Sum-of-squares relaxations for polynomial min–max problems over simple sets

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-03-15 DOI:10.1007/s10107-024-02072-5

引用次数: 0

Abstract

We consider min–max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the variables belong to simple sets (e.g., a hypercube, the Euclidean hypersphere, or a ball), we derive a sum-of-squares formulation based on a primal-dual approach. In the simplest setting, we provide a convergence proof when the degree of the relaxation tends to infinity and observe empirically that it can be finitely convergent in several situations. Moreover, our formulation leads to an interesting link with feasibility certificates for polynomial inequalities based on Putinar’s Positivstellensatz.

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简单集合上多项式最小-最大问题的平方和松弛

摘要我们考虑的是多项式函数的最小-最大优化问题，即多元多项式相对于一个变量子集最大化，由此得到的最大值相对于其余变量最小化。当变量属于简单集合（如超立方体、欧几里得超球面或球）时，我们会根据初等二元方法推导出平方和公式。在最简单的情况下，我们提供了当松弛度趋于无穷大时的收敛性证明，并通过经验观察到，它在几种情况下都可以有限收敛。此外，我们的方法还与基于普提纳正定定理的多项式不等式可行性证明建立了有趣的联系。

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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