arXiv subjects

Wim van Ackooij

Publications and source records attributed to Wim van Ackooij.

9 recordsLinked to original sources

arXiv2026

Joint Design of Time-of-Use Schedules and Price Levels: a Bilevel Model with Client Participation and Load Flexibility

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.

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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 Mérigot, we deduce that the polyhedral approximation of a convex function is equivalent to the quantization of the Monge-Ampère 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.

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arXiv2024

Ergodic control of a heterogeneous population and application to electricity pricing

We consider a control problem for a heterogeneous population composed of agents able to switch at any time between different options. The controller aims to maximize an average gain per time unit, supposing that the population is of infinite size. This leads to an ergodic control problem for a "mean-field" Markov Decision Process in which the state space is a product of simplices, and the population evolves according to controlled linear dynamics. By exploiting contraction properties of the dynamics in Hilbert's projective metric, we prove that the infinite-dimensional ergodic eigenproblem admits a solution and show that the latter is in general non unique. This allows us to obtain optimal strategies, and to quantify the gap between steady-state strategies and optimal ones. In particular, we prove in the one-dimensional case that there exist cyclic policies -- alternating between discount and profit taking stages -- which secure a greater gain than constant-price policies. On numerical aspects, we develop a policy iteration algorithm with "on-the-fly" generated transitions, specifically adapted to decomposable models, leading to substantial memory savings. We finally apply our results on realistic instances coming from an electricity pricing problem encountered in the retail markets, and numerically observe the emergence of cyclic promotions for sufficient inertia in the customer behavior.

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arXiv2023

Probability functions generated by set-valued mappings: a study of first order information

Probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. Probability functions are built up from a random vector belonging to some parameter-dependent subset of the range of a given random vector. In this paper, we investigate first order information of probability functions specified through a convex-valued set-valued application. We provide conditions under which the resulting probability function is indeed locally Lipschitzian as well as subgradient formulae. The resulting formulae are made concrete in a classic optimization setting and put to work in an illustrative example coming from an energy application.

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arXiv2023

Quadratic Regularization of Bilevel Pricing Problems and Application to Electricity Retail Markets

We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case.

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arXiv2023

A Quantization Procedure for Nonlinear Pricing with an Application to Electricity Markets

We consider a revenue maximization model, in which a company aims at designing a menu of contracts, given a population of customers. A standard approach consists in constructing an incentive-compatible continuum of contracts, i.e., a menu composed of an infinite number of contracts, where each contract is especially adapted to an infinitesimal customer, taking his type into account. Nonetheless, in many applications, the company is constrained to offering a limited number of contracts. We show that this question reduces to an optimal quantization problem, similar to the pruning problem that appeared in the max-plus based numerical methods in optimal control. We develop a new quantization algorithm, which, given an initial menu of contracts, iteratively prunes the less important contracts, to construct an implementable menu of the desired cardinality, while minimizing the revenue loss. We apply this algorithm to solve a pricing problem with price-elastic demand, originating from the electricity retail market. Numerical results show an improved performance by comparison with earlier pruning algorithms.

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arXiv2023

Inner Moreau envelope of nonsmooth conic chance constrained optimization problems

Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth. This motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function which covers most of the general classes of probabilistic constraints: $$φ(x)=\mathbb{P}(Φ(x,ξ)\in -\mathcal{K}),$$ where $\mathcal{K}$ is a convex cone of a Banach space. The conic inclusion $Φ(x,ξ) \in - \mathcal{K}$ represents an abstract system of inequalities, and $ξ$ is a random vector. We propose a regularization by applying the Moreau envelope to the scalarization of the function $Φ$. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can thus entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite and probust chance constrained optimization problems.

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arXiv2021

Multi-stage Stochastic Alternating Current Optimal Power Flow with Storage: Bounding the Relaxation Gap

We propose a generic multistage stochastic model for the Alternating Current Optimal Power Flow (AC OPF) problem for radial distribution networks, to account for the random electricity production of renewable energy sources and dynamic constraints of storage systems. We consider single-phase radial networks. Radial three-phase balanced networks (medium-voltage distribution networks typically have this structure) reduce to the former case. This induces a large scale optimization problem, which, given the non-convex nature of the AC OPF, is generally challenging to solve to global optimality. We derive a priori conditions guaranteeing a vanishing relaxation gap for the multi-stage AC OPF problem, which can thus be solved using convex optimization algorithms. We also give an a posteriori upper bound on the relaxation gap. In particular, we show that a null or low relaxation gap may be expected for applications with light reverse power flows or if sufficient storage capacities with low cost are available. Then, we discuss the validity of our results when incorporating voltage regulation devices. Finally, we illustrate our results on problems of planning of a realistic distribution feeder with distributed solar production and storage systems. Scenario trees for solar production are constructed from a stochastic model, by a quantile-based algorithm.

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arXiv2021

On the Convexity of Level-sets of Probability Functions

In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. Even if the original set of inequalities is convex, this favourable property is not immediately transferred to the probabilistically constrained feasible set and may in particular depend on the chosen safety level. In this paper, we provide results guaranteeing the convexity of feasible sets to probabilistic constraints when the safety level is greater than a computable threshold. Our results extend all the existing ones and also cover the case where decision vectors belong to Banach spaces. The key idea in our approach is to reveal the level of underlying convexity in the nominal problem data (e.g., concavity of the probability function) by auxiliary transforming functions. We provide several examples illustrating our theoretical developments.

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