An Efficient Framework for Global Non-Convex Polynomial Optimization over the Hypercube

TL;DR

This paper introduces a new framework for solving high-dimensional non-convex polynomial optimization problems over the hypercube, transforming them into convex reformulations with no spurious local minima, enabling efficient global solutions.

Contribution

It provides a novel reformulation approach and an algorithm that efficiently solves large-scale non-convex polynomial problems with guaranteed global optimality.

Findings

Reformulation yields convex problems with no spurious local minima.

Algorithm achieves polynomial scaling in dimension and degree.

Numerical experiments demonstrate tractability of previously intractable problems.

Abstract

We present a novel efficient theoretical and numerical framework for solving global non-convex polynomial optimization problems. We analytically demonstrate that such problems can be efficiently reformulated using a non-linear objective over a convex set; further, these reformulated problems possess no spurious local minima (i.e., every local minimum is a global minimum). We introduce an algorithm for solving these resulting problems using the augmented Lagrangian and the method of Burer and Monteiro. We show through numerical experiments that polynomial scaling in dimension and degree is achievable for computing the optimal value and location of previously intractable global polynomial optimization problems in high dimension.