标题： 值函数的多面体逼近与测度最优量化之间的对偶性 显示英文标题

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

摘要： 通过具有有限面数的多面函数来近似凸函数是一个基本问题，在多个领域有应用，从减轻最优控制中的维数灾难到双层优化。 我们将这个问题与正测度的最优量化建立了联系。 基于Delalande和Mérigot最近在最优传输中的稳定性结果，我们推导出凸函数的多面近似等价于其Fenchel对偶的Monge-Ampère测度的量化。 这种对偶性启发了一种简单的贪心方法，通过聚类Newton多面体的顶点来计算多面凸函数的简洁近似。 我们在两个应用中评估了我们的算法：1) 一个高维最优控制问题（量子门合成），利用McEneaney基于最大乘法的维数灾难缓解方法；2) 电力定价中的双层优化问题。 数值结果证明了这种方法的效率。 显示英文摘要

摘要： 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.