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33
A Spectral Bundle Method for Semidefinite Programming
- SIAM JOURNAL ON OPTIMIZATION
, 1997
"... A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applica ..."
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Cited by 171 (7 self)
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A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order. We present a method that allows to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored for eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completene...
Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization
- SIAM JOURNAL ON OPTIMIZATION
, 1998
"... We present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational re ..."
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Cited by 119 (11 self)
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We present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interior-point algorithms for approximating the maximum cut semidefinite programs with dimension up-to 3000.
Randomized Heuristics for the Max-Cut Problem
- Optimization Methods and Software
, 2002
"... Given an undirected graph with edge weights, the MAX-CUT problem consists in finding a partition of the nodes into two subsets, such that the sum of the weights of the edges having endpoints in different subsets is maximized. ..."
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Cited by 41 (16 self)
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Given an undirected graph with edge weights, the MAX-CUT problem consists in finding a partition of the nodes into two subsets, such that the sum of the weights of the edges having endpoints in different subsets is maximized.
Numerical Evaluation of SDPA (SemiDefinite Programming Algorithm).
, 1998
"... . SDPA (SemiDefinite Programming Algorithm) is a C++ implementation of a Mehrotra-type primal-dual predictor-corrector interior-point 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 Mehrotra-type primal-dual predictor-corrector interior-point 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 time-consuming parts of SDPA vary with the problem size, the number of constraints and the sparsity of data matrices. Key words Interior-Point Methods, Semidefinite Programming, Numerical Experiments, Sparsity y Department of Architecture and Architectural Systems, Kyoto University, YoshidaHonmati, Sakyou-ku, Kyoto, 606-8501, Japan e-mail: fujisawa@is-mj.archi.kyoto-u.ac.jp ] Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan. e-mail: mituhiro@is.titech.ac.jp ? Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan...
SEMIDEFINITE PROGRAMMING RELAXATIONS FOR THE GRAPH PARTITIONING PROBLEM
, 1999
"... A new semidefinite programming, SDP, relaxation for the general graph partitioning problem, GP, is derived. The relaxation arises from the dual of the (homogenized) Lagrangian dual of an appropriate quadratic representation of GP. The quadratic representation includes a representation of the 0,1 co ..."
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Cited by 31 (6 self)
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A new semidefinite programming, SDP, relaxation for the general graph partitioning problem, GP, is derived. The relaxation arises from the dual of the (homogenized) Lagrangian dual of an appropriate quadratic representation of GP. The quadratic representation includes a representation of the 0,1 constraints in GP. The special structure of the relaxation is exploited in order to project onto the minimal face of the cone of positive semidefinite matrices which contains the feasible set. This guarantees that the Slater constraint qualification holds, which allows for a numerically stable primal-dual interior-point solution technique. A gangster operator is the key to providing an efficient representation of the constraints in the relaxation. An incomplete preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. Only dual feasibility is enforced, which results in the desired lower bounds, but avoids the expensive primal feasibility calculations. Numerical results
A PROJECTED GRADIENT ALGORITHM FOR SOLVING THE MAXCUT SDP RELAXATION
"... In this paper, we present a projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (maxcut) problem. Coupled with a randomized method, this gives a very efficient approximation algorithm for the maxcut problem. We report computational results compar ..."
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Cited by 26 (8 self)
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In this paper, we present a projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (maxcut) problem. Coupled with a randomized method, this gives a very efficient approximation algorithm for the maxcut problem. We report computational results comparing our method with two earlier successful methods on problems with dimension up to 7,000.
A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations
- In The sharpest cut, MPS/SIAM Ser. Optim
, 2001
"... The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with re ..."
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Cited by 22 (3 self)
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The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 0-1 programming problems by odd-cycle inequalities. We also consider separating oddcycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for max-cut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably.
Mixed Linear and Semidefinite Programming for Combinatorial and Quadratic Optimization
, 1999
"... We use the semidefinite relaxation to approximate combinatorial and quadratic optimization problems subject to linear, quadratic, as well as boolean constraints. We present a dual potential reduction algorithm and show how to exploit the sparse structure of various problems. Coupled with randomized ..."
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Cited by 17 (4 self)
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We use the semidefinite relaxation to approximate combinatorial and quadratic optimization problems subject to linear, quadratic, as well as boolean constraints. We present a dual potential reduction algorithm and show how to exploit the sparse structure of various problems. Coupled with randomized and heuristic methods, we report computational results for approximating graph-partition and quadratic problems with dimensions 800 to 10,000. This finding, to the best of our knowledge, is the first computational evidence of the effectiveness of these approximation algorithms for solving large-scale problems.
Lower Bounds and Exact Algorithms for the Graph Partitioning Problem using Multicommodity Flows
, 2001
"... In this paper new and generalized lower bounds for the graph partitioning problem are presented. These bounds base on the well known lower bound of embedding a clique into the given graph with minimal congestion which is equivalent to a multicommodity flow problem where each vertex sends a commodi ..."
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Cited by 15 (1 self)
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In this paper new and generalized lower bounds for the graph partitioning problem are presented. These bounds base on the well known lower bound of embedding a clique into the given graph with minimal congestion which is equivalent to a multicommodity flow problem where each vertex sends a commodity of size one to every other vertex. Our new bounds use arbitrary multicommodity flow instances for the bound calculation, the critical point for the lower bound is the guaranteed cut flow of the instances. Furthermore, a branch&bound procedure basing on these bounds is presented. Finally, upper bounds of the lower bounds are shown which demonstrate the superiority of the presented generalizations; and the new bounds are applied to the Butterfly and Benes network. 1
An Efficient Algorithm for Solving the MAXCUT SDP Relaxation
- School of ISyE, Georgie Tech, Atlanta, GA 30332
, 1998
"... In this paper we present a projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (MAXCUT) problem. Coupled with a randomized method, this gives a very efficient approximation algorithm for the MAXCUT problem. We report computational results compari ..."
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Cited by 11 (1 self)
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In this paper we present a projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (MAXCUT) problem. Coupled with a randomized method, this gives a very efficient approximation algorithm for the MAXCUT problem. We report computational results comparing our method with two earlier successful methods on problems with dimension up to 3000.
