in preparation, we are interested in two learning procedures and their relation to the formation of mean field games (MFGs) equilibria.

The first part is dedicated to the presentation of new results on the fictitious play. Namely, we explore a learning procedure on the controls of players. We prove that this new procedure converges for a class of mean field games of controls (which is to say coefficients depends not only on the distribution of players but also on that of their controls), and possibly with common noise. Besides its interest to explain the formation of mean field games in iterated games, this procedure also suggest a possible alternative approach to learning in MFGs

In the second part, we are concerned with mean field control problems. For such problems, we investigate gradient descent on the space of controls. Under quite general assumptions this gradient descent converges to a mean field game equilibrium, for the associated potential game. This procedure has an interpretation in the formation of MFGs from the optimization problem of a principal agent, moreover it also leads to the presentation of a simple iterative scheme converging to a mean field games equilibrium.

ABSTRACT. Supply-chain optimization has generated a very large literature, but most mathematical models focus on a specific combination of decisions rather than on a unified representation of the complete system. These approaches considerably extend classical supply-chain models. Some authors explicitly introduce the term generalized supply chain. However, "generalize" does not necessarily have the same meaning across the literature. In many cases, generalization concerns the topology of the network, the type of facilities considered, or the integration of several predefined decisions. The underlying production structure generally remains application-dependent.

The approach investigated here adopts a different perspective. We seek to develop a formulation in which transportation, storage and arbitrary transformations of commodities are primitive mathematical operations. A transformation is not restricted to a predefined manufacturing process or to a mixture of materials: a commodity h entering a plant may generate another commodity k, possibly with a yield different from one and with an associated processing time. This unified framework provides a basis for modeling a broad range of integrated supply-chain systems and may facilitate future extensions incorporating additional operational decisions and application-specific constraints representing production, assembly, conversion, dismantling or disposal through the same mathematical object. We present a mathematical programming formulation for this object and some extensions which allow modeling well-known supply chain problems such as multicommodity flow, blending problem, scheduling, etc.

ABSTRACT. Mathematical optimization has been used to model several parts of the supply chain, from production and inventory to transportation, reverse flows and environmental or operational constraints. Some models already combine several of these dimensions.

However, even relatively generic formulation often remain tied to a predefined network structure. We instead investigate whether different supply-chain structures can be represented through the same set of mathematical primitives, without redefining the formulation for each industrial configuration.

Our formulation builds on the transformation-flow concept introduced in. The physical supply chain is described through transportation, storage and commodity transformation. We derive a compact formulation and an equivalent state-expanded network representation, whose structure can be exploited through a decomposition of the problem.

In this presentation, we focus on how this formulation can be extended from the physical description of the supply chain to its economic and industrial operation. We therefore introduce a set of operational mechanisms which are combined within a economic objective based on the value generated by the supply chain and the costs associated with its operation.

We then study whether introducing these mechanisms changes the mathematical structure identified in the original formulation. Several mechanisms can be incorporated while preserving the main structure of the formulation, whereas others introduce additional coupling or integer decisions and require a different treatment.

ABSTRACT. Distributed generation turns households into prosumers, who sell their surplus to the grid at the feed-in tariff and buy their deficit at the retail rate. A local energy community exists to recover the gap between the two, since any exchange between neighbours at a price inside the band benefits both sides. Two questions follow. How to set the price, and how to execute the mechanism while extending as little trust as possible to any party, including the market operator (an entity the French framework makes mandatory). This talk presents the theoretical framework that answers the first question and the proposed infrastructure that answers the second.

ABSTRACT. An invited talk in the session “Black-box Optimization”.

Many real-world optimization problems involve objective functions whose evaluation is computationally expensive, making direct optimization impractical. Surrogate-based optimization addresses this challenge by replacing costly evaluations with predictive models learned from a limited number of samples. Although surrogate models are commonly assessed through predictive accuracy metrics, their suitability for optimization can be overlooked.

This work investigates optimization-oriented surrogate-assisted frameworks for expensive black-box optimization, with a focus on Wind Farm Layout Optimization (WFLO). The objective is not to develop new surrogate models, but rather to study how surrogate representations, optimization formulations, and sequential sampling strategies can be combined to efficiently guide the search towards high-quality solutions. Particular attention is paid to the interaction between model accuracy, and optimization tractability.

The proposed research explores different surrogate families and optimization schemes within iterative design-of-experiments frameworks inspired by Efficient Global Optimization and Sequential Model-Based Optimization. The long-term goal is to identify surrogate-optimization combinations capable of scaling to large and expensive optimization problems while maintaining strong optimization performance.

ABSTRACT. Invited talk in the session titled: Black-box Optimization

The energy transition is reflected in part by the modernization of distribution networks (DNs), which are evolving toward a more active role in power system operations. These active distribution networks (ADNs) are characterized by a high penetration of distributed energy resources (DERs), often of renewable sources, and a flexible network topology enabled by remotely controlled switches. DERs can induce bidirectional power flows that may exacerbate power losses, cause voltage violations, lead to line overloading, and affect frequency stability, among others. DNR can be modelled as a mixed-integer nonlinear optimization problem that seeks a radial topology to minimize power losses through switch operations. The proposed approach for DER integration using DNR combines blackbox optimization (BBO) and a high-accuracy power flow solver for improved feasible topology identification. Its computational burden is reduced by splitting the problem into continuous and binary formulations, solved respectively by BBO and a heuristic combinatorial optimization algorithm adapted to the DN context. To account for uncertainty in load demand and DER generation, the approach is formulated as a two-stage scenario-based stochastic program minimizing expected day-ahead power losses. In the first stage, the heuristic explores the binary decision space to identify a topology. Given this topology, the second stage solves a continuous BBO sub-problem for each scenario, where DER generations provide the recourse to ensure the topology feasibility across all considered scenarios. Each scenario combines load demand and DER generation limits from the same hour, preserving temporal correlation, and is derived from historical demand profiles and solar irradiance data. Spatial correlation is preserved through scaling according to the test DN baseline values. To reduce the computational burden of the second stage, scenario reduction is performed to identify representative clusters of similar scenarios. Inspired by multi-fidelity optimization, the fidelity level is defined by the number of clusters considered and is gradually increased at each iteration. One scenario is randomly sampled from each cluster. Combined with a reduced BBO evaluation budget, it mitigates the computational effort in the early iterations. The proposed method is then evaluated on three DNs of increasing size to assess its scalability.

ABSTRACT. This is an invited talk in the session "black-box optimization".

SPDNets are neural networks whose inputs lie on the manifold of symmetric positive definite matrices. Those networks are particularly useful on Brain-Computer Interface (BCI) tasks. However, they can be fooled by adversarial attacks based on Riemannian projected gradient descent, though designing such attacks requires complete, or white-box, access to the network. In this talk, we propose adversarial attacks on SPDNets assuming that the network is only available in a black-box fashion. Our attacks are based on direct-search and gradient-free algorithms, and, we leverage recent advances in Riemannian black-box optimization to tailor these methods to the Riemannian setting. Our experiments on BCI data show that our methods successfully fool the networks.

ABSTRACT. In this work, we study two-stage stochastic programs in which uncertainty is represented by a finite set of scenarios. Accurately representing the uncertainty requires a large number of scenarios. This makes the extensive formulation intractable, and has motivated a wide range of scenario reduction, clustering and decomposition techniques.

We first propose a matrix-based view of the problem that brings them together. The idea is to aggregate the constraints and the variables of the second stage through two matrices. Each row of a matrix defines an aggregated constraint or an aggregated variable, and its entries give the contribution of each scenario to it. We show that every scenario clustering scheme is a particular choice of these two matrices. Adaptive partition-based methods, which refine a partition of the scenario set guided by dual information, correspond to matrices whose rows have disjoint supports and binary entries. We derive a bound on the approximation error at any first-stage decision. This bound measures the distance between the optimal dual solution and the solutions that the aggregation matrix can reach. For a partition-based method, it is governed by the heterogeneity of dual solutions inside each subset in the partition. In that case we also show that the two matrices can be used together while keeping a relaxation of the original problem, so the partition reduces the number of variables at no cost on the quality of the bound. This property is specific to the partition structure and fixed recourse assumptions, and it gives insights on the advantages of adaptive partition-based methods.

This framework leads us to a new algorithm. We keep only the aggregation of the constraints, and we let the rows of the matrix have arbitrary entries and overlapping supports. A constraint can then contribute to several of the new constraints at the same time. The approximation matrix is now built by factorizing the matrix of the dual solutions collected along the iterations. Unlike adaptive partition-based methods, our algorithm does not require the fixed recourse assumption. Finally, we test our algorithms on instances with up to ten thousand scenarios to validate their performance and practical applicability.

ABSTRACT. The Unit Commitment (UC) problem is a fundamental challenge for energy producers and network operators aiming to determine an optimal generation schedule for electricity production units while satisfying technical and economic constraints. A major source of complexity arises from uncertainty, particularly in forecasting energy demand and renewable generation. To address this issue, decomposition methods have been widely studied to efficiently handle a large number of scenarios.

Recently, a new formulation of the deterministic UC problem based on interval variables has been proposed to better model unit production decisions. Building on this approach, we develop stronger cuts for stochastic UC problems. Interval variables represent the complete online intervals of the units and therefore capture more information than the classical formulation. This additional information allows us to better approximate future operating costs within decomposition algorithms.

Through computational experiments on benchmark instances, we show that the proposed cuts improve convergence and reduce solution times. These results illustrate the benefit of interval variables for solving large-scale stochastic UC problems and their potential for decomposition methods in which decisions are adapted as uncertainty is revealed.

ABSTRACT. This work addresses the numerical resolution of an aggregative stochastic optimization problem in which agents are subject to both common and individual sources of uncertainty. It is motivated by the practical challenge of optimal, coordinated management of a large number of energy dynamical systems under uncertainty.

We propose an extension of the work of \cite{ref2} to a common noise framework, as well as a generalization to problems incorporating risk measures and coupling constraints across agents. We also propose a relaxation of our original problem, recovering the same relaxation gap result as the one established by the authors.

