On the Practical Implementation of a Sequential Quadratic Programming Algorithm for Nonconvex Sum-of-squares Problems

TL;DR

This paper introduces a filter line search algorithm for nonconvex sum-of-squares problems, improving computational efficiency by reducing iterations and time compared to existing methods.

Contribution

It presents a practical implementation of a sequential quadratic programming algorithm tailored for nonconvex SOS problems, addressing a key computational gap.

Findings

Algorithm reduces the number of iterations needed.

Significantly decreases computation time.

Effective on numerical benchmarks.

Abstract

Sum-of-squares (SOS) optimization provides a computationally tractable framework for certifying polynomial nonnegativity. If the considered problem is convex, the SOS problem can be transcribed into and solved by semi-definite programs. However, in case of nonconvex problems iterative procedures are needed. Yet tractable and efficient solution methods are still lacking, limiting their application, for instance, in control engineering. To address this gap, we propose a filter line search algorithm that solves a sequence of quadratic subproblems. Numerical benchmarks demonstrate that the algorithm can significantly reduce the number of iterations, resulting in a substantial decrease in computation time compared to established methods for nonconvex SOS programs