Alternating direction method of multipliers for polynomial optimization

TL;DR

This paper introduces a low-complexity, decentralized algorithm based on the alternating direction method of multipliers for polynomial optimization, offering an efficient alternative to semidefinite relaxation methods.

Contribution

It develops and proves convergence of a novel ADMM-based algorithm specifically tailored for polynomial optimization problems.

Findings

The proposed algorithm is computationally efficient for medium and large problems.

Numerical tests demonstrate competitive performance against existing methods.

The method is straightforward to implement and suitable for decentralized computation.

Abstract

Multivariate polynomial optimization is a prevalent model for a number of engineering problems. From a mathematical viewpoint, polynomial optimization is challenging because it is non-convex. The Lasserre's theory, based on semidefinite relaxations, provides an effective tool to overcome this issue and to achieve the global optimum. However, this approach can be computationally complex for medium and large scale problems. For this motivation, in this work, we investigate a local minimization approach, based on the alternating direction method of multipliers, which is low-complex, straightforward to implement, and prone to decentralization. The core of the work is the development of the algorithm tailored to polynomial optimization, along with the proof of its convergence. Through a numerical example we show a practical implementation and test the effectiveness of the proposed algorithm with respect to state-of-the-art methodologies.