S. Burer, R. D. C. Monteiro, and Y. Zhang, "Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation," TR99-27, Dept. of Computational and Applied Mathematics, Rice University, Houston TX 77005, USA, December 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....optimization problem) Computing lower bounds on the amount of unavoidable interference is a crucial step in this direction. The currently best lower bounds on the amount of unavoidable (co channel) interference are obtained from solving semidefinite programs (using the solvers developed by [1, 7]) These semidefinite programs arise as nonpolyhedral relaxation of a minimum partition problem on complete graphs. The success of this approach is made plausible by revealing structural relations between the feasible set of the semidefinite program and a polytope associated with an integer ....

Samuel Burer, Renato D.C. Monteiro, and Yin Zhang. Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation. Technical Report TR 99--27, Department of Computational and Applied Mathematics, Rice Unviversity, Dec. 1999.

....the smooth but nonconvex problem (D # ) max w,y d T z(w, y) b T y, w 0. The authors then suggest algorithms to solve this problem: a log barrier method and a potential reduction method. A subsequent paper relaxes the requirement that the diagonal of X be fixed. Instead, they require in [11] that the diagonal be bounded below, so the first constraint becomes diag(X) # d. This constraint can be without loss of generality, since it holds for any positive semidefinite matrix if we choose the vector d to be zero. The corresponding change to (D) is that now z must be nonnegative, and ....

S. Burer, R. D. C. Monteiro, and Y. Zhang. Interior point algorithms for semidefinite programming based on a nonlinear programming formulation. Technical Report TR99-27, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1999.

....for exploiting the aggregate sparsity pattern over the data matrices. Besides interior point methods, some other computational methods have been also proposed and intensively studied for solving large scale SDPs; the spectral bundle method [13] and nonlinear programming reformulations of SDPs [4, 5, 34]. Numerical results on large scale SDPs have been reported. They include (i) SDP relaxations of the max cut problem and the graph bisection problem solved by the spectral bundle method [12, 13] the dual scaling method [2] and nonlinear programming reformulations [4] ii) an SDP relaxation of the ....

S. Burer, R. D. C. Monteiro, and Y. Zhang, "Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation," TR99-27, Dept. of Computational and Applied Mathematics, Rice University, Houston TX 77005, USA, December 1999.

No context found.

S. Burer, R. Monteiro, and Y. Zhang. Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation. Computational Optimization and Applications, To appear, 2001.

....the first order barrier algorithm for several classes of linear SDP problems and found that the proposed approach is indeed a viable approach to solving some large scale SDP problems from combinatorial optimization. The implementation details and numerical results are reported in [3] Moreover, in [4] we have extended the application of the proposed transformation to linear SDP problems of a general form (where the diagonal of the primal matrix variable is not required to be fixed) and presented preliminary numerical results. In addition, we have proposed and analyzed a second order potential ....

S. Burer, R. D. C. Monteiro, and Y. Zhang. Interior-point algorithms for semidefinite programming based on a nonlinear formulation. To appear in Computational Optimization and Applications.

....interior feasible set F 0 (D) ae n Theta m Theta p Theta S n into the set n Theta m Theta p , and then apply the log barrier approach to the resulting nonlinear optimization problem in the transformed space. These ideas were first introduced in [4] and [5]. Recall the definition of the interior feasible set for (D) F 0 (D) j f(z; y; u; S) 2 n Theta m Theta p Theta S n : Diag(z) A (y) G (u) Gamma C = Sg: 6 The transformation from (D) to a nonlinear optimization problem consists of two stages. The first stage is ....

.... Gamma0.0029943 Gamma0.0095547 180000 18194 89 93791 fap11 Gamma0.0118932 Gamma0.0296136 180000 39038 87 86135 fap12 Gamma0.2151594 Gamma0.2733099 180000 44984 82 81119 large number of constraints. Recently, we have extended the application of our transformation to general SDP problems [5] where the diagonal of the primal matrix variable need not to be fixed. Preliminary numerical results in [5] indicate that the approach also holds promise for general SDP problems. Acknowledgment We thank Christoph Helmberg for his help in running the code SBmethod, and Andreas Eisenblatter for ....

