An inexact dual logarithmic barrier method for solving sparse semidefinite programs

A dual logarithmic barrier method for solving large, sparse semidefinite programs is proposed in this paper. The method avoids any explicit use of the primal variable X and therefore is well-suited to problems with a sparse dual matrix S. It relies on inexact Newton steps in dual space which are computed by the conjugate gradient method applied to the Schur complement of the reduced KKT system. The linear system is preconditioned by the partial Cholesky factorization of the Schur complement matrix and this allows for the method to be run in a matrix-free scheme. Convergence properties of the method are studied and a polynomial complexity result is extended to the case when inexact Newton steps are employed. A MATLAB-based implementation is developed and preliminary computational results of applying the method to maximum cut problems are reported.