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
In traditional multi-level graph drawing – known as Sugiyama’s framework – the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative merits, and show that drawings that are good w.r.t. verticality … Read more
The single row facility layout problem (SRFLP) is an NP-hard combinatorial optimization problem that is concerned with the arrangement of n departments of given lengths on a line so as to minimize the weighted sum of the distances between department pairs. (SRFLP) is the one-dimensional version of the facility layout problem that seeks to arrange … Read more
Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner … Read more
We present a first order approach for solving semidefinite programs. Goal of this approach is to compute a solution of the SDP up to high accuracy in spite of using only partial second order information. We propose a hybrid approach that uses an accelerated projection method to generate an approximate solution and then switches to … Read more
The recent approach of solving large scale semidefinite programs with a first order method by minimizing an augmented primal-dual function is extended to doubly nonnegative programs. Regularity of the augmented primal-dual function is established under the condition of uniqueness and strict complementarity. The application to the doubly nonnegative cone is motivated by the fact that … Read more