Browsing by Author "Monteiro, Renato D.C."

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    A Computational Study of a Gradient-Based Log-Barrier Algorithm for a Class of Large-Scale SDPs
    (2001-06) Burer, Samuel; Monteiro, Renato D.C.; Zhang, Yin
    The authors of this paper recently introduced a transformation that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, they proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints.
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    Interior-Point Algorithms for Semidefinite Programming Based on A Nonlinear Programming Formulation
    (1999-12) Burer, Samuel; Monteiro, Renato D.C.; Zhang, Yin
    Recently, the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n × n matrix 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 interior point algorithms 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 interior-point algorithms for this type of nonlinear program and establish their global convergence.
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    Solving Semidefinite Programs via Nonlinear Programming, Part I: Transformations and Derivatives
    (1999-09) Burer, Samuel; Monteiro, Renato D.C.; Zhang, Yin
    In this paper, we introduce transformations that convert a large class of linear and/or nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems over "orthants" of the form (R^n)++ × R^N, where n is the size of the matrices involved in the problem and N is a nonnegative integer dependent upon the specific problem. For example, in the case of the SDP relaxation of a MAXCUT problem, N is zero and n, the number of variables of the resulting nonlinear optimization problem, is the number of vertices in the underlying graph. The class of transformable problems includes most, if not all, instances of SDP relaxations of combinatorial optimization problems with binary variables, as well as other important SDP problems. We also derive formulas for the first and second derivatives of the objective function of the resulting nonlinear optimization problem, hence enabling the effective application of existing nonlinear optimization techniques to the solution of large-scale SDP problems.