Strict Complementarity in MaxCut SDP

The MaxCut SDP is one of the most well-known semidefinite programs, and it has many favorable properties. One of its nicest geometric/duality properties is the fact that the vertices of its feasible region 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 $C$ is called a vertex of $C$ if the normal cone of $C$ at $x$ is full-dimensional. We study how often strict complementarity holds or fails for the MaxCut SDP when a vertex of the feasible region 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 at the boundary of such normal cone. In this regard, the MaxCut SDP displays 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 the negative semidefinite rank-one matrices in the boundary of the normal cone at any vertex, the probability that strict complementarity holds lies in $(0,1)$. We also 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.