Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant
This work addresses computational efficiency and accuracy in polynomial optimization, which is incremental but offers practical improvements for tasks such as verifying neural network robustness.
The paper tackles polynomial optimization problems on nonnegative orthant sets by proposing a hierarchy of semidefinite relaxations using even symmetry and factor width, achieving convergence at rate O(ε^{-c}) and showing in applications like neural network robustness certification that it provides better bounds and runs hundreds of times faster than standard methods.
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 Pólya'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 Pólya's Positivstellensatz provides better bounds and runs several hundred times faster than the standard Moment-SOS hierarchy.