Occupation Measure Relaxations in Variational Problems: The Role of Convexity

IF 2.6 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Optimization Pub Date : 2024-05-07 DOI:10.1137/23m1557088

Didier Henrion, Milan Korda, Martin Kruzik, Rodolfo Rios-Zertuche

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引用次数: 0

Abstract

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1708-1731, June 2024.
Abstract. This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming reformulation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving around the convexity of the data, are simple and apply in very general settings that may be of arbitrary dimensions and may include pointwise and integral constraints, thereby considerably strengthening the existing results. Our conditions are also extended to optimal control problems. In addition, we demonstrate how these results can be applied in nonconvex settings, showing that the occupation measure relaxation is at least as strong as the convexification using the convex envelope; in doing so, we prove that a certain weakening of the occupation measure relaxation is equivalent to the convex envelope. This opens the way to application of the occupation measure relaxation in situations where the convex envelope relaxation is known to be equivalent to the original problem, which includes problems in magnetism and elasticity.

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变分问题中的占位测量松弛：凸性的作用

SIAM 优化期刊》，第 34 卷第 2 期，第 1708-1731 页，2024 年 6 月。摘要本研究探讨了变化微积分问题的占优度量松弛，它是一种无限维线性规划重构，可通过半有限优化问题的层次进行数值逼近。我们要解决的问题是这种松弛与原始问题的等价性。我们的主要结果为这种等价提供了充分条件。这些条件围绕数据的凸性展开，非常简单，而且适用于非常普遍的情况，可能是任意维度，可能包括点约束和积分约束，从而大大加强了现有结果。我们的条件还可以扩展到最优控制问题。此外，我们还证明了如何将这些结果应用于非凸环境，证明了占用度量松弛至少与使用凸包络的凸化一样强；在此过程中，我们证明了占用度量松弛的某种弱化等同于凸包络。这为在已知凸包络松弛等同于原始问题的情况下应用占测度松弛开辟了道路，其中包括磁性和弹性问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Optimization 数学-应用数学

CiteScore

5.30

自引率

9.70%

发文量

101

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.