A Certificate of Unboundedness for Polynomial Optimization Problems

TL;DR

This paper introduces a pre-processing algorithm that efficiently determines whether a polynomial optimization problem is unbounded from below, aiding in understanding the problem's asymptotic behavior before solving.

Contribution

The authors present a novel, simple pre-processing method to identify unbounded polynomial optimization problems, improving prior approaches that rely on compactness assumptions.

Findings

The algorithm can quickly detect unboundedness in polynomial problems.

It provides insights into the problem's asymptotic geometry.

It potentially reduces computational effort in global optimization.

Abstract

Global polynomial optimization methods typically rely on compactness of the feasible region in order to find solutions. These methods can incur considerable computational expense and most commercially available solvers do not verify the existence of a solution prior to undergoing global search. In this manuscript we propose a simple pre-processing algorithm to determine if an arbitrary polynomial optimization problem is unbounded from below thereby providing information about the problem's asymptotic geometry prior to solving the problem if a solution can be found.