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

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.