On the exactness for polynomial optimization strengthened with Fritz John conditions

TL;DR

This paper extends polynomial non-negativity representations on semi-algebraic sets using Fritz John conditions, enabling exact computation of polynomial optimization problems through finitely converging semidefinite programs.

Contribution

It introduces new polynomial representations under general conditions, leading to semidefinite programs that finitely converge to the exact optimal value of polynomial optimization problems.

Findings

Semidefinite programs finitely converge to the optimal value.

Exact minimal value computation for polynomials over convex semi-algebraic sets.

Generalization of polynomial non-negativity representations.

Abstract

We utilize the same technique as in [arXiv:2205.04254 (2022)] to provide some representations of polynomials non-negative on a basic semi-algebraic set, defined by polynomial inequalities, under more general conditions. Based on each representation, we obtain semidefinite programs which return a sequence of values that finitely converges to the optimal value of a given polynomial optimization problem under generic assumption. Consequently, we can compute exactly the minimal value of any polynomial over a basic convex semi-algebraic set which is defined by the inequalities of concave polynomials.