Browsing by Author "Monteiro, Renato"

Now showing 1 - 3 of 3
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    Maximum Stable Set Formulations and Heuristics Based on Continuous Optimization
    (2000-12) Burer, Samuel; Monteiro, Renato; Zhang, Yin
    The stability number for a given graph G is the size of a maximum stable set in G. The Lovasz theta number provides an upper bound on the stability number and can be computed as the optimal value of the Lovasz semidefinite program. In this paper, we show that restricting the matrix variable in the Lovasz semidefinite program to be rank-one or rank-two yields a pair of continuous, nonlinear optimization problems each having the global optimal value equal to the stability number. We propose heuristics for obtaining large stable sets in G based on these new formulations and present computational results indicating the effectiveness of the heuristics.
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    Rank-Two Relaxation Heuristics for Max-Cut and Other Binary Quadratic Programs
    (2000-11) Burer, Samuel; Monteiro, Renato; Zhang, Yin
    Semidefinite relaxation for certain discrete optimization problems involves replacing a vector-valued variable by a matrix-valued one, producing a convex program while increasing the number of variables by an order of magnitude. As useful as it is in theory, this approach encounters difficulty in practice as problem size increases. In this paper, we propose a rank-two relaxation approach and construct continuous optimization heuristics applicable to some binary quadratic programs, including primarily the Max-Cut problem but also others such as the Max-Bisection problem. A computer code based on our rank-two relaxation heuristics is compared with two state-of-the-art semidefinite programming codes. We will report some rather intriguing computational results on a large set of test problems and discuss their ramifications.
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    Solving Semidefinite Programs via Nonlinear Programming, Part II: Interior Point Methods for a Subclass of SDPs
    (1999-10) Burer, Sam; Monteiro, Renato; Zhang, Yin
    In Part I of this series of papers, we have introduced transformations which convert a large class of linear and nonlinear semidefinite programs (SDPs) 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. In doing so, we have effectively reduced the number of variables and constraints. In this paper, we develop interior point methods for solving a subclass of the transformable linear SDP problems where the diagonal of a matrix variable is given. These new interior point methods have the advantage of working entirely within the space of the transformed problem while still maintaining close ties with the original SDP. Under very mild and reasonable assumptions, global convergence of these methods is proved.