A Simplified/Improved HKM Direction for Certain Classes of Semidefinite Programming

Semidefinite Programming (SDP) provides strong bounds for many NP-hard combinatorial problems. Arguably the most popular/efficient search direction for solving SDPs using a primal-dual interior point (p-d i-p) framework is the HKM direction. This direction is a Newton direction found from the linearization of a symmetrized version of the optimality conditions. For many of the SDP relaxations of NP-hard problems, a simple primal-dual feasible starting point is available. In theory, the Newton type search directions maintain feasibility. However, in practice it is assumed that roundofferror must be taken into account and the residuals are used to recover feasibility. We introduce preprocessing for SDP to obtain a modified HKM direction. This direction attains exact primal and dual feasibility throughout the iterations while: setting the residuals to 0; reducing the arithmetic expense; maintaining the same iteration counts for well-conditioned problems; and reducing the iteration count for illconditioned problems. We apply the technique to the Max-Cut, Lovász Theta, and Quadratic Assignment problems. We include an illustration on an ill-conditioned two dimensional problem, a discussion on convergence, and a similar simplification for the Monteiro-Zhang family of search directions. This paper can serve as an introduction to both the HKM search direction and preprocessing (presolve) for SDP. ∗Department of Mathematics, University of Klagenfurt, Austria, e-mail: [email protected] †Department of Mathematics, University of Klagenfurt, Austria, e-mail: [email protected] ‡Research supported by The Natural Sciences and Engineering Research Council of Canada. E-mail [email protected] 0 URL for paper and MATLAB programs: http://orion.math.uwaterloo.ca/ ̃hwolkowi/henry/reports/ABSTRACTS.html