A sublevel moment-SOS hierarchy for polynomial optimization

We introduce a sublevel Moment-SOS hierarchy where each SDP relaxation can be viewed as an intermediate (or interpolation) between the d-th and \((d+1)\)-th order relaxations of (dense or sparse version). With flexible choice determining size (level) number (depth) subsets in relaxation, one is able to obtain different improvements compared based on machine memory capacity. In particular, we provide numerical experiments for \(d=1\) various types problems both combinatorial optimization (Max-Cut, Mixed Integer Programming) deep learning (robustness certification, Lipschitz constant neural networks), standard Lasserre’s its variant) computationally intractable. our results, lower bounds from improve bound Shor’s (first relaxation) are significantly closer optimal value best-known lower/upper bounds.