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

TL;DR

This paper introduces a sum-of-squares relaxation method for polynomial min-max problems over simple sets, providing convergence proofs and empirical evidence of finite convergence, while linking to polynomial inequality certificates.

Contribution

It develops a primal-dual sum-of-squares approach for polynomial min-max problems over simple sets, with convergence analysis and connections to Positivstellensatz.

Findings

Convergence proof as relaxation degree tends to infinity.

Empirical finite convergence in several cases.

Connection to polynomial inequality feasibility certificates.

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.