Optimality conditions for homogeneous polynomial optimization on the unit sphere

TL;DR

This paper proves that for generic homogeneous polynomial optimization on the sphere, points satisfying first and second order optimality are local minimizers, ensuring finite convergence of Lasserre's hierarchy.

Contribution

It establishes that under generic conditions, optimality conditions guarantee local minimality and finite convergence of the hierarchy.

Findings

Feasible points satisfying optimality are local minimizers for generic polynomials.

Lasserre's hierarchy converges finitely for generic homogeneous polynomials.

Addresses an open issue raised by Lasserre (2021).

Abstract

In this note, we prove that for homogeneous polynomial optimization on the sphere, if the objective $f$ is generic in the input space, all feasible points satisfying the first order and second order necessary optimality conditions are local minimizers, which addresses an issue raised in the recent work by Lasserre (Optimization Letters, 2021). As a corollary, this implies that Lasserre's hierarchy has finite convergence when $f$ is generic.