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Maximum stable set formulations and heuristics based on continuous optimization
 MATH. PROGRAM., SER. A 94: 137–166 (2002)
, 2002
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Interiorpoint algorithms for semidefinite programming based on a nonlinear formulation
 COMP. OPT. AND APPL
, 2002
"... 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 matrixvalued function of a certain form into the positivity constraint on n scalar variables while keeping t ..."
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Cited by 18 (7 self)
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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 matrixvalued 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 firstorder interiorpoint 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 firstorder and secondorder interiorpoint algorithms for this type of nonlinear program and establish their global convergence. Computational results demonstrating the effectiveness of the firstorder method are also presented.
A Computational Study of a GradientBased LogBarrier Algorithm for a Class of LargeScale SDPs
 Mathematical Programming Series B
, 2001
"... The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrixvalued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the so ..."
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Cited by 17 (3 self)
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The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrixvalued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interiorpoint methods to handle efficiently. Based on the transformation, they proposed a globally convergent, firstorder (i.e., gradientbased) logbarrier 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 largescale SDPs and is particularly effective for problems with a large number of constraints.
DSDP5 user guide — software for semidefinite programming
 Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 2005
"... DSDP implements the dualscaling algorithm for semidefinite programming. The source code if this interiorpoint solver, written entirely in ANSI C, is freely available. The solver can be used as a subroutine library, as a function within the MATLAB environment, or as an executable that reads and wri ..."
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Cited by 10 (0 self)
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DSDP implements the dualscaling algorithm for semidefinite programming. The source code if this interiorpoint solver, written entirely in ANSI C, is freely available. The solver can be used as a subroutine library, as a function within the MATLAB environment, or as an executable that reads and writes to files. Initiated in 1997, DSDP has developed into an efficient and robust general purpose solver for semidefinite programming. Although the solver is written with semidefinite programming in mind, it can also be used for linear programming and other constraint cones. The features of DSDP include: • a robust algorithm with a convergence proof and polynomially bounded complexity under mild assumptions on the data, • primal and dual solutions, • feasible solutions when they exist or approximate certificates of infeasibity, • initial points that can be feasible or infeasible, • relatively low memory requirements for an interiorpoint method, • sparse and lowrank data structures,
On The Slater Condition For The SDP Relaxations Of Nonconvex Sets
, 2000
"... We prove that all results determining the dimension and the ane hull of feasible solutions of any combinatorial optimization problem, and various more general nonconvex optimization problems, directly imply the existence of Slater points for a very wide class of semidefinite programming relaxations ..."
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Cited by 8 (2 self)
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We prove that all results determining the dimension and the ane hull of feasible solutions of any combinatorial optimization problem, and various more general nonconvex optimization problems, directly imply the existence of Slater points for a very wide class of semidefinite programming relaxations of these nonconvex problems. Our proofs are very concise, constructive and elementary.
On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász’s Theta Function
, 2004
"... We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász’s theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its subgraphs. We develop a ..."
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Cited by 4 (1 self)
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We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász’s theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its subgraphs. We develop an efficient, polynomialtime algorithm to extract a maximum stable set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warmstart strategy
A Heuristic for the Stability Number of a Graph based on Convex Quadratic Programming and Tabu Search, special issue of the
 Journal of Mathematical Sciences, Aveiro Seminar on Control Optimization and Graph Theory, Second Series
"... Abstract. Recently a characterization of the Lovász theta number based on convex quadratic programming was established. As a consequence of this formulation, we introduce a new upper bound on the stability number of a graph which slightly improves the theta number. Like this number, the new bound ca ..."
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Abstract. Recently a characterization of the Lovász theta number based on convex quadratic programming was established. As a consequence of this formulation, we introduce a new upper bound on the stability number of a graph which slightly improves the theta number. Like this number, the new bound can be characterized as the minimum of a function whose values are the optimum values of convex quadratic programs. This paper is mainly oriented to the following question: how can the new bound be used to approximate the maximum stable set for large graphs? With this in mind we present a twophase heuristic for the stability problem which begins by computing suboptimal solutions using the new bound de
nition. In the second phase a multistart tabu search heuristic is implemented. The results of applying this heuristic to some DIMACS benchmark graphs are reported. 1.
Math. Program., Ser. B 95: 359–379 (2003) Digital Object Identifier (DOI) 10.1007/s1010700203537
"... A computational study of a gradientbased logbarrier algorithm for a class of largescale SDPs † ..."
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A computational study of a gradientbased logbarrier algorithm for a class of largescale SDPs †