Simple Efficient Solutions for Semidefinite Programming

This paper provides a simple approach for solving a semidefinite program, SDP. As is common with many other approaches, we apply a primal-dual method that uses the perturbed optimality equations for SDP, Fμ(X, y, Z) = 0, where X, Z are n × n symmetric matrices and y ∈ Rm. However, as in Kruk et al [19], we look at this as an overdetermined system of nonlinear (bilinear) equations on vectors X, y, Z which has a zero residual at optimality. We do not use any symmetrization on this system. That the vectors X, Z are symmetric matrices is ignored. What is different in this paper is a preprocessing step that results in one single, well-conditioned, overdetermined bilinear equation. We do not form a Schur complement form for the linearized system. In addition, asymptotic q-quadratic convergence is obtained with a crossover approach that switches to affine scaling without maintaining the positive (semi)definiteness of X and Z. This results in a marked reduction in the number and the complexity of the iterations. We use the long accepted method for nonlinear least squares problems, the GaussNewton method. For large sparse data, we use an inexact Gauss-Newton approach with a preconditioned conjugate gradient method applied directly on the linearized equations in matrix space. This does not require any operator formations into a Schur complement system or matrix representations of linear operators. ∗Research supported by The Natural Sciences and Engineering Research Council of Canada. E-mail [email protected] A modified version of this paper, Semidefinite Programming and Some Closest Matrix Approximation Problems, was presented at the 1st Annual McMaster Optimization Conference (MOPTA 01), August 2-4, 2001 Hamilton, Ontario. 1 URL for paper and MATLAB programs: http://orion.math.uwaterloo.ca/ ̃hwolkowi/henry/reports/ABSTRACTS.html