Strict Complementarity in Semidefinite Optimization with Elliptopes Including the MaxCut SDP

Abstract

The MaxCut approximation algorithm by Goemans and Williamson is one of the most celebrated results in semidefinite optimization, and the corresponding MaxCut semidefinite optimization problem (SDP) has many favourable properties. The feasible regions of this class of SDPs are known as elliptopes, and they have been studied extensively. One of their nicest geometric/duality properties is the fact that their vertices correspond exactly to the cuts of a graph, as proved by Laurent and Poljak in 1995. Recall that a boundary point $x$ of a convex set $\mathscr{C}$ is called a vertex of $\mathscr{C}$ if the normal cone of $\mathscr{C}$ at $x$ is full-dimensional. Semidefinite programs over elliptopes were also exploited by Goemans and Williamson and by Nesterov to develop approximation algorithms for the Maximum-2-Satisfiability problem and for nonconvex quadratic optimization problems, respectively. We study how often strict complementarity holds or fails for SDPs over elliptopes when a vertex is optimal, i.e., when the SDP relaxation is tight. While strict complementarity is known to hold when the objective function is in the interior of the normal cone at any vertex, we prove that it fails generically (in a context of Hausdorff measure and Hausdorff dimension) at the boundary of such normal cones. In this regard, SDPs over elliptopes display the nastiest behavior possible for a convex optimization problem. We also study strict complementarity with respect to two classes of objective functions. We show that, when the objective functions are sampled uniformly from a class of negative semidefinite rank-one matrices in the boundary of the normal cone at any vertex, the probability that strict complementarity holds lies in $(0,1)$. To complete our study with a spectral-graph-theory-based viewpoint of the data for the MaxCut SDP, we extend a construction due to Laurent and Poljak of weighted Laplacian matrices for which strict complementarity fails. Their construction works for complete graphs, and we extend it to cosums of graphs under some mild conditions.