Then, motivated by our practical problems, we restrict our attention to problems with noise having a finite number of scenarios and a scenario tree representation. Generalizing Rockafellar's expansion lemma \cite[Lemma~1]{ref1}, we obtain an equivalent reformulation of the relaxed problem. It is particularly well-suited to numerical resolution via the Douglas--Rachford Splitting (DRS) algorithm, a well-known method for minimizing the sum of two convex functions. Indeed, the first part of the objective becomes separable with respect to both the agents and scenarios, and the second one admits a numerically tractable proximal operator. In particular, at each iteration of the algorithm, independent and parallelizable subproblems are solved. Therefore, the combination of the relaxation and the resulting decomposition allows us to control the complexity of our aggregative problem of interest as the number of agents and scenarios grows.

References:

[1] R. T. Rockafellar, Progressive decoupling of linkages in optimization and variational in equalities with elicitable convexity or monotonicity.

[2] A. Seguret, C. Alasseur, J. F. Bonnans, A. De Paola, N. Oudjane, and V. Trovato, Decomposition of convex high dimensional aggregative stochastic control problems.

ABSTRACT. First-order methods are widely used in optimization and machine learning, and their behavior is often analyzed through the spectrum of worst-case convergence rates. Obtaining such guarantees is often difficult, time-consuming, and error-prone. Starting with the work of Drori and Teboulle (2014), novel techniques have been used to gain numerical insights, leading to the release of various performance estimation (PE) software.

In this talk, I will show how various computer-aided techniques can be used to study first-order optimization methods in a systematic way. From performance estimation problems with automated Lyapunov discovery to symbolic regression and computer algebra systems, novel tools completely reshape the way we approach the theory of optimization.

As a main example, I will focus on error feedback methods used with compressed communication in distributed optimization. While error feedback has been widely studied, existing theory often provides non-tight, and thus unreliable, bounds. I will present tight analyses with matching lower bounds~\cite{thomsen2025,thomsen2026} that allow a fair comparison between error feedback schemes and standard compressed gradient descent, and help explain when error feedback is useful and when it is not.

Overall, the talk aims to show how various computer-aided proofs can lead to clearer and more reliable insights into first-order optimization methods.

ABSTRACT. [session title: “Recent Advances in Systematic and Computer-Aided Methods in Optimization”]

A fixed-coefficient first-order method performing $N$ gradient evaluations can be represented as $x_{k+1}=x_k-\frac1L \sum_{i=0}^k H_{k,i}\nabla f(x_i)$, where $H \in \mathbb{R}^{N \times N}$ is a lower-triangular matrix. Recent works design optimal methods by numerically optimizing coefficients in $H$ \cite{dasguptavanparysryu2024,kamri2025numerical}. In this representation, computing an iterate may require storing all past gradients. However, many methods, including standard accelerated methods, admit a much more compact recurrence that stores only a few summaries of the past. In this work we focus on \emph{memory-$r$ methods}, where the $r$ stored vectors are collected in a state matrix $S_k\in\mathbb R^{d\times r}$ and updated linearly: \[ x_{k+1}=x_k- \tfrac1L \left(\nabla f(x_k) d_k + S_k c_k \right) \text{ with the state computed as } S_0=0, S_{k+1}= S_k A_k + \nabla f(x_k) b_k \] with $A_k\in\mathbb R^{r\times r}$, $b_k\in\mathbb R^{1\times r}$, $c_k\in\mathbb R^{r\times1}$, and $d_k\in\mathbb R$ for all $k$. We show that the minimum memory required to implement the method described by $H$ is $r=\max_k \rank H_{k:N-1,\;0:k-1} $, and we give an algorithm to recover $(A_k,b_k,c_k,d_k)$ from $H$.

The $H$-dual of a method, obtained by anti-transposing matrix $H$, was shown in some situations to transfer convergence guarantees between different performance metrics \cite{kim2023timereversed,shuwang2026unified}. We show that taking the $H$-dual of a memory-$r$ method gives another memory-$r$ method described by \[ (A_k,b_k,c_k,d_k) \xleftrightarrow{\ \text{H-dual}\ }(\widetilde{A}_k,\widetilde{b}_k,\widetilde{c}_k,\widetilde{d}_k) = (A_{N-1-k}^\top, c_{N-1-k}^\top, b_{N-1-k}^\top, d_{N-1-k}) \quad \text{for all $k$.} \]

Shu and Wang \cite{shuwang2026unified} recently considered the design of an optimized method for $L$-smooth $\mu$-strongly convex functions, targeting the terminal distance relative to the initial gradient norm. For $N=5$ and $q=\mu/L=1/11$, we show that their numerically optimized method is memory-$1$, up to rounding errors. Reconstructing its compact recurrence allows us to identify the coefficients in closed form and extend the method to every $N\ge1$ and $q\in(0,1)$. We also show that this new method satisfies $\norm{x_N-x_\star}\le\frac{1+\sqrt{q}}{2\mu}\,(1-\sqrt{q})^N\,\norm{\nabla f(x_0)}$ for all $N\ge1$.

ABSTRACT. [session title: “Recent Advances in Systematic and Computer-Aided Methods in Optimization”]

In smooth convex optimization, the gradient norm is a directly observable measure of stationarity. Accelerating a first-order method that minimizes the gradient norm is known to be more delicate than accelerating the minimization of function values. Optimal accelerated methods such as OGM-G \cite{kimfessler2021} exist for any prescribed finite horizon, but their coefficients depend explicitly on that horizon, i.e. on the number of iterations.

We ask what kind of acceleration remains possible when the stopping horizon is unknown to the method, i.e. for horizon-independent methods. Diakonikolas and Wang \cite{diakonikolaswang2022} conjectured that, for any nonadaptive horizon-independent linear-span first-order method, an $\Omega(N^{-1})$ lower bound on the squared gradient norm must hold at every horizon $N$. Recent work \cite{tsai2026} proves such a pointwise obstruction for the more restricted class of nonadaptive gradient descent with positive stepsizes, supporting this conjecture. We show, however, that the conjecture is false for the broader linear-span class: a single horizon-independent method can achieve near-$N^{-2}$ last-iterate guarantees on a density-one set of horizons. What remains true is that an $\Omega(N^{-1})$ lower bound must occur infinitely often. More precisely, if $\G_N(\A)$ denotes the bound on the squared gradient norm after $N$ iterations for a method $\A$, we prove that any method in the class satisfies $\limsup_{N\to\infty}N\G_N(\A) \ge 1/2$. This bound is sharp: the constant $1/2$ is exactly attained by the horizon-independent gradient-descent schedule of \cite{rotaruglineurpatrinos2026}.

We further show that the two extreme behaviors cannot coexist for the same method: if the bound for method $\A$ satisfies $\G_{N_j}(\A)=o(N_j^{-1})$ along some subsequence, then we also necessarily have $\limsup_N N\G_N(\A)=\infty$. Thus, horizon independence does not rule out acceleration at almost every horizon, but it prevents uniformly accelerated last iterates. In contrast, best-so-far output admits a uniform $O(N^{-2})$ guarantee.

ABSTRACT. This talk explores the potential, strategies, and challenges of high-frequency algorithmic trading in continuous intraday power markets. We examine key market characteristics and explore the design of profitable high-frequency trading strategies. Additionally, we discuss the opportunities for multi-market trading, integrating the intraday market with ancillary service markets.

Session: Optimal trading in intraday electricity markets (Teemu Pennanen)

ABSTRACT. Optimal management of battery storage in the continuous intraday power market requires accounting for future order book uncertainties, rather than only reacting to their current state as myopic policies do. We propose a simple probabilistic model of the order book before gate closure to compute dynamic trading policies. We first reconstruct limit order books from orderlevel data for the French and German markets. We then document several stylized facts of the order books. Building on these facts, we calibrate a Markov chain on the order book state before gate closure. This chain serves as the stochastic process in an optimisation model solved by Stochastic Dual Dynamic Programming (SDDP), from which we derive a dynamic trading policy. This work connects statistical analysis of order books with stochastic control for battery storage in European intraday power markets.

ABSTRACT. We study optimal management of Battery Energy Storage Systems in intraday power markets. A popular approach is the so-called ``rolling intrinsic'' strategy, which optimises a deterministic trading strategy against the available intraday price quotes. This results in deterministic optimisation problems that are particularly easy to solve, but the obtained ``intrinsic value'' misses the value of optionality of a battery, which may be significant in the face of price uncertainty. We propose stochastic optimisation models that 1) allow for dynamic trading strategies that may react to the arrival of new information on prices, etc., 2) can be solved efficiently using techniques of dynamic programming and convex optimisation, and 3) outperform the deterministic formulations, which are obtained as special cases.

ABSTRACT. Semidefinite relaxations are among the strongest tools available for solving difficult optimization problems, but they become costly as the dimension grows. Cutting-plane methods address this bottleneck by starting from a cheaper outer approximation and enforcing positive semidefiniteness progressively through valid inequalities. We place this strategy in a general framework by normalizing positive semidefinite separators with a symmetric gauge, i.e., a norm invariant under permutations and sign changes of the coordinates.

Given a symmetric matrix X̄ and a gauge φ, the resulting separation problem admits a closed-form optimal value given by −φ°(λ⁻(X̄)), where φ° is the dual gauge and λ⁻(X̄) is the negative spectral part.

This approach extends and unifies cut constructions that had previously appeared separately. The ℓ1 gauge recovers the classical eigencut, support restrictions give sparse eigencuts, while the ℓ∞ gauge yields nuclear cuts. Beyond these special cases, within the ℓp norms, each 1 < p < ∞ defines a new intermediate cut. More generally, gauges can be designed to enforce properties such as low rank or restricted support, yielding families with different tightness-cost trade-offs.

We embed the oracle in a Kelley-type algorithm, prove convergence to the semidefinite optimum, and test it on instances of box-constrained quadratic programming (BoxQP) and sparse principal component analysis (sparse PCA).

ABSTRACT. The moment-SOS (Sum-Of-Squares) hierarchy, originally introduced by Lasserre in [1], provides a powerful framework for solving nonconvex polynomial optimization problems to global optimality by constructing a sequence of convex semidefinite relaxations. When applied to nonlinear Partial Differential Equations (PDEs), this approach allows to compute rigorous bounds on a specified, possibly nonlinear, functional of their solutions [2]. Such bounds are particularly valuable in optimal control, as they can provide quantitative guarantees on the performance of a given controller relative to the global optimum. Furthermore, the numerical pseudo-moments generated by the hierarchy can be exploited to synthesize nonlinear feedback control laws [3].

