Henry Wolkowicz
Department of Combinatorics and Optimization
University of Waterloo
Waterloo, Ontario N2L 3G1
This talk 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, Fu(X,y,Z)=0, where X,Z are n by n symmetric matrices and y in Rm. However, 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 approach 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 Gauss-Newton 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.
To illustrate the approach, we apply it to the well known SDP relaxation of the Max-Cut problem. The cost of an iteration consists almost entirely in the solution of a (generally well-conditioned) least squares problem of size n2 by n+1C2. This system consists of one large (sparse) block with nC2 columns (one CG iteration cost is one sparse matrix multiplication) and one small (dense) block with n columns (one CG iteration cost is one matrix row scaling). Exact primal and dual feasibility are guaranteed throughout the iterations. We include the derivation of the optimal diagonal preconditioner. The numerical tests show that the algorithm exploits sparsity and obtains q-quadratic convergence.