arXiv subjects

Luce Brotcorne

Publications and source records attributed to Luce Brotcorne.

9 recordsLinked to original sources

arXiv2025

Duality between polyhedral approximation of value functions and optimal quantization of measures

Approximating a convex function by a polyhedral function that has a limited number of facets is a fundamental problem with applications in various fields, from mitigating the curse of dimensionality in optimal control to bi-level optimization. We establish a connection between this problem and the optimal quantization of a positive measure. Building on recent stability results in optimal transport, by Delalande and M\'erigot, we deduce that the polyhedral approximation of a convex function is equivalent to the quantization of the Monge-Amp\`ere measure of its Legendre-Fenchel dual. This duality motivates a simple greedy method for computing a parsimonious approximation of a polyhedral convex function, by clustering the vertices of a Newton polytope. We evaluate our algorithm on two applications: 1) A high-dimensional optimal control problem (quantum gate synthesis), leveraging McEneaney's max-plus-based curse-of-dimensionality attenuation method; 2) A bi-level optimization problem in electricity pricing. Numerical results demonstrate the efficiency of this approach.

math.OC↗

arXiv2025

Game Theory and Multi-Agent Reinforcement Learning for Zonal Ancillary Markets

We characterize zonal ancillary market coupling relying on noncooperative game theory. To that purpose, we formulate the ancillary market as a multi-leader single follower bilevel problem, that we subsequently cast as a generalized Nash game with side constraints and nonconvex feasibility sets. We determine conditions for equilibrium existence and show that the game has a generalized potential game structure. To compute market equilibrium, we rely on two exact approaches: an integrated optimization approach and Gauss-Seidel best-response, that we compare against multi-agent deep reinforcement learning. On real data from Germany and Austria, simulations indicate that multi-agent deep reinforcement learning achieves the smallest convergence rate but requires pretraining, while best-response is the slowest. On the economics side, multi-agent deep reinforcement learning results in smaller market costs compared to the exact methods, but at the cost of higher variability in the profit allocation among stakeholders. Further, stronger coupling between zones tends to reduce costs for larger zones.

cs.MA↗

arXiv2025

Learning in Conjectural Stackelberg Games

We extend the formalism of Conjectural Variations games to Stackelberg games involving multiple leaders and a single follower. To solve these nonconvex games, a common assumption is that the leaders compute their strategies having perfect knowledge of the follower's best response. However, in practice, the leaders may have little to no knowledge about the other players' reactions. To deal with this lack of knowledge, we assume that each leader can form conjectures about the other players' best responses, and update its strategy relying on these conjectures. Our contributions are twofold: (i) On the theoretical side, we introduce the concept of Conjectural Stackelberg Equilibrium -- keeping our formalism conjecture agnostic -- with Stackelberg Equilibrium being a refinement of it. (ii) On the algorithmic side, we introduce a two-stage algorithm with guarantees of convergence, which allows the leaders to first learn conjectures on a training data set, and then update their strategies. Theoretical results are illustrated numerically.

cs.GT↗

arXiv2021

A Rolling Horizon Approach for a Bilevel Stochastic Pricing Problem for Demand-Side Management

To guarantee the well-functioning of electricity distribution networks, it is crucial to constantly ensure the demand-supply balance. To do this, one can control the means of production, but also influence the demand: demand-side management becomes more and more popular as the demand keeps increasing and getting more chaotic. In this work, we propose a bilevel model involving an energy supplier and a smart grid operator (SGO): the supplier induces shifts of the load controlled by the SGO by offering time-dependent prices. We assume that the SGO has contracts with consumers and decides their consumption schedule, guaranteeing that the inconvenience induced by the load shifts will not overcome the related financial benefits. Furthermore, we assume that the SGO manages a source of renewable energy (RE), which leads us to consider a stochastic bilevel model, as the generation of RE is by nature highly unpredictable. To cope with the issue of large problem sizes, we design a rolling horizon algorithm that can be applied in a real context.

math.OC↗

arXiv2020

Group design in group testing for COVID-19 : A French case-study

Group testing is a screening strategy that involves dividing a population into several disjointed groups of subjects. In its simplest implementation, each group is tested with a single test in the first phase, while in the second phase only subjects in positive groups, if any, need to be tested again individually. In this paper, we address the problem of group testing design, which aims to determine a partition into groups of a finite population in such a way that cardinality constraints on the size of each group and a constraint on the expected total number of tests are satisfied while minimizing a linear combination of the expected number of false negative and false positive classifications. First, we show that the properties and model introduced by Aprahmian et al. can be extended to the group test design problem, which is then modeled as a constrained shortest path problem on a specific graph. We design and implement an ad hoc algorithm to solve this problem. On instances based on Santé Publique France data on Covid-19 screening tests, the results of the computational experiments are very promising.

stat.ME↗

arXiv2020

Complexity of near-optimal robust versions of multilevel optimization problems

Near-optimality robustness extends multilevel optimization with a limited deviation of a lower level from its optimal solution, anticipated by higher levels. We analyze the complexity of near-optimal robust multilevel problems, where near-optimal robustness is modelled through additional adversarial decision-makers. Near-optimal robust versions of multilevel problems are shown to remain in the same complexity class as the problem without near-optimality robustness under general conditions.

math.OC↗

arXiv2019

A Bilevel Approach to Optimal Price-Setting of Time-and-Level-of-Use Tariffs

Time-and-Level-of-Use (TLOU) is a recently proposed pricing policy for energy, extending Time-of-Use with the addition of a capacity that users can book for a given time frame, reducing their expected energy cost if they respect this self-determined capacity limit. We introduce a variant of the TLOU defined in the literature, aligned with the supplier interest to prevent unplanned over-consumption. The optimal price-setting problem of TLOU is defined as a bilevel, bi-objective problem anticipating user choices in the supplier decision. An efficient resolution scheme is developed, based on the specific discrete structure of the lower-level user problem. Computational experiments using consumption distributions estimated from historical data illustrate the effectiveness of the proposed framework.

math.OC↗

arXiv2019

Robust Bilevel Optimization for Near-Optimal Lower-Level Solutions

Bilevel optimization problems embed the optimality of a subproblem as a constraint of another optimization problem. We introduce the concept of near-optimality robustness for bilevel optimization, protecting the upper-level solution feasibility from limited deviations from the optimal solution at the lower level. General properties and necessary conditions for the existence of solutions are derived for near-optimal robust versions of general bilevel optimization problems. A duality-based solution method is defined when the lower level is convex, leveraging the methodology from the robust and bilevel literature. Numerical results assess the efficiency of exact and heuristic methods and the impact of valid inequalities on the solution time.

math.OC↗

arXiv2016

Achieving an optimal trade-off between revenue and energy peak within a smart grid environment

We consider an energy provider whose goal is to simultaneously set revenue-maximizing prices and meet a peak load constraint. In our bilevel setting, the provider acts as a leader (upper level) that takes into account a smart grid (lower level) that minimizes the sum of users' disutilities. The latter bases its decisions on the hourly prices set by the leader, as well as the schedule preferences set by the users for each task. Considering both the monopolistic and competitive situations, we illustrate numerically the validity of the approach, which achieves an 'optimal' trade-off between three objectives: revenue, user cost, and peak demand.

math.OC↗