An inexact dual logarithmic barrier method for solving sparse semidefinite programs

Technical Report ERGO-16-004

S. Bellavia, J. Gondzio and M. Porcelli

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 method may take advantage of low-rank representations of matrices A_i to perform implicit matrix-vector products with the Schur complement matrix and to compute only specific parts of this matrix. This allows the construction of the partial Cholesky factorization of the Schur complement matrix which serves as a good preconditioner for it and permits 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 and matrix completion problems are reported.

Key words: Semidefinite programming, dual logarithmic barrier method, inexact Newton method, preconditioning

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Written: July 29, 2016, revised March 31, 2017.