The Hierarchy of Local Minimums in Polynomial Optimization

TL;DR

This paper introduces a hierarchy-based method for identifying local minimums in polynomial optimization problems using semidefinite relaxations, with proven finite convergence under certain conditions.

Contribution

It develops a new hierarchy of semidefinite relaxations for computing local minimums and critical values, with finite convergence guarantees and procedures for complete local minimum identification.

Findings

Finite convergence of the hierarchy under generic conditions

Procedures for computing all local minimums

Extensions to constrained problems

Abstract

This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we construct a sequence of semidefinite relaxations, based on optimality conditions. We prove that each constructed sequence has finite convergence, under some generic conditions. A procedure for computing all local minimums is given. When there are equality constraints, we have similar results for computing the hierarchy of critical values and the hierarchy of local minimums. Several extensions are discussed.