Although the lift-and-project operators of Lovász and Schrijver have been the subject of intense study, their M(K , K ) operator has received little attention. We consider an application of this operator to the stable set problem. We begin with an initial linear programming (LP) relaxation consisting of clique and non-negativity inequalities, and then apply the operator to obtain a stronger ext...

We propose the moment cone relaxation for a class of polynomial optimization problems (POPs) to extend the results on the completely positive cone programming relaxation for the quadratic optimization (QOP) model by Arima, Kim and Kojima. The moment cone relaxation is constructed to take advantage of sparsity of the POPs, so that efficient numerical methods can be developed in the future. We es...

This problem is NP-hard, so there is no known algorithm to determine the solution in polynomial time. Instead we seek to compute an approximate solution in polynomial time. One strategy for accomplishing this, first pioneered by Goemans & Williamson in Ref. [3], is to employ a semidefinite relaxation. This technique consists of three steps: first construct a related problem that is a semidefini...

We consider 3-partitioning the vertices of a graph into sets S1, S2 and S3 of specified cardinalities, such that the total weight of all edges joining S1 and S2 is minimized. This problem is closely related to several NP-hard problems like determining the bandwidth or finding a vertex separator in a graph. We show that this problem can be formulated as a linear program over the cone of complete...

This paper presents a semidefinite relaxation technique for computing a minimal bounding ellipsoid that contains the set of static responses of an uncertain truss. We assume that both member stiffnesses and external forces are uncertain but bounded. By using a combination of the quadratic embedding technique of the uncertainty and the Sprocedure, we formulate a semidefinite programming (SDP) pr...

In this lecture, we provide another class of relaxations, called Semidefinite Programming Relaxation. These serve as relaxations for several NP-hard problems, in particular, for problems that can be expressed as strict quadratic programs. The relaxed problems, together with techniques like randomized rounding, give good approximation algorithms to hard combinatorial problems. We will illustrate...

Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: A difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constrained is dropped. For instance, the integrality gap ...

The first instance, rank one matrix completion, was known to be solved by non-linear propagation algorithms without stability guarantees. The thesis closes the line of work on this problem by introducing a stable algorithm based on two levels of semidefinite programming relaxation. For this algorithm, recovery of the unknown matrix is first certified in the absence of noise, at the information ...