A novel exact approach to polynomial optimization

TL;DR

This paper introduces a new convex relaxation method for polynomial optimization that improves bounds and scalability over existing SOS-based approaches, enabling solutions for larger problems with guaranteed optimality.

Contribution

The authors develop a sum of linear times convex (SLC) decomposition approach that always exists and yields tighter bounds, enhancing polynomial optimization solutions.

Findings

Outperforms state-of-the-art methods like BARON and SOS.

Solves problems with up to 40 variables and degree 3 in under an hour.

Provides guaranteed optimal solutions for larger polynomial problems.

Abstract

Polynomial optimization problems represent a wide class of optimization problems, with a large number of real-world applications. Current approaches for polynomial optimization, such as the sum of squares (SOS) method, rely on large-scale semidefinite programs, and therefore the scale of problems to which they can be applied is limited and an optimality guarantee is not always provided. Moreover, the problem can have other convex nonlinear parts, that cannot be handled by these approaches. In this paper, we propose an alternative approach for polynomial optimization. We obtain a convex relaxation of the original polynomial optimization problem, by deriving a sum of linear times convex (SLC) functions decomposition for the polynomial. We prove that such SLC decompositions always exist for arbitrary degree polynomials. Moreover, we derive the SLC decomposition that results in the tightest lower bound, thus improving significantly the quality of the obtained bounds in each node of the spatial Branch and Bound method. In the numerical experiments, we show that our approach outperforms state-of-the-art methods for polynomial optimization, such as BARON and SOS. We show that with our method, we can solve polynomial optimization problems to optimality with 40 variables and degree 3, as well as 20 variables and degree 4, in less than an hour.