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.
翻译:MaxCut SDP是最著名的半无限期方案之一,它有许多优点。它的精美几何/质量特性之一是,其可行区域的脊椎与图表的剪切完全吻合,如Laurent和Poljak1995年所证明的,1995年Laurent和Poljak所证明的。我们回顾,如果正常的C$为x美元,通常的Convex值为x美元,则称为C$的顶点为美元。我们研究,当可行的区域的一个顶端是最佳的,即SDP的松动性时,MaxCut SDP的严格互补性经常保持或失效。当目标功能处于任何垂直的正常锥体内时,人们知道严格的互补性是保持的,但我们发现,在这种正常锥体的边界边界的边界上,它一般地不能达到。在这方面, MaxCut SDP表现出最坏的直观行为,以至于螺旋的平面。我们还研究与两个目标值的平面功能之间的严格互补性,即当SDP在任何正常的平面的平面功能下,我们显示正常的平面的平整。我们把平面的平面的平面的平整。