Strengthening SONC Relaxations with Constraints Derived from Variable Bounds

TL;DR

This paper enhances SONC relaxations for polynomial optimization by incorporating variable bounds, improving their applicability in branch-and-bound algorithms for constrained problems.

Contribution

It introduces a novel extension of SONC relaxations that leverages variable bounds, facilitating their integration into branch-and-bound methods.

Findings

Extension improves applicability to constrained problems

Variable bounds significantly enhance relaxation strength

Enables integration of SONC into branch-and-bound algorithms

Abstract

Certificates of polynomial nonnegativity can be used to obtain tight dual bounds for polynomial optimization problems. We consider Sums of Nonnegative Circuit (SONC) polynomials certificates, which are well suited for sparse problems since the computational cost depends only on the number of terms in the polynomials and does not depend on the degrees of the polynomials. This work is a first step to integrating SONC-based relaxations of polynomial problems into a branch-and-bound algorithm. To this end, the SONC relaxation for constrained optimization problems is extended in order to better utilize variable bounds, since this property is key for the success of a relaxation in the context of branch-and-bound. Computational experiments show that the proposed extension is crucial for making the SONC relaxations applicable to most constrained polynomial optimization problems and for integrating the two approaches.