Duality between polyhedral approximation of value functions and optimal quantization of measures
Abstract
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.
Cite
@article{arxiv.2509.04101,
title = {Duality between polyhedral approximation of value functions and optimal quantization of measures},
author = {Abdellah Bulaich Mehamdi and Wim van Ackooij and Luce Brotcorne and Stéphane Gaubert and Quentin Jacquet},
journal= {arXiv preprint arXiv:2509.04101},
year = {2025}
}