However, a central challenge for nonlinear PDEs is the possible existence of a relaxation gap between the original nonlinear, nonconvex problem and its convex relaxation over measures. Such a gap may compromise the convergence guarantees of numerical schemes based on the moment-SOS hierarchy.

In this talk, we will present recent theoretical results proving the absence of a relaxation gap for scalar nonlinear parabolic PDEs of the reaction-diffusion type [4]. We will also present numerical applications of the moment-SOS hierarchy to optimal control problems governed by reaction-diffusion PDEs and the Navier–Stokes equations.

References [1] J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM Journal on Optimization, 11(3):796–817, 2001. [2] M. Korda, D. Henrion, J. B. Lasserre, Moments and convex optimization for analysis and control of nonlinear PDEs, Handbook of Numerical Analysis, 23:339–366, Elsevier, 2022. [3] C. Lebarbé, É. Flayac, M. Fournié, D. Henrion, M. Korda, Optimal control of 1D semilinear heat equations with moment-SOS relaxations, 2025 European Control Conference (ECC), 731–736, IEEE, 2025. [4] C. Lebarbé, É. Flayac, M. Fournié, D. Henrion, M. Korda, Concentration of measure-valued solutions for semilinear parabolic equations, arXiv:2605.23678, 2026.

ABSTRACT. Polynomial optimization encompasses problems for which the objective function and constraints are encoded by multivariate polynomials. In the general case, such problems may have multiple local minima and can be numerically challenging. Given a specific polynomial optimization problem (POP), the moment-SOS hierarchy introduces a sequence of semi-definite programs whose values converge to the infimum of the POP. This approach has shown promising numerical performances, and technical conditions to enforce convergence in finitely many steps of hierarchy are known. Yet, even when finite convergence holds, there still lacks a systematic understanding of the amount of steps required for the hierarchy to converge.

In this talk, we introduce a parametrized quadratic univariate POP for which the moment-SOS hierarchy always converges in finitely many iterations. Interestingly, the number of steps required to achieve convergence can be arbitrarily large, as the parameter tends to a limit value. In the second part of the talk, we also discuss some examples of converging sparse moment-SOS hierarchies without the standard running intersection property.

ABSTRACT. We introduce a method for solving Hamilton-Jacobi equations, both inviscid and viscous, by minimizing the squared residuals of monotone finite-difference discretizations on grids of varying resolution. In contrast to the uniform-grid setting of Esteve-Yagüe, Tsai, and Massucco (Finite-difference least square methods for solving Hamilton-Jacobi equations using neural networks, Journal of Computational Physics, 524:113721, 2025), our analysis and numerical framework accommodate possibly non-uniform grids. The method is designed to leverage neural networks and modern GPUs to solve these equations in higher dimensions; consequently, the setting for our analysis is the minimization of the residual functionals via gradient-based optimization. We establish a well-posedness theory for this approach: any critical point of the finite-difference loss solves the monotone scheme together with the prescribed Dirichlet boundary conditions, and the error of an approximation is controlled by its residual. We then derive the rate of convergence of the gradient flow that minimizes the residual in several settings, noting that the rate may depend on the grid resolution and the domain dimension. Building on this foundation, we propose a multi-level training algorithm that exploits the faster convergence available on the coarser grids of the discretization. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee convergence to the unique viscosity solution as the grid is refined. We illustrate the approach on eikonal equations, level-set problems, and a Hamilton-Jacobi-Isaacs equation arising from a stochastic differential game, in dimensions up to eight (Esteve-Yagüe, Bokanowski, and Tsai, Solving Hamilton-Jacobi equations by residual minimization of monotone finite-difference discretizations, preprint, 2026, https://arxiv.org/abs/2601.21764).

ABSTRACT. Least-squares residual minimization approximates a differential equation through an optimization problem. This paper examines the implications needed to pass from stationary points and minimizing sequences of the residual functional to the intended solution. Starting from classical range--adjoint geometry, we study how boundary conditions and parameterized variations determine what can be inferred about the residual at a stationary point. Smooth spurious critical points can occur even for well-posed problems; for linear equations, a compatibility condition yields an exact projection characterization. For piecewise-smooth trial functions, we identify conditions under which the mean gradient of the sampled loss differs from the continuous-loss gradient, including an explicit example with identically vanishing sampled gradient and nonzero continuous derivative. We then examine two further issues: for hyperbolic conservation laws, transonic minimizing sequences can lose regularity and force parameter escape, while an exactly vanishing Hamilton--Jacobi residual does not necessarily lead to the viscosity solution. Favorable results for a viscous model and uniformly convex Monge--Ampère critical points identify conditions that restore the corresponding implications. Together, these results distinguish the roles of stationarity, sampling, compactness, and solution admissibility in interpreting residual minimization, with particular relevance to physics-informed neural networks.

ABSTRACT. Solving high-dimensional partial differential equations (PDEs) constitutes a core task in many engineering and scientific disciplines. While classical methods such as the finite difference, finite element, or spectral element method suffer from the curse of dimensionality, physics-informed neural networks and the deep Galerkin method effectively leverage machine learning models to approximate solutions of known PDEs by minimizing residuals corresponding to the PDE operator alongside external constraints. While second-order optimizers empirically outperform their first-order counterparts in both speed and accuracy, the convergence theory for training such models is essentially confined to gradient descent. In this paper, we close this gap and develop a convergence analysis for the regularized Newton as well as Gauss-Newton method. As the number of hidden units tends to infinity, the training dynamics converge in probability, with explicit rates, to the solution of a deterministic limit equation involving the Newton neural tangent kernel associated with the PDE operator. Although the limiting kernel admits no spectral gap, we prove that, in the overparameterized limit, the PDE residual vanishes as training proceeds, and, under mild assumptions, the neural network converges to the PDE solution. We further show that, provided a sufficiently small regularization parameter, the projection of the residual onto any fixed number of leading eigenfunctions decays exponentially fast, demonstrating that this hyperparameter acts as a spectral filter: it lifts the convergence rates of high-frequency components towards those of low-frequency ones, thereby mitigating the spectral bias inherent in gradient descent. Mathematical challenges arise from the implicitly defined parameter update, involving a potentially indefinite Hessian, and from the PDE operator entering the kernel.

Part of the parallel session: "Neural Network approximations and related techniques for high-dimensional PDEs and Control Problems".

ABSTRACT. Stabilizer theory defines another novel perspective on mixers. We propose to introduce the main definitions and provide an application framework for this theory to design mixers describing potential solutions in the context of constrained function minimization. Building upon the work of [1][2], we formalize an algorithmic description of the theory to enable its practical application to optimization problems. The key advantage is to construct a superposition restricted solely to quantum states satisfying the constraints, thereby achieving an efficient modeling. We propose an iterative constraint-traversal method that iteratively generates the mixer space at each step. Numerical examples are provided to model systems with 5 constraints.

ABSTRACT. The travelling salesman problem (TSP) is one of the most studied problems in operations research. Many works focus on this problem both in classical and quantum optimization. For a N city TSP, we propose an improvement over the work of (Baldazzi et al., 2026) which came up with a 2 log_2⁡(N). Our proposal consists in a new mixer allowing us to put into superposition only valid TSP tours at the cost of an ancilla qubit. As in the original work, the resolution of the problem is through a variational quantum circuit (VQA). Examples with five and eight cities are solved and two different methods to extract the solutions are given.

ABSTRACT. This talk discusses quantum approaches to multi-objective optimization. Current literature on the topic is relatively sparse and can be loosely structured around three lines of research. The first one focuses on sampling the solutions of scalarized single-objective reformulations of the problem. From the quantum computing perspective, such approaches include both gate-based and annealing-based algorithms. The second line of work adapts variational algorithms, most notably Quantum Approximate Optimization Algorithm (QAOA), to multi-objective setting. Finally, there are approaches relying on Grover’s search subroutine. We outline the key algorithms present in the literature and discuss their advantages and limitations. To highlight the resulting computational dynamics, we consider a few numerical examples of multi-objective problem instances and discuss possible approaches to sampling locally- and globally-nondominated solutions.

ABSTRACT. Microgrid energy management is a sequential decision-making problem under uncertainty, where photovoltaic generation and electricity demand are stochastic. Model Predictive Control (MPC) is a widely adopted approach where future net load is first forecasted, typically using machine learning models trained with symmetric losses such as Mean Squared Error (MSE), and then used in a deterministic optimization problem. However, prediction accuracy is not necessarily aligned with operational performance when electricity tariffs induce asymmetric costs. Decision-Focused Learning (DFL) addresses this limitation by training predictive models directly with respect to the downstream optimization objective. In this work, we compare Predict-then-Optimize (PtO) and direct-policy learning approaches for microgrid control on the EMSx benchmark dataset. We evaluate four learning-based controllers grouped in two families. The PtO family includes conventional MPC trained with a forecasting loss and Decision-Focused Forecasting (DFF), which optimizes forecasting models end-to-end through a differentiable optimization layer. The direct-policy family includes imitation learning of anticipative controller and Learning to Optimize (L2O), which learns battery control actions directly from operating costs through a differentiable projection layer that reconstructs feasibility. We also investigate a recent soft-radial projection mechanism that provides a differentiable closed-form projection onto the battery feasibility set, enabling solver-free training. Experiments across multiple electricity tariffs show that decision-focused approaches consistently outperform prediction-focused baselines in terms of operating cost. The improvements are particularly significant under tariffs that strongly penalize grid imports. Among the best performing methods, L2O combined with soft-radial projection achieves the most competitive operating costs while requiring the lowest training and inference times. These results demonstrate the potential of decision-focused learning as an efficient and scalable alternative for practical microgrid energy management.

