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)

ization problems. Nonlinear programming did not receive as much attention in this respect until the recent work by Goemans and Williamson [62]. They use semidefinite programs, which are nonlinear programs, to obtain approximation solutions with much better approximation factors. For example, the best previously known approximation algorithm for MAXCUT, which was invented twenty years ago, has approximation factor 0.5 [137]. The algorithm of Goemans and Williamson dramatically improves the approximation factor to 0.878. Inspired by the work on MAXCUT, Karger, Motwani, and Sudan [86] adapt the same technique and obtain the currently best approximation algorithm for coloring a k-colorable graph with the fewest possible number of colors. The approximation ratio is improved by a factor of \Omega\Gamma n 2=k ) over the best previously known result [29]. Later Karger and Blum give the best known approximation algorithm for color

Abstract. A semidefinite program (SDP) is an optimization problem over n × n symmetric matrices where a linear function of the entries is to be minimized subject to linear equality constraints, and the condition that the unknown matrix is positive semidefinite. Standard techniques for solvingSDP’s require O(n 3) operations per iteration. We introduce subspace algorithms that greatly reduce the cost os solving large-scale SDP’s. We apply these algorithms to SDP approximations of graph partitioningproblems. We numerically compare our new algorithm with a standard semidefinite programming algorithm and show that our subspace algorithm performs better.