Abstract
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.
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The authors would like to thank the anonymous referees for their suggestions, which lead to significant improvement of the manuscript.
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The work of the first and the third author was supported by Gruppo Nazionale per il Calcolo Scientifico (GNCS-INdAM) of Italy. The work of the second author was supported by EPSRC Research Grant EP/N019652/1. Part of the research was conducted during a visit of the second author at Dipartimento di Ingegneria Industriale, UNIFI, the visit was supported by the University of Florence Internationalisation Plan.
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Bellavia, S., Gondzio, J. & Porcelli, M. An inexact dual logarithmic barrier method for solving sparse semidefinite programs. Math. Program. 178, 109–143 (2019). https://doi.org/10.1007/s10107-018-1281-5
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DOI: https://doi.org/10.1007/s10107-018-1281-5