A correlative sparse Lagrange multiplier expression relaxation for polynomial optimization

Abstract

In this paper, we consider polynomial optimization with correlative sparsity. For such problems, one may apply the correlative sparse sum of squares (CS-SOS) relaxations to exploit the correlative sparsity, which produces semidefinite programs of smaller sizes, compared with the classical moment-SOS relaxations. The Lagrange multiplier expression (LME) is also a useful tool to construct tight SOS relaxations for polynomial optimization problems, which leads to convergence with a lower relaxation order. However, the correlative sparsity is usually corrupted by the LME based reformulation. We propose a novel parametric LME approach which inherits the original correlative sparsity pattern, that we call correlative sparse LME (CS-LME). The reformulation using CS-LMEs can be solved using the CS-SOS relaxation to obtain sharper lower bound and convergence is guaranteed under mild conditions. Numerical examples are presented to show the efficiency of our new approach.

Safety quantification for nonlinear and time-delay systems using occupation measures

“…One direction is to try and reduce the complexity of analysis of discrete-time dynamics by finding sparse and exploitable structures other than switching. Correlative sparsity is generally incompatible with robust counterparts, but further investigation should lead to cases in which imposing that the multipliers ζ have a CSP [98]. Another avenue is to incorporate warm starts into SDP solvers so system estimates can be updated as more data gets added to D. Chapter 7…”