On Polynomial Optimization over Non-compact Semi-algebraic Sets

TL;DR

This paper extends the hierarchy of semidefinite programming relaxations to polynomial optimization problems over non-compact semi-algebraic sets, providing conditions for finite convergence and numerical verification.

Contribution

It introduces a method to approximate solutions to polynomial optimization over non-compact sets using SDP hierarchies with verifiable Archimedean conditions.

Findings

Hierarchy of SDP relaxations converges finitely in generic cases.

Archimedean condition can be checked numerically via SDP.

Method extends standard hierarchy to non-compact semi-algebraic sets.

Abstract

We consider the class of polynomial optimization problems $\inf \{f(x):x\in K\}$ for which the quadratic module generated by the polynomials that define $K$ and the polynomial $c-f$ (for some scalar $c$) is Archimedean. For such problems, the optimal value can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically. Moreover, the Archimedean condition (as well as a sufficient coercivity condition) can also be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions the now standard hierarchy of SDP-relaxations extends to the non-compact case via a suitable modification.