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31
Numerical Evaluation of SDPA (SemiDefinite Programming Algorithm).
, 1998
"... . SDPA (SemiDefinite Programming Algorithm) is a C++ implementation of a Mehrotratype primaldual predictorcorrector interiorpoint method for solving the standard form semidefinite program and its dual. We report numerical results of large scale problems to evaluate its performance, and investiga ..."
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Cited by 36 (12 self)
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. SDPA (SemiDefinite Programming Algorithm) is a C++ implementation of a Mehrotratype primaldual predictorcorrector interiorpoint method for solving the standard form semidefinite program and its dual. We report numerical results of large scale problems to evaluate its performance, and investigate how major timeconsuming parts of SDPA vary with the problem size, the number of constraints and the sparsity of data matrices. Key words InteriorPoint Methods, Semidefinite Programming, Numerical Experiments, Sparsity y Department of Architecture and Architectural Systems, Kyoto University, YoshidaHonmati, Sakyouku, Kyoto, 6068501, Japan email: fujisawa@ismj.archi.kyotou.ac.jp ] Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 1528552, Japan. email: mituhiro@is.titech.ac.jp ? Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 1528552, Japan...
Solving Graph Bisection Problems With Semidefinite Programming
 INFORMS Journal on Computing
, 1997
"... . An exact solution method for the graph bisection problem is presented. We describe a branchandbound algorithm which is based on a cutting plane approach combining semidefinite programming and polyhedral relaxations. We report on extensive numerical experiments which were performed for various cl ..."
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Cited by 33 (2 self)
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. An exact solution method for the graph bisection problem is presented. We describe a branchandbound algorithm which is based on a cutting plane approach combining semidefinite programming and polyhedral relaxations. We report on extensive numerical experiments which were performed for various classes of graphs. The results indicate that the present approach solves general problem instances with 80 \Gamma 90 vertices exactly in reasonable time, and provides tight approximations for larger instances. Our approach is particularly well suited for special classes of graphs as planar graphs and graphs based on grid structures. 1. Introduction We consider the problem of partitioning the vertices of a graph into two components. Given is an undirected edgeweighted graph G(V; E), where V denotes the vertex set consisting of n vertices, and E the edge set. The weight of the edges are given by the Laplace matrix L, which is defined through the adjacency matrix A of the graph by L := Diag(Ae...
Sums of random symmetric matrices and quadratic optimization under orthogonality constraints
 MATHEMATICAL PROGRAMMING (2006), ONLINE FIRST ISSUE, DOI
"... Let Bi be deterministic real symmetric m × m matrices, and ξi be independent random scalars with zero mean and “of order of one” (e.g., ξi ∼ N (0, 1)). We are interested to know under what conditions “typical norm ” of the random matrix SN = N� ξiBi is of order of 1. An evident necessary condition ..."
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Cited by 30 (3 self)
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Let Bi be deterministic real symmetric m × m matrices, and ξi be independent random scalars with zero mean and “of order of one” (e.g., ξi ∼ N (0, 1)). We are interested to know under what conditions “typical norm ” of the random matrix SN = N� ξiBi is of order of 1. An evident necessary condition is E{S 2 N to N� B i=1 2 i i=1} � O(1)I, which, essentially, translates � I; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of SN is ≤ O(1)m 1 6: Prob{�SN �> Ωm 1/6} ≤ O(1) exp{−O(1)Ω 2} for all Ω> 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints Opt = max Xj∈Rm×m � F (X1,..., Xk) : XjX T j = I, j = 1,..., k �, where F is quadratic in X = (X1,..., Xk). We show that when F is convex in every one of Xj, a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices Xj: Opt ≤ Opt(SDP) ≤ O(1)[m1/3 + ln k]Opt.
A Near Maximum Likelihood Decoding Algorithm for MIMO Systems Based on SemiDefinite Programming
, 2005
"... In MultiInput MultiOutput (MIMO) systems, MaximumLikelihood (ML) decoding is equivalent to finding the closest lattice point in an Ndimensional complex space. In general, this problem is known to be NP hard. In this paper, we propose a quasimaximum likelihood algorithm based on SemiDefinite Pr ..."
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Cited by 28 (4 self)
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In MultiInput MultiOutput (MIMO) systems, MaximumLikelihood (ML) decoding is equivalent to finding the closest lattice point in an Ndimensional complex space. In general, this problem is known to be NP hard. In this paper, we propose a quasimaximum likelihood algorithm based on SemiDefinite Programming (SDP). We introduce several SDP relaxation models for MIMO systems, with increasing complexity. We use interiorpoint methods for solving the models and obtain a nearML performance with polynomial computational complexity. Lattice basis reduction is applied to further reduce the computational complexity of solving these models. The proposed relaxation models are also used for soft output decoding in MIMO systems.
Fixing Variables in Semidefinite Relaxations
 SIAM J. MATRIX ANAL. APPL
, 1996
"... The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations as the corresponding Lagrange multipliers are usually not available. We propose a general technique for computing reasonable Lagrange multipliers to constraints which are not ..."
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Cited by 27 (2 self)
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The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations as the corresponding Lagrange multipliers are usually not available. We propose a general technique for computing reasonable Lagrange multipliers to constraints which are not part of the problem description. Its specialization to the semidefinite f\Gamma1; 1g relaxation of quadratic 01 programming yields an efficient routine for fixing variables. The routine offers the possibility to exploit problem structure. We extend the traditional bijective map between f0; 1g and f\Gamma1; 1g formulations to the constraints such that the dual variables remain the same and structural properties are preserved. In consequence the fixing routine can efficiently be applied to optimal solutions of the semidefinite f0; 1g relaxation of constrained quadratic 01 programming, as well. We provide numerical results showing the efficacy of the approach.
Graph partitioning and continuous quadratic programming
 SIAM J. DISCRETE MATH
, 1999
"... A continuous quadratic programming formulation is given for mincut graph partitioning problems. In these problems, we partition the vertices of a graph into a collection of disjoint sets satisfying specified size constraints, while minimizing the sum of weights of edges connecting vertices in dif ..."
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Cited by 22 (8 self)
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A continuous quadratic programming formulation is given for mincut graph partitioning problems. In these problems, we partition the vertices of a graph into a collection of disjoint sets satisfying specified size constraints, while minimizing the sum of weights of edges connecting vertices in different sets. An optimal solution is related to an eigenvector (Fiedler vector) corresponding to the second smallest eigenvalue of the graph’s Laplacian. Necessary and sufficient conditions characterizing local minima of the quadratic program are given. The effect of diagonal perturbations on the number of local minimizers is investigated using a test problem from the literature.
INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION
, 1995
"... Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivale ..."
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Cited by 16 (9 self)
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Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.
Strong Duality and Minimal Representations for Cone Optimization
, 2008
"... The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is ..."
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Cited by 14 (2 self)
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The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs; duality and characterizations of optimality that hold without any CQ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQand failure of strict complementarity. One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ. We include a discussion on obtaining these representations efficiently.
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.