Semidefinite relaxations for semi-infinite polynomial programming

TL;DR

This paper presents semidefinite relaxation techniques and an exchange algorithm to efficiently solve semi-infinite polynomial programming problems, including those with noncompact index sets, demonstrated by numerical experiments.

Contribution

It introduces new SDP relaxation methods and an exchange algorithm tailored for semi-infinite polynomial programming with both compact and noncompact index sets.

Findings

The proposed algorithms are effective in practice.

Semidefinite relaxations improve solution quality.

Numerical results validate the efficiency of the methods.

Abstract

This paper studies how to solve semi-infinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.