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S. Burer, R. D. C. Monteiro, and Y. Zhang. Interior-Point Algorithms for Semidefinite Programming Based on a Nonlinear Formulation. Submitted to Computational Optimization and Applications .

....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. Interior-Point Algorithms for Semidefinite Programming Based on A Nonlinear Programming Formulation. Technical Report TR99-27, Department of Computational and Applied Mathematics, Rice University,

.... on semidefinite programming and the classical interior point algorithms to solve them, we refer the reader to [14] Even though there have been some recent advances in solving (1) for graphs having more than 1,000 vertices and 100,000 edges using non traditional approaches (see, for example, [2, 3, 4, 9]) solving (1) for large scale instances is still a formidable challenge. Besides providing a high quality upper bound on #(G) can the Lovasz theta SDP (1) be utilized in some way to provide a high quality lower bound on #(G) More specifically, can (1) be exploited to find large stable sets in ....

S. Burer, R. D. C. Monteiro and Y. Zhang. Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation. Technical Report TR9920 27, Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA, December 1999.

....method in particular, a first order method can be used. Burer and Monteiro [4] improved upon the idea of Homer and Peinado by simply noting that, without loss of generality, V can be required to be lower triangular in accordance with the Cholesky factorization. Then, in a series of papers [6, 7, 5], Burer, Monteiro, and Zhang showed how one could apply the idea of Cholesky factorization in the dual SDP space to transform any SDP into a nonlinear optimization problem over a simple feasible set. They also provided a globally convergent, first order log barrier algorithm to solve SDPs via this ....

S. Burer, R.D.C. Monteiro, and Y. Zhang. Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation. manuscript, School of ISyE, Georgia Tech, Atlanta, GA, 30332, USA, December 1999. To appear in Computational Optimization and Applications.

....of the SDP relaxation approach with respect to the problem size. There have been a great deal of research e#orts towards improving the e#ciency of SDP solvers, including works on exploiting sparsity in more traditional interior point methods [1, 8, 12, 13, 22] and works on alternative methods [3, 4, 5, 6, 15, 16, 23, 24]. Indeed, the e#ciency of SDP solvers has been improved significantly in the last few years. Nevertheless, the scalability problem still remains. Can the scalability problem of the SDP relaxation be overcome Can the SDP relaxation approach ever become competitive in approximating large scale ....

S. Burer, R. D. C. Monteiro, and Y. Zhang. Interior-Point Algorithms for Semidefinite Programming Based on A Nonlinear Programming Formulation. Technical Report TR9927, Department of Computational and Applied Mathematics, Rice University,

.... on semidefinite programming and the classical interior point algorithms to solve them, we refer the reader to [14] Even though there have been some recent advances in solving (1) for graphs having more than 1,000 vertices and 100,000 edges using non traditional approaches (see, for example, [2, 3, 4, 9]) solving (1) for large scale instances is still a formidable challenge. Besides providing a high quality upper bound on ff(G) can the Lov asz theta SDP (1) be utilized in some way to provide a high quality lower bound on ff(G) More specifically, can (1) be exploited to find large stable sets ....

S. Burer, R. D. C. Monteiro and Y. Zhang. Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation. Technical Report TR9927, Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA, December 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, 8, 12, 13, 23] and works on alternative methods [3, 4, 5, 6, 15, 16, 24, 25]. Indeed, the efficiency of SDP solvers has been improved significantly in the last few years. Nevertheless, the scalability problem still remains. Can the scalability problem of the SDP relaxation be overcome Can the SDP relaxation approach ever become competitive in approximating large scale ....

S. Burer, R. D. C. Monteiro, and Y. Zhang. Interior-Point Algorithms for Semidefinite Programming Based on A Nonlinear Programming Formulation. Technical Report TR9927, Department of Computational and Applied Mathematics, Rice University,

No context found.

S. Burer, R. D. C. Monteiro, and Y. Zhang, "Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation," TR99-27, Dept. of Computational and Applied Mathematics, Rice University, Houston TX 77005, USA, December 1999.

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