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.

中文翻译：

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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

我们考虑多项式函数的最小-最大优化问题，其中多元多项式相对于变量子集最大化，而所得的最大值相对于剩余变量最小化。当变量属于简单集合（例如，超立方体、欧几里得超球面或球）时，我们基于原对偶方法导出平方和公式。在最简单的设置中，我们提供了当松弛程度趋于无穷大时的收敛证明，并根据经验观察到它在几种情况下可以有限收敛。此外，我们的公式与基于 Putinar 的 Positivstellensatz 的多项式不等式的可行性证明建立了有趣的联系。