Given a graph G = ([n], E) and w ∈ R , consider the integer program maxx∈{±1}n ∑ ij∈E wijxixj and its canonical semidefinite programming relaxation max ∑ ij∈E wijv T i vj , where the maximum is taken over all unit vectors vi ∈ R. The integrality gap of this relaxation is known as the Grothendieck constant κ(G) of G. We present a closed-form formula for the Grothendieck constant of K5-minor free...

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of a sum of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We disscuss effective methods to get a simpler representation of a “sparse” polynomial as a sum of squares of sparse pol...

Several signal recovery tasks can be relaxed into semidefinite programs with rank-one minimizers. A common technique for proving these programs succeed is to construct a dual certificate. Unfortunately, dual certificates may not exist under some formulations of semidefinite programs. In order to put problems into a form where dual certificate arguments are possible, it is important to develop c...

Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance problem, this is a bundle of spectrahedral shadows surrounding the given variety. We characterize the algebraic boundary of this region and we derive a formula...

Mobile communication is a key technology in today’s information age. Despite the ongoing improvements in equipment design, interference remains a limiting factor for the use of radio communication. The author investigates in his PhD thesis how to largely prevent interference in GSM networks by carefully assigning the available frequencies to the installed base stations. The topic is addressed f...

In this paper we extend a recent algorithm for solving the sensor network localization problem (SNL ) to include instances with facial reduction to include the intersection of more than two faces. In particular, we continue to exploit the implicit degeneracy in the semidefinite programming (SDP ) relaxation of SNL . This is a preliminary working paper, and is a work in progress.

This paper addresses inclusion of behaviors and its verification. It is shown that verifying inclusion of frequency domain behaviors defined by polynomial frequency domain equalities and inequalities amounts to proving emptiness of some basic semialgebraic sets. A semidefinite programming relaxation method for solving this problem is outlined. Some applications are given to illustrate the use o...

This paper explores generalizations of the Goemans-Williamson randomization technique. It establishes a simple equivalence of binary linear feasibility problems and max-cut problems and presents an analysis of the semidefinite max-cut relaxation for the case of a single linear equation. Numerical examples for feasible random binary problems indicate that the randomization technique is efficient...

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We disscuss effective methods to obtain a simpler representation of a “sparse” polynomial as a sum of squares of sparse p...