ABSTRACT. Decision-making under uncertainty involves making a decision at time where all of the elements are not known. If, instead, contextual data is available, it is possible, with data-driven methods, to predict the missing elements. There are thus two challenges: prediction, using machine learning, and decision, using constrained optimization. The predict-then-optimize sequential paradigm proposes to predict the missing elements, with the machine learning model aiming at minimizing the error between the predicted and the true parameters. However this method has been proven to be structurally sub-optimal [1]. Decision-focused learning proposes that the machine learning model learns to minimize a function of regret over the final decision, under strong differentiability conditions [2]. Predictive maintenance is one of these cases, with a failure prediction problem and a main tenance planning optimization problem. The maintenance planning optimization problem is modeled as a mixed-integer linear program aiming at minimizing the total maintenance cost under resources and potentiality constraints. Decision-focused learning is not applicable in this case because the uncertainty concerns the set of decision variables. The set of tasks to plan is indeed not fixed. Furthermore, there is an intermediary step between prediction and decision consisting in transforming the failure predictions into predictive tasks. This paper proposes a model integrating prediction and optimization to serve a decision-aid tool for farm operators. The integrated model is original and integrates a function of regret over the decision as the score for the hyperparameters selection of the machine learning model. Experiments are carried out based on data from industrial partners, namely wind and solar production assets operators, and show that, on the tested instances, a predictive maintenance model allows to avoid curative maintenance. Furthermore, the proposed integrated model reduces over-maintenance over a sequential model.

ABSTRACT. We study the dynamic replenishment problem of a single store facing stochastic demand under production quotas. At each period, the store chooses the replenishment quantities of multiple items subject to shared quotas, coupling decisions across the catalog. Demand is endogenous: the available assortment depends on past replenishment decisions and, in turn, influences future demand. The resulting Markov Decision Process has combinatorial state and action spaces and endogenous uncertainty, making dynamic programming intractable. On the other hand, the Sample Average Approximation (SAA) of the corresponding multi-stage stochastic program does not scale with the number of items, limiting its applicability to large catalogs.

We leverage recent advances in contextual stochastic optimization, which learn policies composed of a statistical model followed by a combinatorial optimization layer. The model predicts parameters for the objective of a mixed-integer program, leveraging its structure to learn high-quality and feasible decisions.

We therefore design a combinatorial optimization layer for dynamic replenishment and show that appropriate parameter choices recover many classical inventory policies, including base-stock and $(s,S)$ policies and their periodic and capped variants, the capacitated multi-item newsvendor, and constrained assortment optimization under a multinomial logit choice model. We train this policy by extending the primal-dual algorithm for contextual stochastic combinatorial optimization of Bouvier et al. (2026) to the multi-stage setting.

Numerical experiments on both academic instances and industrial data from a car manufacturer show that the learned policy achieves performance comparable to SAA while requiring four orders of magnitude less computing time. It also outperforms state-of-the-art inventory management baselines.

ABSTRACT. Polynomial optimization problems (POPs) arise in numerous domains such as energy systems and water networks, yet their nonconvexity poses major computational challenges. This talk focuses on exploiting sparsity structures to enable scalable global optimization of large-scale POPs. We develop formulations that leverage correlative sparsity to construct structured semidefinite and second-order cone relaxations of significantly reduced dimensionality. We further explore machine learning models that learn variable dependencies in order to guide sparsity-pattern selection. By exploiting chordal extensions of the sparsity graph and decomposing the semidefinite matrices, the resulting sparse relaxations achieve tight bounds at tractable computational cost. Computational results on optimal power flow instances demonstrate the effectiveness of the approach.

ABSTRACT. This talk presents Romain Cosson's thesis, which was defended in team Argo of Inria Paris in July 2025, under the supervision of Laurent Massoulié, and which was selected for the PGMO PhD award 2026. The thesis focuses on the design and analysis of online algorithms, which make decisions based on information revealed progressively. It is centered on problems in which several decision-makers must coordinate to perform tasks with limited information, such as exploring an unknown structure. A central problem studied in the thesis is collective tree exploration, which models the exploration of an unknown environment by multiple mobile agents (see an illustration here). The thesis also investigates other problems, such as ultrametric traversal (illustration here), metrical task systems, and the k-server problem. The thesis proposes new algorithms, improves performance guarantees, and establishes new lower bounds. It also investigates various aspects of online-algorithm complexity, such as average-case analysis, communication complexity, computational complexity, and randomness.

ABSTRACT. Many no-regret dynamics are known to eliminate strictly dominated strategies. Nevertheless, we show on a 3 x 2 example that a standard no-regret dynamics, the projected gradient algorithm, may fail to eliminate pure strategies strictly dominated by other pure strategies. This holds to an open set of games, and extends to the optimistic variant of the projected gradient algorithm. This may occur because, by contrast with other major no-regret dynamics such as the exponential weight algorithm or potential based no-regret dynamics, the projected gradient algorithm is not a perturbed version of continuous fictitious play. Our results show that, though achieving no-regret seems harder than avoiding irrational behaviour, examples of no-regret distributions putting weight on dominated strategies are not mere intellectual curiosities: standard no-regret algorithms may really lead to such distributions of play.

ABSTRACT. We consider regularized mean field optimization problems, where the regularization is typically given by an entropy, a Fisher information term, or an Lp-norm. We develop a conditional gradient algorithm tailored to this class of problems and provide a unified convergence analysis across several regimes. In particular, we establish convergence guarantees in the non-convex and convex settings, as well as under relative strong convexity and L-smoothness assumptions. In the latter regimes, we derive improved convergence rates, including sublinear and linear rates under appropriate assumptions.

We illustrate the effectiveness of the proposed method through a range of numerical experiments, including the minimization of maximum mean discrepancy (MMD), single-layer neural network training, and multi-layer neural network training. These experiments demonstrate the applicability of the conditional gradient approach to a variety of regularized mean field optimization problems.

ABSTRACT. Invited session : Linearly Controlled Systems

Let $\mathbb T^d=\mathbb R^d/2\pi\mathbb Z^d$ be the $d$-dimensional torus. Translations of the torus are generated by constant vector fields. Let $f$ be a non-constant divergence-free vector field on $\mathbb T^d$. It generates a one-parameter group of volume-preserving diffeomorphisms. In this paper, we try to understand which transformations of $\mathbb T^d$ can be reached if we perturb $f$ by constant fields (with the constant depending on time). Indeed, we study the flows generated by the control system $\dot x=f(x)+u(t)$, and we characterize the subgroup of volume-preserving diffeomorphisms that can be approximated by such flows when the control parameter varies within the class of piecewise constant functions. The most accurate results concern $d=2$ and $d=3$. Then we study the consequences of these results on the control of ensembles of points evolving under the same controlled dynamics.

ABSTRACT. We study the small and large scale geometry of sub-Finsler metrics on Lie groups. A subFinsler distance can be viewed as the value function of a minimum-time optimal control problem for a driftless system that is linear in the controls. By two theorems of Mitchell and Pansu, under suitable hypotheses, both the tangent cone and the asymptotic cone of Lie groups equipped with left-invariant sub-Finsler metrics are Carnot groups. In this talk, we present quantitative estimates on the rate of convergence to these limiting spaces, obtained in [1].

ABSTRACT. In this talk, we study Pontryagin-type optimality conditions for optimal control problems of non-local continuity equations over the Wasserstein space P_2(R^d). Whenever the underlying data is smooth, the theory for addressing these problems is well-established in the literature. However, the class of smooth maps over P_2(R^d) is rather scarce. In fact, even the squared Wasserstein distance fails to be smooth unless strong assumptions are imposed on the measures.

We overcome this difficulty by introducing a new map-based subdifferential on P_2(R^d), in the spirit of Clarke’s subdifferential in Banach spaces. Using this new object, we establish a Pontryagin maximum principle for state-constrained control problems in which both the cost and constraint functionals are only locally Lipschitz.

ABSTRACT. Remark: This talk is invited in the session “Mixed Integer Programming and Graphs for Challenging Problems”.

In an industrial context, this talk addresses a multi-period scheduling problem involving the routing of agents required to perform localized tasks, some of which require the simultaneous presence of multiple agents. The objective is to maximize the number of completed tasks, weighted by their priority.

The proposed model relies on two key assumptions: a time discretization that facilitates the handling of synchronization constraints, and a spatial aggregation of tasks by city, within which travel distances are neglected.

The aggregation of agents under the assumption of homogeneity is justified. A heuristical aggregation of tasks, leading to a formulation whose complexity is independent of the number of optional tasks, is presented.

ABSTRACT. Invited talk in the session “Mixed Integer Programming and Graphs for Challenging Problems”.

This work studies adversarial sonar placement for maritime surveillance on discrete hexagonal grids: given a fixed number of sonars, the objective is to maximize the minimum detection cost of an evading submarine that optimally plans a path to a fixed goal. In general, bilevel problems are computationally demanding because they nest a planner inside the decision model and typically lack exact baselines for comparison. However, when the inner problem is a shortest path LP on a graph, the graph incidence matrix is totally unimodular and the follower's problem admits an exact, efficiently solvable LP formulation. This setting provides a provable benchmark for evaluating surrogate approaches. Building on this benchmark, we train compact ReLU surrogates to predict shortest path costs from sensor placement configurations and exploit their piecewise-linearity to embed trained networks exactly into single-level mixed-integer linear programs. Computational experiments show that the resulting surrogate-based MILPs can be solved to obtain robust sensor placements.

We further extend the problem to a setting in which one of the sonars is initially unknown to the evader and its location is revealed when the evader approaches it, allowing the evader to adapt its remaining path. The same surrogate-based pipeline can be used, but the MILP formulation must account for the additional information-dependent decisions. Although an exact formulation can be constructed, solving the extended problem becomes computationally prohibitive at the considered instance sizes. This motivates the development of tractable surrogate-based formulations for the extended adversarial sonar placement problem.

