A bounded degree SOS hierarchy for polynomial optimization

TL;DR

This paper introduces a new semidefinite relaxation hierarchy for polynomial optimization that maintains fixed matrix sizes, ensures finite convergence for certain convex problems, and enables efficient implementation, improving upon existing methods.

Contribution

It proposes a bounded degree SOS hierarchy combining advantages of LP and SOS relaxations, with fixed matrix sizes and finite convergence properties.

Findings

Fixed matrix size in relaxations simplifies computation

Finite convergence for certain convex problems

Preliminary results show promising performance on non-convex problems

Abstract

We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem $(P):\:f^{\ast}=\min \{\,f(x):x\in K\,\}$ on a compact basic semi-algebraic set $K\subset\R^n$. This hierarchy combines some advantages of the standard LP-relaxations associated with Krivine's positivity certificate and some advantages of the standard SOS-hierarchy. In particular it has the following attractive features: (a) In contrast to the standard SOS-hierarchy, for each relaxation in the hierarchy, the size of the matrix associated with the semidefinite constraint is the same and fixed in advance by the user. (b) In contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems. Finally (c) some important techniques related to the use of point evaluations for declaring a polynomial to be zero and to the use of rank-one matrices make an efficient implementation possible. Preliminary results on a sample of non convex problems are encouraging.