In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth … Read more
In this paper we describe an algorithm to approximately solve a class of semidefinite programs called covering semidefinite programs. This class includes many semidefinite programs that arise in the context of developing algorithms for important optimization problems such as sparsest cut, wireless multicasting, and pattern classification. We give algorithms for covering SDPs whose dependence on … Read more
Let A(n, d) be the maximum number of 0, 1 words of length n, any two having Hamming distance at least d. We prove A(20, 8) = 256, which implies that the quadruply shortened Golay code is optimal. Moreover, we show A(18, 6) ≤ 673, A(19, 6) ≤ 1237, A(20, 6) ≤ 2279, A(23, 6) … Read more
Let P be a pointed, polyhedral cone in R_n. In this paper, we study the cone C = cone{xx^T: x \in P} of quadratic forms. Understanding the structure of C is important for globally solving NP-hard quadratic programs over P. We establish key characteristics of C and construct a separation algorithm for C provided one … Read more
In this paper, we study polynomial-time interior-point algorithms in view of information geometry. We introduce an information geometric structure for a conic linear program based on a self-concordant barrier function. Riemannian metric is defined with the Hessian of the barrier function. We introduce two connections $\nabla$ and $\nabla^*$ which roughly corresponds to the primal and … Read more
In this paper, we analyze two popular semidefinite programming \SDPb relaxations for quadratically constrained quadratic programs \QCQPb with matrix variables. These are based on \emph{vector-lifting} and on \emph{matrix lifting} and are of different size and expense. We prove, under mild assumptions, that these two relaxations provide equivalent bounds. Thus, our results provide a theoretical guideline … Read more
The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in $L_1$, $L_\infty$ and $L_2$ norm. Extensive numerical experiments … Read more
The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace f(A) can attain for a tuple of matrices A? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not … Read more
We consider the problem of minimizing a polynomial on the hypercube [0,1]^n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmuedgen (1991). The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates … Read more
The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then … Read more