ABSTRACT. Mixed Integer Programming and Graphs for Challenging Problems

Optimal pump scheduling in Drinking Water Distribution Networks (DWDNs) seeks the pump on/o prole over a discrete horizon minimising energy cost subject to demand, tank level and pressure requirements. The formulation is a Mixed-Integer Nonlinear Program: head losses follow the nonconvex HazenWilliams law, pump curves are nonlinear, and binary scheduling variables are coupled across time by tank dynamics. Monolithic solution is impractical beyond small benchmarks; the literature has responded either with simplied hydraulics, which weakens feasibility guarantees, or with metaheuristics, which give no optimality information. We study a decomposition-based framework that keeps the full hydraulic model and distributes the computational eort instead of simplifying the physics. Graph partitioning is applied to the network topology to produce subnetworks joined by a small set of boundary nodes; the intuition, which we test rather than assume, is that hydraulic coupling concentrates on few links, so a good cut yields balanced and weakly coupled subproblems. The Alternating Direction Method of Multipliers (ADMM) then acts as the coordination layer: each subnetwork solves its own MINLP and consistency of heads and ows at the boundary nodes is enforced through the dual updates, with binary pump variables handled inside the loop rather than relaxed away. ADMM has no convergence guarantee on a nonconvex mixed-integer problem, so the iterates are used as a heuristic: the output is post-processed into a feasible schedule and assessed against reference solutions, and what we report is a measured gap, not a certied bound. Performance is also sensitive to the partition number of parts, balance criterion, edge weighting and to the penalty parameter ρ, with no single choice dominating across networks. Characterising which topological features make a DWDN amenable to this decomposition is, in our view, the open question rather than a settled step. Experiments and discussion. The framework is evaluated on benchmark networks of in- creasing size and topological complexity, against the monolithic MINLP where tractable and against a non-partitioned baseline, reporting solution cost, constraint violation after recovery, and wall-clock time, together with a sensitivity study over the number of partitions and ρ. The talk focuses on where the decomposition helps, where it does not, and what the observed failure modes reveal about the interaction between network structure and distributed optimisation. Di- rections under investigation include partition selection informed by hydraulic rather than purely topological edge weights, and warm-starting the subproblems across a rolling horizon.

ABSTRACT. This work studies and develops projection-free algorithms for online learning with linear optimization oracles (a.k.a.~Frank--Wolfe) for handling the constraint set, and for convex loss functions. More precisely, this work (i)~shows how to exploit semidefinite programming to jointly design and analyze online Frank--Wolfe-type algorithms numerically in a variety of settings, (ii)~leverages those design techniques to propose an improved (optimized) variant of an online Frank--Wolfe algorithm along with its conceptually simple potential-based proof, and (iii)~extends this proof to its anytime version, which benefits from a similar $O(T^{3/4})$ regret rate without requiring knowledge of the time horizon $T$ in advance. We are not aware of other direct regret guarantees for an anytime version of online Frank--Wolfe without using the classical doubling trick.

Based on the semidefinite technique, we conclude with strong numerical evidence suggesting that no pure online Frank--Wolfe algorithm within our model class can have a regret guarantee better than $O(T^{3/4})$ without additional assumptions, that the current algorithms do not have optimal constants, and that multiple linear optimization rounds do not generally help to obtain better regret bounds.

ABSTRACT. Performance estimation has put first-order optimization in a remarkable position: for many algorithms and problem classes, we can now compute tight worst-case convergence rates and extract numerical certificates of optimality. The harder question is no longer whether a proof exists, but whether it can be made simple. In many cases, the available certificate is a sum-of-squares identity with coefficients so intricate that checking the algebra by hand is itself a test of patience. This talk introduces several notions of proof complexity for first-order methods and asks how they can be used to search for simpler proofs. The main idea is to exploit the proof structure of problems in first-order optimization, together with heuristics inspired by sparse optimization, to turn numerical certificates into short, interpretable arguments. We also discuss methods that automatically recover the simplest proof within a chosen proof class, along with examples. The goal is to bridge the gap between computational proof discovery and human-readable mathematics.

ABSTRACT. opt-syn (https://opt-syn.github.io/) is a MATLAB package for the automated analysis and synthesis of first-order algorithms. Optimization and fixed-point algorithms can be interpreted through the lens of dynamical systems as the interconnection of linear systems and operator nonlinearities. The convergence of well-posed optimization algorithms can then be connected to disturbance rejection/output regulation properties of these nonlinear systems. This dynamical systems formulation allows for the analysis and synthesis of optimization algorithms by solving robust control problems. Analysis bounds performance measures such as worst-case linear convergence rates, and gains for robust or stochastic error amplification. Synthesis tries to design optimization algorithm satisfying these desired performance constraints. Analysis and Synthesis may be performed for algorithms arising in dynamic environments, including cases with constant or time-varying delays, channel memory, and cross-talk. The Analysis and Synthesis problems are posed using the frameworks of Integral Quadratic Constraints, and are solved through Linear Matrix Inequality methods.

ABSTRACT. We consider probability maximization problems of the form

max_{x ∈ X} φ(x) := P(ξ ≤ h(x)),

where X ⊂ Rⁿ is a closed feasible set, h: Rⁿ → Rᵐ is a C¹,¹ mapping, and ξ is a random vector with continuously differentiable density and marginal distributions F_j, j = 1, ..., m. After establishing conditions under which φ is a C¹,¹ function, we propose MAJU (Marginal Assessment of Joint Uncertainty), an algorithm that exploits the marginal distributions F_j to construct a local approximation model at the current stability center x̂ᵏ:

M_k(x) = φ(x̂ᵏ) + Σ_{j=1}^m w_j^k [F_j(h_j(x)) − F_j(h_j(x̂ᵏ))].

Here, wᵏ ∈ [0,1]ᵐ is a vector of weights selected by the algorithm, which defines the next iterate as a stationary point of the proximal subproblem

max_{x ∈ X} M_k(x) − (L_k/2) ‖x − x̂ᵏ‖².

A serious-step/no-step acceptance rule, together with the update of L_k > 0, promotes stability of the method and ensures monotone ascent of the objective function.

We illustrate the performance of MAJU on three benchmark probability maximization problems, where it exhibits competitive performance compared with other known NLP methods applicable to C¹,¹ objectives.

ABSTRACT. We present ProbLiM (Probabilistic Linear Minimization), an open-source Julia toolbox for linear problems involving joint affine chance constraints under elliptically symmetric uncertainty. At its core is a spherical-radial oracle for estimating the joint probability and its gradient under several elliptically symmetric distributions. ProbLiM integrates this oracle with three specialized optimization methods, referred to as the Proximal, Feasible, and Penalty methods, through a unified computational interface. ProbLiM also provides interfaces to a level bundle method and IPOPT, offering alternative approaches for solving the considered problems. We describe the main components of ProbLiM, discuss practical aspects of their implementation, and illustrate its use on several linear problems with joint affine chance constraints. The numerical results highlight the computational efficiency, robustness, and flexibility of the three specialized methods across the considered problem classes.

ABSTRACT. We study linear complementarity problems (LCPs) under uncertainty, which we model using chance constraints. Since the complementarity condition of the LCP is an equality constraint, it is required to consider relaxations, which naturally leads to optimization problems in which the relaxation parameters are minimized for given probability levels. We focus on these optimization problems and first study the continuity of the related probability functions and the compactness of the feasible sets. This leads to existence results for both types of models: one with a joint chance constraint and one with separate chance constraints for both uncertainty-affected conditions of the LCP. For both, we prove the differentiability of all probability functions and derive respective gradient formulae. For the separate case, we prove convexity of the respective optimization problem and use the gradient formulae to derive necessary and sufficient optimality conditions. In a small example of a Cournot oligopoly among energy producers, we finally illustrate our theoretical findings.

ABSTRACT. Chance constraints are widely used in optimization under uncertainty. This paper aims to show the eventual convexity (EV) of chance constraints with skewed generalized hyperbolic (GH) random variables. We prove that the densities of GH distributions are $\alpha$-decreasing, and obtain exact convex reformulations for separable jointly chance constraints with GH distributions. We provide numerical results to compute the $\alpha$-decreasing threshold parameters and show the EV of the feasible set together with the computational tractability to solve the associated optimization problems.

ABSTRACT. We investigate stochastic simple bilevel optimization with smooth and possibly nonconvex upper- and lower-level objectives. Existing stochastic extensions of dynamic barrier gradient descent (DBGD) \cite{cao_complexity_2025} either obtain fast convergence under an unverifiable trajectory-dependent ``rare-visit'' assumption, or remove this assumption at a substantially higher oracle cost \cite{ahmadi_stochastic_2026}. We show that a simple denominator-only regularization of the DBGD multiplier eliminates the need for such an assumption while preserving fast convergence rates \cite{cortild_stochastic_2026a}. Specifically, our method achieves $(\varepsilon, \varepsilon)$-stationarity in $\mathcal O(\varepsilon^{-2})$ iterations using $\mathcal O(\varepsilon^{-4})$ upper-level and $\mathcal O(\varepsilon^{-7})$ lower-level stochastic gradients, which improves upon the best assumption-free complexities.

ABSTRACT. The concept of projected solution for nonself generalized Nash games and quasi-variational inequalities has been defined in \cite{ASV} and this approach has attracted attention in different context (see e.g. \cite{BL24,BC21,CGL23,CGMS24,GCCY24,CC25,CCCG25}). Nevertheless the computation of projected solutions has been only recently investigated (\cite{BCL25,BCL26,R26,BMR26}). Our aim in this talk is to present a new numerical approach and its application for the modelling of bid processes in electricity markets.

ABSTRACT. Addressing infeasibility in non-cooperative games has become an important topic, as many problems across different applications face this issue. We propose a new solution concept for generalized games with possibly infeasible individual constraints. A solution is defined as the limit of a sequence of generalized Nash equilibria induced by games with penalty terms relaxing the individual constraints. Existence is established for a broad range of games and we provide conditions allowing to characterize a ψ-penalized solution as a strategy profile maximizing every player’s utility over all her penalty minimizing strategies. A variation of Divide-the-Dollar serves as an illustrative example.

ABSTRACT. Computing the market equilibrium involves solving a set of variational inequalities, which is nontrivial to do accurately. The problem is here formulated as a large-scale Equilibrium Problem with Equilibrium Constraints (EPEC). We propose a solution based on nonlinear programming, where the EPEC is interpreted as a particular instance of a Mathematical Program with Complementarity Constraints (MPCC). We develop a package to formulate the market equilibrium directly in the JuMP modeling language and implement automatic procedures to reformulate the problem as an MPCC. In particular, we compare two specific reformulations for the resulting MPCC: (i) the Scholtes relaxation, which reformulates the MPCC as a (nearly degenerate) nonlinear program, allowing it to be solved with a nonlinear interior-point solver such as Ipopt; (ii) a reformulation of the complementarity constraints using SOS1 constraints, as is standard in the literature. We test both approaches numerically on several practical EPEC instances: (i) a tariff-setting problem given by EDF, (ii) investment under risk in incomplete markets, (iii) wholesale market equilibrium with market power, (iv) wholesale market equilibrium with day-ahead/real-time arbitrage. We assess the strengths and weaknesses of each method.

