Interior-Point Algorithms for Semidefinite Programming Based on a Nonlinear Formulation

Authors:
Samuel Burer;Renato D. C. Monteiro;Yin Zhang
Affiliations:
Department of Management Sciences, University of Iowa, Iowa City, IA 52242, USA. samuel-burer@uiowa.edu;School of ISyE, Georgia Tech., Atlanta, Georgia 30332, USA. monteiro@isye.gatech.edu;Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA. zhang@caam.rice.edu
Venue:
Computational Optimization and Applications
Year:
2002

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Abstract

Recently in Burer et al. (Mathematical Programming A, submitted), the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n × n matrix-valued function of a certain form into the positivity constraint on n scalar variables while keeping the number of variables unchanged. Based on this transformation, they proposed a first-order interior-point algorithm for solving a special class of linear semidefinite programs. In this paper, we extend this approach and apply the transformation to general linear semidefinite programs, producing nonlinear programs that have not only the n positivity constraints, but also n additional nonlinear inequality constraints. Despite this complication, the transformed problems still retain most of the desirable properties. We propose first-order and second-order interior-point algorithms for this type of nonlinear program and establish their global convergence. Computational results demonstrating the effectiveness of the first-order method are also presented.