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.
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Burer, S., Monteiro, R.D. & Zhang, Y. Interior-Point Algorithms for Semidefinite Programming Based on a Nonlinear Formulation. Computational Optimization and Applications 22, 49–79 (2002). https://doi.org/10.1023/A:1014834318702
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DOI: https://doi.org/10.1023/A:1014834318702