The crossing number of a graph is the minimal number of edge crossings achievable in a drawing of the graph in the plane. The crossing numbers of complete and complete bipartite graphs are long standing open questions. In a 2-page drawing of a graph, all vertices are drawn on a circle, and no edge may … Read more
We propose an algorithm that computes the epsilon-correlated equilibria with global-optimal (i.e., maximum) expected social welfare for single stage polynomial games. We first derive an infinite-dimensional formulation of epsilon-correlated equilibria using Kantorovich polynomials and re-express it as a polynomial positivity constraint. In addition, we exploit polynomial sparsity to achieve a leaner problem formulation involving Sum-Of-Squares … Read more
A closed convex cone K is called nice, if the set K^* + F^\perp is closed for all F faces of K, where K^* is the dual cone of K, and F^\perp is the orthogonal complement of the linear span of F. The niceness property plays a role in the facial reduction algorithm of Borwein … Read more
The accelerated proximal gradient (APG) method, first proposed by Nesterov, and later refined by Beck and Teboulle, and studied in a unifying manner by Tseng has proven to be highly efficient in solving some classes of large scale structured convex optimization (possibly nonsmooth) problems, including nuclear norm minimization problems in matrix completion and $l_1$ minimization … Read more
Farkas’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality … Read more
We derive a new semidefinite programming relaxation for the general graph partition problem (GPP). Our relaxation is based on matrix lifting with matrix variable having order equal to the number of vertices of the graph. We show that this relaxation is equivalent to the Frieze-Jerrum relaxation [A. Frieze and M. Jerrum. Improved approximation algorithms for … Read more
The rank function $\rank(\cdot)$ is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of $\rank(\cdot)$, and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the … Read more
Examples of weakly infeasible semidefinite programs are useful to test whether semidefinite solvers can detect infeasibility. However, finding non trivial such examples is notoriously difficult. This note shows how to use Lasserre’s semidefinite programming relaxations for polynomial optimization in order to generate examples of weakly infeasible semidefinite programs. Such examples could be used to test … Read more
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
We present a semide nite programming (SDP) approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This Multi-Level Vertical Ordering (MLVO) problem is a quadratic ordering problem and conceptually related to the well-studied problem of Multi-Level Crossing Minimization (MLCM). In contrast … Read more