S. Burer, R. Monterio, and Y. Zhang, Solving a class of semidefinite programs via nonlinear programming, Mathematical Programming A, 93 (2002), pp. 97--122.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....SDPs. Currently, there are two main classes of first order methods. In [14] the dual SDP was first formulated as a non smooth convex optimization problem and was solved by a spectral bundle method based on standard non smooth optimization techniques. On the other hand, Burer, Monterio, and Zhang, [2] converted the dual SDP into a nonconvex nonlinear program in IR , and used log barrier methods to solve the resulting nonlinear program. However, there are recent advances in using second order methods to solve large SDPs. In [27] Toh and Kojima constructed preconditioners for the SCE ....

....7.5e 03 6.2e 00 3.7e 10 17 9.0e 11 7.0e 16 1.8e 06 7.8e 03 6.2e 00 1.0e 12 18 3.0e 12 7.7e 16 1.7e 06 7.3e 03 6.2e 00 3.0e 13 Table 4: Same as Table 3 for the SDP problem mcp250 1, but with fixed constraints removed. The approximate optimal solution has a relative dual gap of 5.7e 14. in [2]. This SDP has a semdefinite variable in 52 and a linear variable in IR 1160 . The number of constraints is m = 1378. The approximate optimal solution is strictly complementary with min i 10 4 . We have p = 48, q = 4 for the semidefinite block, and p = 30, q = 1130 for the linear ....

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S. Burer, R. Monterio, and Y. Zhang, Solving a class of semidefinite programs via nonlinear programming, Mathematical Programming A, 93 (2002), pp. 97--122.

.... uses a spectral bundle method to solve the nondi#erentiable convex program, a penalty approach by Kocvara and Stingl [19] low rank factorizations of Burer and Monteiro [10] and transformation to a constrained nonlinear program proposed by Burer and Monteiro [9] and Burer, Monteiro, and Zhang [11]. A discussion and comparison of these methods can be found in [24] Some of these methods are particularly well suited for large scale problems [21] In particular, the spectral bundle method and low rank factorizations have solved solved some large instances of SDP. However, these methods lack ....

S. Burer, R. D. C. Monteiro, and Y. Zhang. Solving a class of semidefinite programs via nonlinear programming. Mathematical Programming, 93(1):97--122, 2002.

....Computational Study of a Gradient Based Log Barrier Algorithm for a Class of Large Scale SDPs y Samuel Burer z Renato D.C. Monteiro x Yin Zhang June 1, 2001 Abstract The authors of this paper recently introduced a transformation [4] 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 ....

.... have proven capable of obtaining moderate accuracy in a reasonable amount of time for large scale problems [3, 16, 17] Based on a nonlinear transformation, we recently proposed a first order, log barrier method for solving a class of large scale SDPs and established its global convergence [4]. The main purpose of this paper is to study the implementation issues for this algorithm and to report our computational results. This paper is organized as follows. In Section 2, we introduce the class of SDPs to be considered and describe three types of such SDPs that will be used to test the ....

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S. Burer, R. D. C. Monteiro, and Y. Zhang. Solving a class of Semidefinite Programs via Nonlinear Programming. Submitted to Mathematical Programming (Series A).

....Algorithms for Semidefinite Programming Based on a Nonlinear Formulation Samuel Burer y Renato D.C. Monteiro z Yin Zhang x December 1999 Revised May 2001 Abstract Recently in [5], the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n Theta 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 ....

....needed for computing the Cholesky factor of the dual slack matrix S. Since S is sparse whenever the underlying graph is sparse, Vavasis s observation may potentially lead to efficient, gradient based implementations of the classical log barrier method that can exploit sparsity. In our recent paper [5], we showed how a class of linear and nonlinear SDPs can be reformulated into nonlinear optimization problems over very simple feasible sets of the 2 form n Theta m , where n is the size of matrix variable X , m is a problem dependent, nonnegative integer, and n is the ....

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S. Burer, R. D. C. Monteiro, and Y. Zhang. Solving a Class of Semidefinite Programs via Nonlinear Programming. Submitted to Mathematical Programming A. (Also see Technical Report TR99-17, Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA, September 1999.)

....of the SDP relaxation approach with respect to the problem size. There have been a great deal of research efforts towards improving the efficiency of SDP solvers, including works on exploiting sparsity in more traditional interior point methods [1, 9, 16, 17, 29] and works on alternative methods [5, 6, 7, 20, 21, 30, 31]. Indeed, the efficiency of SDP solvers has been improved significantly in the last few years. Nevertheless, the scalability problem still remains. On the other hand, computational studies have continued to affirm that the quality of bounds produced by the SDP relaxation is quite high. For ....

S. Burer, R. D. C. Monteiro, and Y. Zhang. Solving a Class of Semidefinite Programs via Nonlinear Programming. Submitted to Mathematical Programming A. (Also Technical Report TR99-17, Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA, September 1999.)

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