ABSTRACT. This work proposes to use the spectral properties of the graph Laplacian matrix to balance load and optimize power flows in transmission lines. We develop two models: a quadratic formulation that minimizes the squared norm of line flows, solved efficiently through Karush-Kuhn-Tucker (KKT) conditions, and a linear formulation that directly minimizes the maximum line power across the entire system, promoting more balanced line loading. Computational experiments on Pandapower-adapted benchmark networks, including a large-scale 2848-bus system, demonstrate reductions of up to 39\% in maximum line power, along with significant computational efficiency gains. The linear formulation can be naturally extended to incorporate realistic operational constraints, including line capacity limits, generation bounds, and phase angle restrictions. These results highlight the potential of spectral methods as an effective tool for enhancing load balancing, operational security, and resilience in modern power transmission networks.

ABSTRACT. Data-driven surrogate models enable faster security assessment of power grids compared to repeated steady-state power-flow calculations. Training such surrogate models poses a challenge of building representative labelled training data, because historical measurements are scarce due to privacy constraints. Also, constraint-violating operating conditions are rare in historical measurements, leaving few samples near the classification boundary; this imbalance in training data limits the ability of historical data alone to train a classifier for rare violation events. Synthetic-data generation strategies have therefore been proposed, while deep active learning can reduce the number of power-flow-labelled points by selecting informative samples.

Here, we propose a novel training data generation strategy that samples active power independently for each load from a truncated Weibull distribution. Its parameters control the frequency of higher-consumption operating points and are tuned to obtain a balanced proportion of constraint-satisfying and constraint-violating samples. Unlike iterative power-flow-based generation, the procedure is independent of the evaluation oracle and can therefore be combined with an active-learning-based labelling process.

The method is evaluated against uniform and iterative sampling on low- and medium-voltage test cases. Testing uses realistic residential load profiles. The proposed Weibull strategy achieves higher F1-scores across the reported cases with substantially fewer labelled points. In particular, surrogate models achieve high recall; this is desirable for security assessment because false alarms are preferable to missed constraint violations. Once trained, the ML surrogate evaluates operating points several orders of magnitude faster than conventional power-flow assessment.

ABSTRACT. In order to preserve frequency stability, Transmission System Operators (TSOs) must maintain generation and load balance at all time. To do this, they rely on reserves, often procured in advance, with varying activation times -- some activated automatically and others manually. Since the beginning of 2026 this obligation has been extended to include all renewable energy facilities with an installed capacity above 10 MW. As a result, the french Balancing Mechanism (Méchanisme d'Ajustement) receives a growing volume of offers, which real-time operators increasingly struggle to fully exploit. Moreover, 300+ of these balancing entities are spread across multiple sites, adding a further challenge to processing and analyzing the offers.

The core is a branch-and-check decomposition: a master MILP selects and volume-allocates generation offers to meet a signed volume target at minimum cost. Congestion management is enforced through DC load-flow checks that lazily generate linear feasibility cuts based on Power Transfer Distribution Factors (PTDFs), an adaptively-scoped N-1 watchlist, and a final AC validation of the retained operating point. Post-contingency line flows are reconstructed analytically from a single base-case sensitivity computation using Line Outage Distribution Factors (LODF), avoiding a full sensitivity solve per contingency; adaptive electrical neighborhoods, cut-edge exclusion, and low-impact pruning bound the monitored perimeter, and post-incident temporary/permanent (TATL/PATL) loading limits are enforced. Congestion and post-contingency analysis are done using PowSyBl library.

The formulation is multi-horizon and can handle several types of offers: a single dispatch decision is validated simultaneously across the current situation and forecast horizons to ensure offers' constraints, each with its own network snapshot, sensitivities and contingency set, and supports both upward and downward offers through a unified signed-volume model with typed offer structures (indivisible, prorated, stacked).

The tool has been deployed as an interactive web application to be tested by real-time operators. It returns least-cost redispatch, a per-offer / per-area breakdown ranked by marginal price, forecast available margins per horizon, an itemized computation-time decomposition, and voltage-security diagnostics. Results on a national-scale network highlight that security and precision must be traded against computation time. Instead of an exhaustive N-1 analysis over all horizons, the method makes this trade-off explicit through adaptive contingency scoping and perimeter reduction, keeping runtimes compatible with real-time operation.

ABSTRACT. Network design is classically performed offline, before the network starts operating. This is restrictive when the environment is stochastic and evolves during operation: a transport network, a communication network, or a service system may benefit from adapting its topology to requests and costs that are only progressively revealed. We introduce the problem of online design of dynamic networks: the network itself is a decision variable that evolves over time, rather than a fixed infrastructure over which online routing decisions are subsequently made.

We represent the evolving network as a time-expanded graph built over a fixed substrate graph. A stochastic environment generates requests that have to be routed on the graph being constructed. Network modifications are formulated as an impulsive control strategy: at each intervention, the controller selects which time-expanded edge to add and when it should become active. Under natural assumptions, we prove that optimal interventions can be restricted to specific discrete decision epochs. This reduces the continuous-time control problem to a Markov decision process, while retaining the temporal structure needed to route requests over the evolving network.

The resulting state-action space is nevertheless combinatorial: every decision opens many possible future construction trajectories. Moreover, some parts of the network must be committed in advance so that users or routing algorithms can plan future paths. We address these two difficulties with a rolling-horizon method based on Monte Carlo Tree Search (MCTS). Search is guided by a neural-network policy learned by imitation from a limited set of exemplary network evolutions, while a prediction model simulates future environmental dynamics. The learned policy guides MCTS toward promising network structures, while MCTS still determines the online control decisions.

We illustrate the approach on dynamic public transport. Instead of extending each vehicle trajectory independently, as in dynamic vehicle-routing methods, the controller constructs a structured network of bus lines on the fly, allowing passenger journeys that include transfers between lines. Experiments driven by real New York City trip requests show that this network-wide structure nearly doubles the service rate relative to a state-of-the-art high-capacity ride-sharing method under comparable operating conditions. Comparisons with static bus-network design also show the value of adapting the network online. Additional case studies on evolvable complex systems and the k-server problem demonstrate that the formulation is not transport-specific. The work thus connects stochastic optimal control, combinatorial search, and learning to make network topology an online operational decision.

ABSTRACT. Overparameterization is a defining feature of modern machine learning, but understanding how it affects the dynamics and implicit regularization of Stochastic Gradient Descent (SGD) remains a major challenge. In this talk, we study a tractable form of overparameterization arising from symmetries: different parameter values can represent the same function, and the corresponding transformations of the parameters can be described by an isometric group action. Such symmetries appear naturally in modern architectures, including transformers. We will first present recent work, published at COLT 2026, in which we model SGD for functions with such symmetries by a continuous-time Langevin dynamics whose noise covariance is given by the orthogonal projection onto the normal space to the group orbit. Thus, the noise acts only in directions that modify the represented function, while directions tangent to the orbit corresponding to different parameterizations of the same function are not affected. We show that, when the initial and target distributions are invariant under the group action, this apparently anisotropic Langevin dynamics is equivalent in law to standard Langevin dynamics with isotropic noise, but with an additional drift term given by the gradient of the negative logarithm of the volume of the group orbit. In other words, the geometry of the parameter space itself induces an implicit regularization: the dynamics is biased towards regions where the symmetry orbits have smaller volume. We establish this result by coupling the dynamics with an auxiliary stochastic process evolving on the group, and identify the additional drift with the mean curvature of the group orbits. Wewillthenturntoongoingworkonoverparameterizedmatrixsensing, aclassicaltoymodel for non-convex learning. Here, a more refined description of the SGD noise reveals a strong numerical bias towards low-rank solutions. Interestingly, unlike the low-rank bias observed in gradient flow, this effect persists even when the initialization is not close to zero. Finally, we will discuss a broader question: how do different sources of overparameterization interact? In particular, can the geometric overparameterization induced by symmetries interact with the more familiar notion of having more parameters than training examples, and together explain some of the implicit regularization observed in SGD?

ABSTRACT. Ample empirical evidence in deep neural network training suggests that a variety of optimizers tend to find nearly global optima. In this work, we adopt the reversed perspective that convergence to an arbitrary point is assumed rather than proven, focusing on the consequences of this assumption. From this viewpoint, in line with recent advances on the edge-of-stability phenomenon, we argue that different optimizers effectively act as eigenvalue filters determined by their hyperparameters.

Specifically, standard gradient descent inherently avoids the sharpest minima, whereas Sharpness-Aware Minimization (SAM) algorithms go even further by actively favoring wider basins. Inspired by these insights, we establish a generalized Hadamard--Perron stable manifold theorem applying to general semialgebraic $C^2$ functions, without requiring additional non-degeneracy conditions or global Lipschitz bound assumptions. We leverage this theoretical framework to propose two novel algorithms—Two-step USAM and Hessian USAM—that exhibit enhanced eigenvalue filtering and promote wider minima. Numerical experiments on feed-forward neural networks and ResNet architectures support our theoretical conclusions.

ABSTRACT. Despite recent algorithmic advances, we still lack principled ways to leverage the well-documented rescaling symmetries in ReLU neural network parameters. While two properly rescaled weights implement the same function, the training dynamics can be dramatically different. To offer a fresh perspective on exploiting this phenomenon, we build on the recent path-lifting framework, which provides a compact factorization of ReLU networks. We introduce a geometrically motivated criterion to rescale neural network parameters which minimization leads to a conditioning strategy that aligns a kernel in the path-lifting space with a chosen reference. We derive an efficient algorithm to perform this alignment. In the context of random network initialization, we analyze how the architecture and the initialization scale jointly impact the output of the proposed method. Numerical experiments illustrate its potential to speed up training.

