Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant

TL;DR

This paper introduces a new hierarchy of semidefinite relaxations for polynomial optimization on the nonnegative orthant, offering adjustable matrix sizes, faster computation, and improved bounds for specific applications like neural network robustness.

Contribution

It extends Pólya's Positivstellensatz to create a flexible hierarchy of relaxations with arbitrarily chosen matrix sizes, ensuring convergence and enhanced efficiency over traditional methods.

Findings

Hierarchy converges at rate O(ε^{-c}) for sets with interior.

Method outperforms Moment-SOS in speed by several hundred times.

Provides tighter bounds for neural network robustness and singular value computations.

Abstract

We consider polynomial optimization problems (POP) on a semialgebraic set contained in the nonnegative orthant (every POP on a compact set can be put in this format by a simple translation of the origin). Such a POP can be converted to an equivalent POP by squaring each variable. Using even symmetry and the concept of factor width, we propose a hierarchy of semidefinite relaxations based on the extension of P\'olya's Positivstellensatz by Dickinson-Povh. As its distinguishing and crucial feature, the maximal matrix size of each resulting semidefinite relaxation can be chosen arbitrarily and in addition, we prove that the sequence of values returned by the new hierarchy converges to the optimal value of the original POP at the rate $O(\varepsilon^{-c})$ if the semialgebraic set has nonempty interior. When applied to (i) robustness certification of multi-layer neural networks and (ii) computation of positive maximal singular values, our method based on P\'olya's Positivstellensatz provides better bounds and runs several hundred times faster than the standard Moment-SOS hierarchy.