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 … Read more

Completely positive semidefinite rank

An $n\times n$ matrix $X$ is called completely positive semidefinite (cpsd) if there exist $d\times d$ Hermitian positive semidefinite {matrices} $\{P_i\}_{i=1}^n$ (for some $d\ge 1$) such that $X_{ij}= {\rm Tr}(P_iP_j),$ for all $i,j \in \{ 1, \ldots, n \}$. The cpsd-rank of a cpsd matrix is the smallest $d\ge 1$ for which such a representation … Read more


Utilizing dual descriptions of the normal cone of convex optimization problems in conic form, we characterize the vertices of semidefinite representations arising from Lovász theta body, generalizations of the elliptope, and related convex sets. Our results generalize vertex characterizations due to Laurent and Poljak from the 1990’s. Our approach also leads us to nice characterizations … Read more

Complexity of the positive semidefinite matrix completion problem with a rank constraint

We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most $k$. We show that this problem is $\NP$-hard for any fixed integer $k\ge 2$. Equivalently, for $k\ge 2$, it is $\NP$-hard to test membership in the … Read more

Computing the Grothendieck constant of some graph classes

Given a graph $G=([n],E)$ and $w\in\R^E$, consider the integer program ${\max}_{x\in \{\pm 1\}^n} \sum_{ij \in E} w_{ij}x_ix_j$ and its canonical semidefinite programming relaxation ${\max} \sum_{ij \in E} w_{ij}v_i^Tv_j$, where the maximum is taken over all unit vectors $v_i\in\R^n$. The integrality gap of this relaxation is known as the Grothendieck constant $\ka(G)$ of $G$. We present … Read more