ABSTRACT. Preconditioning, Approximation, Symmetries \\[2em] Due to properties of the model parametrization or of the loss, many machine learning problems can be reformulated as the minimization of a composition of functions, $\mathcal{L} = L\circ g$, where $g$ is said to factorize the original objective function $\mathcal{L}$. While such optimization problems are usually addressed in the domain of the original objective $\mathcal{L}$, refereed to as the parameter space, we investigate the opportunities offered by a modification of the optimization domain using a change of variable with $g$, which also translates as the intent of optimizing the function $L$ on the range of $g$ rather than $\mathcal{L}$ on the parameter space. For an arbitrary $g$, we connect this approach with differential geometry through the Riemannian gradient of $L$ on the manifold of the image of $g$, and identify the associated update of parameters as a preconditioned version of gradient descent, which, even for non-injective $g$ as in the context of a quotient manifold, is associated with a well-defined Riemannian metric on the parameter space \cite{SmoothMani,ref}. While the computational challenges associated with such approaches are known in the literature -- namely, the computation of the preconditioner via the inverse of a large matrix $S$ and its application to the usual gradient vector -- we show that the change of variable viewpoint offers a fresh perspective and opens new computational approaches. Indeed, it naturally allows direct approximation of $S^{-1}$ via a criterion whose minimization can be performed without ever computing $S^{-1}$ or any inverse, and points out new easy-to-apply preconditioners that exploit the linear invariances of the original objective function. We illustrate the consequences of the approach on optimization problems involving ReLU networks and their rescaling symmetries \cite{ref2,ref3}.

ABSTRACT. This presentation addresses robust optimization problems with risk-averse and almost-sure constraints and makes several key contributions. First, we introduce a unified optimization framework that jointly integrates almost-sure constraints, risk-averse constraints, and variance penalization of decision variables. The variance penalization mechanism, inspired by machine learning to mitigate overfitting, is designed to improve the out-of-sample robustness of stochastic optimization solutions.

Second, using operator-splitting techniques, we derive a dedicated algorithm for this framework: the Risk-Averse Variance-Penalized Progressive Hedging Algorithm (RAVP-PHA). Each iteration follows the same decomposition structure as the original Progressive Hedging Algorithm: one part corresponds to the parallelization of deterministic optimization subproblems, while the other part consists of a closed-form projection step.

Third, we instantiate this framework and its solving algorithm in the context of stochastic optimal control problems with linear dynamics, ensuring convexity, and with both pure-state and mixed constraints, providing a complete solution methodology.

Finally, the effectiveness of the proposed modeling framework and solving algorithm is demonstrated on an energy management application involving a battery system (EMS). Using real consumption and production data from a commercial facility, we show that combining risk-averse constraints with variance penalization significantly improves performance in terms of cost reduction and constraint satisfaction, outperforming standard Mode Predictive Control (MPC) strategies. We show that most of the improvement over MPC is due to variance penalization.

ABSTRACT. We study stochastic optimization problems in which a decision-maker can acquire partial information about uncertainty before making operational decisions. Selecting what information to collect is combinatorial, and each selection induces a different conditional stochastic optimization problem. This combination makes probing-enhanced stochastic programs particularly challenging, because many closely related subproblems must be solved repeatedly while exploring alternative information structures.

We develop structural and algorithmic tools that exploit the relationships among these information structures. First, we characterize probing decisions through the scenario partitions they induce and derive dominance relations that identify redundant information sources and reduce the effective search space. Second, we establish stability and solution-reuse properties under changes in the information partition, allowing decisions computed for previously solved subproblems to be inherited or certified. Building on these results, we propose an approximate information-relaxation branch-and-bound algorithm that combines dominance-based pruning with a hierarchical evaluation of aggregated subproblems. Exact optimization is used only when inherited solutions and progressively refined bounds cannot provide the required certificate.

We apply the framework to a multi-item capacitated lot-sizing problem in which demand-related signals can be selectively observed before setup and production decisions are made. Computational experiments show that dominance rules reduce the average number of explored nodes by up to 47.6%. With a moderate approximation tolerance, only 16.2% of block subproblems require exact evaluation. The proposed method achieves runtime reductions exceeding a factor of twenty while keeping gaps below 0.83%, and it produces feasible bounds for instances with up to 3,200 scenarios that the exact baseline generally cannot handle. A complementary analysis shows that the relative operational gain from optimal probing reaches 45.13%, highlighting both the computational and managerial value of selective information acquisition.

ABSTRACT. This presentation proposes to study powers of graphs as robust structures. The first part aims at tackling complexity-theoretic questions related to powers of subgraphs and the second focuses on integer programming for solving related problems while keeping an eye on power of Hamiltonian cycles as a case study.

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The $k$-th power $G^k$ of a graph $G = (V,E)$ is the graph obtained from $G$ by adding edges between pairs of vertices at distance at most $k$ in $G$. Seeking a power of a given subgraph might provide a more robust solution than searching for the subgraph itself: we first establish the robustness guarantee provided by the $k$-th power of a Hamiltonian cycle in terms of edge deletion while deleting any edge in a Hamiltonian cycle yields a non-Hamiltonian graph.

\medskip Given an existence problem $\Pi$ such as "Does the graph $G$ contain a subgraph satisfying property $P$" for some fixed property $P$, we consider its empowered variant $\Pi^R$: "Does the graph $G$ contain a subgraph $H$ such that $\sqrt[k]{H}$ exists and satisfies property $P$". While computing the power of a graph is can be done in polynomial time, checking whether $\sqrt[k]{G}$ exists is NP-hard~\cite{MOTWANI199481,rootchordal}. Therefore, for some property $P$, an encoding of $H$ might not be a polynomially checkable certificate~\cite{rootchordal} and therefore might be harder than checking $P$ itself. However, checking whether $\sqrt[k]{H}$ satisfies $P$ and whether $G$ contains $H$ is as hard as checking $P$ (when $P$ is nontrivial). We formalize this idea and show that $\Pi^R$ is no harder than $\Pi$ in the polynomial hierarchy, making it compatible with other robustness criteria. Based on the theory of SSP-reduction~\cite{grune2025completenesspolynomialhierarchynatural} we also establish $\Sigma^2_p$-completeness of the problem of finding a min-max-interval-regret-$k$-th-power of a Hamiltonian cycle.

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Finally, we present several integer linear programs for solving the problem of finding a minimum weight $k$-th power of a Hamiltonian cycle. Moreover, we propose a dedicated formulation for the special case $k=2$ and show experimentally that it outperforms the generic formulations on TSPLIB instances. However, it is unclear how to generalize it to other values of $k$. We show that its naïve generalization already fails for $k=3$.

ABSTRACT. In decision problems under incomplete information, actions --- identified with payoff vectors indexed by (finite) states of nature --- and beliefs (probability distributions) are naturally paired by bilinear duality. We exploit this duality to analyze the interpersonal comparison of the value of information, using concepts and tools from convex analysis. We identify a decision-maker (DM) who maximizes expected payoff either with a closed convex positively homogeneous function over beliefs (expected payoff), or with a closed convex lower set (random payoff set). Equipped with two suitable operations --- that we interpret economically --- we obtain two isomorphic diods. Then, we show that one DM values information more than another DM if and only if their random payoff sets are solution of an algebraic equation involving the Minkowski addition, and their star-differences.

ABSTRACT. We study the joint design of a time-of-use (ToU) tariff (the partition of the day into pricing blocks and the price level attached to each block) by a profit-maximizing electricity retailer facing flexible clients who may refuse the offer in favour of an outside option. The problem is modeled as a bilevel program in which each client, at the lower level, reshapes its consumption under pointwise bounds and a fixed daily energy total, and simultaneously chooses a participation level penalized by a quadratic regularizer. We prove the client’s best response is computable in loglinear time. We derive two equivalent single-level reformulations: a closed-form continuous one tackled via stochastic gradient ascent (using a continuous reparametrization of block start times and durations) and a mixed-integer nonlinear program solved via dedicated solvers. The stochastic ascent scheme provides Clarke-critical accumulation points almost surely, in a scalable way. On a case study from the French retail electricity market with 5000 simulated households, the gradient method matches a commercial mixed-integer solver on the pointwise problem to within 0.3% of the optimal profit while running two to eight times faster on 100 representative clients. Restricting unconstrained pointwise prices to admissible ToU schedules yields an estimated profit loss of 1.3%- 13.8% for two-period tariffs and 0.6%-6.6% for three-period tariffs on the instances considered.

ABSTRACT. This work investigates a pricing competition framework among multiple Charging Stations (CSs). Each CS acts as a strategic player aiming to maximize its profit by attracting Electric Vehicles (EVs) through personalized charging offers. EV users select a CS based on both the proposed charging price and the travel distance to the station in order to maximize their utility. CSs seek flexibility from EV users by encouraging delayed charging schedules, enabling them to shift charging to periods of lower electricity prices. Accordingly, each EV is offered multiple charging options, such as immediate or flexible charging, each associated with different prices. Therefore, EV users must decide not only which charging station to visit but also which charging option to choose. To resolve situations in which multiple charging stations provide the same utility to an EV, a ranking preference mechanism is introduced. The resulting interaction is modeled as a game-theoretic framework, and the uniqueness of the equilibrium as well as the convergence of the Best-Response Dynamics algorithm are analyzed.

ABSTRACT. Reaching greenhouse gas reduction targets and improving air quality in the transport sector requires combining several policy levers, whose joint impact is far from straightforward. Low emission zones (LEZ), which restrict access to the most polluting vehicles, and electric vehicle (EV) purchase incentives are two policies that have been extensively studied in the literature. Yet, they are traditionally evaluated separately, ignoring their simultaneous interactions and effects on the vehicle fleet. This work proposes a backcasting methodology to support the combined implementation of these two policies. Rather than forecasting the effects of a given policy, backcasting starts from a set of predefined targets (e.g. environmental, budgetary) and searches for the combination of policy levers that achieves them. In our work, we use the Île-deFrance region as a case study, where the vehicle fleet is modeled through its renewal dynamics, and the LEZ implementation schedule together with the EV purchase incentive are formulated as discrete time-dependent control variables. The resulting optimal control problem is large scale and nonlinear, which adds even more complexity. Three criteria are considered in this study: (i) an emission indicator (CO2, NOx or PM), (ii) the budget which is captured by the total cost of EV purchase incentives, and (iii) a societal indicator given by the number of vehicles whose replacement is forced by the LEZ schedule, reflecting the acceptability of LEZ restrictions. Any of the three criteria can be minimized as an objective, or set as a constraint whenever maximum thresholds can be chosen by decision-makers. The criteria set as an objective can be optimized jointly in a multi-objective setting, or combined into a single mono-objective criterion, with weights chosen by the decision-maker according to their priorities. The results presented in this study focus on the unconstrained multi-objective formulation, solved with the genetic algorithm NSGA-II, as a first step to map out the Pareto front of achievable trade-offs. Once the decision-maker has situated their priorities on this front, thresholds or weights can be fixed for a more targeted formulation. The purpose of the Île-de-France case study is to illustrate this Backcasting methodology as a decision-support tool, rather than to recommend a specific policy setting. It shows how these trade-offs can guide the combination of several transport policy instruments, under multiple requirements. While NSGA-II provides a first set of results, further work is needed to explore other resolution methods for this complex optimal control problem, and to improve computational efficiency.

