this paper is organized as follows. First, we discuss the formulation of the semidefinite programming problem used by CSDP. We then describe the predictor corrector algorithm used by CSDP to solve the SDP. We discuss the storage requirements of the algorithm as well as its computational complexity. Finally, we present results from the solution of a number of test problems. 2 The SDP Problem We consider semidefinite programming problems of the form max tr (CX)

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. Recently, a number of primal-dual interior-point methods for semidefinite programming have been developed. To reduce the number of floating point operations, each iteration of these methods typically performs block Gaussian elimination with block pivots that are close to singular near the optimal solution. As a result, these methods often exhibit complex numerical properties in practice. We consider numerical issues related to some of these methods. Our error analysis indicates that these methods could be numerically stable if certain coefficient matrices associated with the iterations are well-conditioned, but are unstable otherwise. With this result, we explain why one particular method, the one introduced by Alizadeh, Haeberly and Overton is in general more stable than others. We also explain why the so called least squares variation, introduced for some of these methods, does not yield more numerical accuracy in general. Finally, we present results from our numerical experiments ...

this article is to provide an overview of the state of the art of

SDPLIB is a collection of semidefinite programming (SDP) test problems. The problems are drawn from a variety of applications, including truss topology design, control systems engineering, and relaxations of combinatorial optimization problems. The current version of the library contains a total of 93 SDP problems encoded in a standard format. It is hoped that SDPLIB will stimulate the development of improved software for the solution of SDP problems. 1 Introduction Semidefinite programming (SDP) is an important new area in optimization. Applications of semidefinite programming include truss topology design, control systems engineering, and relaxations of combinatorial optimization problems such as graph partitioning problems and quadratic assignment problems. [1, 3, 14, 19]. A number of software packages for solving semidefinite programming problems are available [2, 8, 9, 11, 17, 18]. Collections of test problems in various areas of optimization have been developed in recent years, ...

. We consider the denitions of nondegeneracy and strict complementarity given in [5] for semidenite programming (SDP) and their obvious extensions to mixed semidenite{quadratic programming (SDQP). We show that a solution to SDQP satises strict complementarity and primal and dual nondegeneracy if and only if the Jacobian of the Newton system determined by the optimality conditions is nonsingular at the solution. 1. Mixed Semidefinite{Quadratic Programs We begin by introducing some notation, based on [3]. Let N = (N 1 ; : : : ; N s ) and n = (n 1 ; : : : ; n q ) denote two vectors of positive integers. We dene the following spaces. Let S N S N1 S Ns denote the space of real, symmetric, block diagonal matrices with block sizes N 1 ; : : : ; N s . Let S N i + denote the semidenite cone of positive semidenite N i N i matrices and let S N + S N1 + S Ns + . For a matrix XS 2 S N we write XS 0 to denote that XS 2 S N + . Let Q n R ...

We consider mixed semidenite{quadratic{linear programs. These are linear optimization problems with three kinds of cone constraints, namely: the semidenite cone, the quadratic cone and the nonnegative orthant. We outline a primal{dual path following method to solve these problems and highlight the main features of SDPpack, a Matlab package which solves such programs. We give some examples where such mixed programs arise, and provide numerical results on benchmark problems. 1 Introduction We consider the following mixed semidenite{quadratic{linear program (SQLP): max b T y (1) s:t: F (k) 0 + P m i=1 y i F (k) i 0; k = 1; : : : ; L (2) k(c (k) A (k) ) T yk (k) (g (k) ) T y; k = 1; : : : ; M (3) (A (0) ) T y c (0) (4) where y 2 R m and F (k) i 2 S nk ; i = 0; : : : ; m; k = 1; : : : ; L A (k) 2 R mpk ; c (k) 2 R pk ; g (k) 2 R m ; (k) 2 R; k = 1; : : : ; M A (0) 2 R mp0 ; c (0) 2 R p0 : The rst set of constraints (2)...

In this paper we summarize recent results on finding tight semidefinite programming relaxations for the Max-Cut problem and hence tight upper bounds on its optimal value. Our results hold for every instance of Max-Cut and in particular we make no assumptions on the edge weights. We present two strengthenings of the well-known semidefinite programming relaxation of Max-Cut studied by Goemans and Williamson. Preliminary numerical results comparing the relaxations on several interesting instances of Max-Cut are also presented.

It is well known that many of the optimization problems which arise in applica-tions are “hard”, which usually means that they are NP-hard. Hence much research has been devoted to finding “good” relaxations for these hard problems. Usually a “good” relaxation is one which can be solved (either exactly or within a prescribed numerical tolerance) in polynomial-time. Nesterov and Nemirovskii showed that by this criterion, many convex optimization problems are good relaxations. This thesis presents new convex relaxations for two such hard problems, namely the Maximum-Cut (Max-Cut) problem and the VLSI (Very Large Scale Integration of electronic circuits) layout problem. We derive and study the properties of two new strengthened semidefinite pro-