Alternating direction method of multipliers for polynomial optimization
V. Cerone, S. M. Fosson, S. Pirrera, D. Regruto
TL;DR
Non-convex polynomial optimization is challenging to solve globally, with SDP-based hierarchies offering global optima at high computational cost. The paper introduces ADMM4POP, an ADMM-based local minimization method that reformulates a POP into a quadratic form with a non-convex algebraic constraint set and decouples updates into a convex x-update and a z-update via per-constraint projections onto the set defined by x_i x_j = x_k. A convergence result is established for a relaxed version of the problem, and two numerical experiments show competitive performance and significantly reduced runtime compared to interior-point and SDP approaches, while maintaining accuracy. The work suggests a scalable, decentralized framework for POPs with potential for parallelization on large-scale problems and lays groundwork for extensions to constrained settings and broader POP classes.
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.