ABSTRACT. The estimation of Distribution Locational Marginal Costs (DLMCs), i.e., the marginal costs for serving real and reactive power demand at a certain time and location, can serve as a means for efficient coordination of distributed energy resources in active distribution networks. DLMC computation can benefit from the computation of voltage and current sensitivities with respect to nodal real and reactive net demand, yet detailed network models are often unavailable in practice. We derive a parameterized formulation of these sensitivities that explicitly captures their dependence on net demand, based on the LinDistFlow model. Building on this formulation, we propose an optimization based approach that learns network parameters and bias terms from measurement data or high-fidelity power flow outputs. Numerical results on standard test feeders, and on an actual distribution feeder demonstrate the effectiveness of our data driven approach and its potential to support dynamic and adaptive distribution network tariffs. Additional results on an actual distribution feeder that employ an iterative coordinate descent algorithm illustrate that the coordinated charging of electric vehicles, which adapt their charging profiles to these tariffs, can achieve most of the system-wide efficiency gains in just a few iterations.

ABSTRACT. As one of the largest postal operators, La Poste is directly confronted with large-scale combinatorial optimization problems, particularly variants of the Vehicle Routing Problem (VRP). Among the best-performing heuristics for such problems is Hybrid Genetic Search (HGS), which — like many state-of-the-art combinatorial optimization methods — combines a genetic algorithm with local search procedures, following the general paradigm known as memetic algorithms. Beyond these classical approaches, quantum computing has emerged as a new perspective on combinatorial optimization. Current hardware, however, remains in the Noisy Intermediate-Scale Quantum (NISQ) era, where only a small number of physical qubits are available. This constraint motivates the use of LogQ encoding, which represents solutions using exponentially fewer qubits than direct encodings, making it compatible with near-future hardware — but at the cost of significantly restricting the freedom available in designing quantum encodings and algorithmic operations. Genetic algorithms are naturally suited to this constraint, as they operate through generic variation and selection operators rather than requiring rich circuit structures or gradient access. Combined with the strong classical track record of genetic approaches such as HGS, this makes the genetic paradigm a well-motivated candidate for building an optimization algorithm within the LogQ framework.

ABSTRACT. In this paper, we prove that the method for handling the third class of problems should not be limited to penalize the quadratic violation of the constraints of the original model but also to penalize the linear violation of these constraints. In mixed-integer and nonlinear programming, this combined method is well-known augmented Lagrangian method (ALM) introduced by Powell and Hestenes in 1969. ALM alternates updates of the primal variable by minimizing the Augmented Lagrangian function and the Lagrangian multiplier by dual gradient ascent. The augmented Lagrangian variant of the QUBO formulation has been recently considered to solve the quadratic knapsack problem (QKP) using a quantum annealer.

Further, we characterize the penalty terms of the augmented Lagrangian function of general classes of linearly constrained problems that meets the properties of a QUBO. Then, we illustrate its application to the max k-cut problem and other QKP variants that remain challenging with quantum algorithms. Handling inequality constraints through the introduction of slack variables—typically encoded as sums of binary variables—entails a substantial increase in the number of binary variables in the resulting QUBO reformulation. We therefore develop and present alternative formulations for penalizing inequality constraint violations that mitigate this growth.

ABSTRACT. We consider a finite-horizon classical Lady-in-the-Lake pursuit-evasion differential game in which an evading agent moves inside a circular region with velocity bound \(\mu\), while a boundary-constrained pursuer seeks interception. The game is formulated through a Hamilton-Jacobi-Isaacs (HJI) equation, whose repeated numerical solution can become computationally demanding when the game parameters change. We develop a Physics-Informed Neural Network (PINN) formulation that approximates the HJI value function by combining the viscous HJI residual with dynamic-programming consistency. The proposed framework is then extended to a parametric PINN in which the Lady's velocity \(\mu\) and terminal horizon \(T\) are provided as network inputs, allowing a single trained model to represent a continuous family of game configurations. The learned solutions are validated against a classical grid based Semi-Lagrangian numerical solver for fixed and previously unsampled parameter combinations. Finally, we consider an adaptive execution scenario in which the Lady's velocity changes unexpectedly during the game. Rather than retraining the network or performing a new numerical HJI solve, the updated feedback strategy is obtained by direct evaluation of the trained parametric PINN using the new velocity and remaining horizon. Numerical experiments for both one-pursuer and two-pursuer configurations demonstrate the accuracy of the learned value functions and illustrate the potential of parametric PINNs for rapid policy adjustment in changing pursuit-evasion environments.

ABSTRACT. Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here, we investigate the potential role of quantum circuits for solving PDEs. We design a parameterized quantum circuit (PQC) for implementing a polynomial based on tensor rank decomposition, reducing the quantum resource complexity from exponential to polynomial when the corresponding tensor rank is moderate. Building on this circuit, we develop a Quantum Physics-Informed Neural Network (QPINN) and a Quantum-inspired PINN, both of which guarantee the existence of an approximation of the PDE solution, and this approximation can be represented as a polynomial that incorporates tensor rank decomposition. Numerical experiments are conducted on the Hamilton--Jacobi--Bellman (HJB) PDE arising from the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk-free asset. Unlike static quantum portfolio optimization, our QPINN approximates a continuous-time value function, enabling repeated evaluation and strategy recovery from its derivatives without retraining. The results show that our quantum models achieve lower losses and approximation errors than a classical fully connected PINN while using substantially fewer trainable parameters. Our quantum models further outperform a classical PINN constructed to share a similar inductive bias, providing experimental evidence of quantum-induced improvement in the tested settings and highlighting a resource-efficient pathway toward classical and near-term quantum solvers for PDEs with exploitable solution structure.

ABSTRACT. Two intertwined challenges increasingly arise in modern optimization: the growing complexity and dimensionality of optimization problems, which place increasing demands on classical computational resources, and the prevalence of complex-valued decision variables in applications such as signal processing, communications, control, and quantum systems. Quantum computing offers alternative representations and search mechanisms for addressing such problems, while continuous-variable (CV) quantum systems provide a natural setting for continuous optimization through their infinite-dimensional Hilbert spaces.

In this work, we propose the Complex Continuous-Variable Quantum Approximate Optimization Algorithm (CCV-QAOA), a variational quantum framework for optimizing directly over complex-valued decision variables while also accommodating real-valued problems. The approach exploits the natural correspondence between the real and imaginary components of complex variables and the position--momentum quadratures of CV systems, allowing each complex variable to be represented within a single qumode and reducing the number of required qumodes. We evaluate CCV-QAOA on convex quadratic problems, constrained quadratic programs, the non-convex Styblinski--Tang benchmark, and complex quartic landscapes. We further investigate the effects of circuit depth and cutoff dimension on algorithmic performance.

ABSTRACT. We present two new methods for building aperiodic canons. We introduce an Integer Linear Programming and a SAT formulation to the generalization of the Aperiodic Tiling Rhythmic Canons, a well known problem in mathematical music.

ABSTRACT. The structure of an optimization problem determines what methods can be used to solve it. One key distinction arises immediately at problem formulation: is there a single objective function, or are there multiple? Since this choice determines whether a complete ordering exists between potential solutions, it influences all subsequent algorithm design choices. As a result, multi-objective optimization is often analyzed as its own subdomain, with its own set of practices, which can diverge from those present in the single-objective setting.

One key difference between the settings lies in how we analyze the performance of an algorithm. When dealing with multiple objectives, we need to judge not just a single point, but rather a set of tradeoff solutions. This means that concepts like a best-so-far solution require some changes to translate. In the talk, we will showcase how we tackle this in the IOHinspector framework, making use of an unbounded archive to regain monotonicity for anytime analysis.

Next, we will discuss how the design of benchmark problems, with a focus on the structure of the objective space, can lead to potential bias in the way multi-objective optimization algorithms are designed. By placing the focus only on the objective space, the characteristics of the search space risk being ignored, resulting in characteristics which can (unintentionally) be exploited by common, but biased, mutation operators.

We will finally zoom in on algorithm restarts. While these are considered natural in single-objective optimization, most multi-objective optimizers opt to exclude them. This can in part be explained by the more exploratory focus induced by the tradeoff solutions present in the population. However, through a set of experiments, we illustrate that even these methods can benefit from independent restarts when faced with sufficiently challenging multi-modal problems.

ABSTRACT. This study introduces a decision-epoch-based dynamic electric autonomous dial-a-ride problem (Dyn-EADARP), in which incoming requests are collected and processed at periodic decision epochs. A key decision is not only how to serve requests, but also when to dispatch them. At each decision epoch, the service provider decides which requests to serve and which to postpone, while jointly determining vehicle routes, schedules, and charging decisions. Unexecuted parts of existing plans can be revised as new information becomes available. To solve this problem, we develop an ML--CO policy following the combinatorial optimization augmented machine learning (COAML) framework, which combines a statistical model with a combinatorial optimization layer for decision making. The statistical model predicts prizes for available requests, and a prize-collecting E-ADARP uses these prizes to jointly determine request selection, routing, scheduling, and charging. The statistical model is trained directly to improve the decisions produced by the optimization layer. Computational experiments on 320 test instances demonstrate the efficiency of ML--CO, which solves instances with nearly 500 requests in about 6 seconds on average. It achieves 9.8%--16.8% lower objective values than benchmark policies and an average gap of 4.8% to the anticipative reference. The results provide several managerial insights. First, serving requests immediately is not always best, as selectively postponing some requests can create better ride-sharing opportunities. Second, revising existing plans preserves operational flexibility and substantially improves solution quality. Finally, more frequent decision making does not necessarily improve performance, highlighting the importance of choosing an appropriate decision frequency.