  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.
