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

• 成果类型: Article
• 署名作者: Bach, Francis
• 署名单位: Inria; Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-024-02072-5
• 发表日期: 2025
• 页码: 475-501
• 关键词: semidefinite programming relaxations; optimization

摘要

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.