Improved Lower Bounds for Global Polynomial Optimisation

TL;DR

This paper introduces a branch-and-bound algorithm that enhances lower bounds for global polynomial optimization using SONC/SAGE, and proposes a heuristic for finding candidate minima, achieving high success rates in test cases.

Contribution

The paper develops a new algorithm to improve lower bounds in polynomial optimization and introduces a heuristic for global minimum estimation based on SONC decomposition.

Findings

Achieves small duality gaps in most test cases

Solves about 70% of cases optimally

Algorithm is fixed-parameter tractable in the number of variables

Abstract

We present a branch-and-bound algorithm to improve the lower bounds obtained by SONC/SAGE. The running time is fixed-parameter tractable in the number of variables. Furthermore, we describe a new heuristic to obtain a candidate for the global minimum of a multivariate polynomial, based on its SONC decomposition. Applying this approach to thousands of test cases, we mostly obtain small duality gaps. In particular, we optimally solve the global minimisation problem in about 70% of the investigated cases.