In this paper we study interior point (IP) methods for solving optimal design problems. In particular, we propose a primal IP method for solving the problems with general convex optimality criteria and establish its global convergence. In addition, we reformulate the problems with A-, D- and E-criterion into linear or log-determinant semidefinite programs (SDPs) and … Read more
Recently, researchers have been interested in studying the semidefinite programming (SDP) relaxation model, where the matrix is both positive semidefinite and entry-wise nonnegative, for quadratically constrained quadratic programming (QCQP). Comparing to the basic SDP relaxation, this doubly-positive SDP model possesses additional O(n2) constraints, which makes the SDP solution complexity substantially higher than that for the … Read more
In this article we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and the dual theory of moments. A crucial ingredient is a semidefinite … Read more
The goal of this paper is an easy and self-contained presentation of optimality conditions for nonlinear semidefinite programs concentrating on the differences between nonlinear semidefinite programs and nonlinear programs. Citation Technical Report, Department of Mathematics, Universit\”at D\”usseldorf. Article Download View Elementary optimality conditions for nonlinear SDPs
We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let F denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on F is dominated by an alternative … Read more
We derive a new semidefinite programming bound for the maximum k-section problem. For k=2 (i.e. for maximum bisection), the new bound is least as strong as the well-known bound by Frieze and Jerrum [A. Frieze and M. Jerrum. Improved Approximation Algorithms for MAX k-CUT and MAX BISECTION. Algorithmica, 18(1): 67-81, 1997]. For k > 2 … Read more
In this chapter we study a class of semidefinite programming relaxations of combinatorial problems. These relaxations are derived using the theory of coherent configurations in algebraic combinatorics. Citation Draft version of a chapter for “Handbook on SDP II” (M. Anjos and J. Lasserre, eds.), Springer. Article Download View Relaxations of combinatorial problems via association schemes
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
Abstract. NCSOStools is a Matlab toolbox for – symbolic computation with polynomials in noncommuting variables; – constructing and solving sum of hermitian squares (with commutators) programs for polynomials in noncommuting variables. It can be used in combination with semidefi nite programming software, such as SeDuMi, SDPA or SDPT3 to solve these constructed programs. This paper provides … 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