Results 1 - 10 of 36

Monotonicity of primal-dual interior-point algorithms for semidefinite programming problems

by Masakazu Kojima, Levent Tuncel , 1998
"... We present primal-dual interior-point algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly imp ..."
Abstract - Cited by 216 (35 self) - Add to MetaCart
We present primal-dual interior-point algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.

Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization

by Steven J. Benson, Yinyu Ye, Xiong Zhang - 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 ..."
Abstract - Cited by 119 (11 self) - Add to MetaCart
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.
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Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework

by Mituhiro Fukuda, Masakazu Kojima, Kazuo Murota, Kazuhide Nakata - SIAM JOURNAL ON OPTIMIZATION , 1999
"... A critical disadvantage of primal-dual interior-point methods against dual interiorpoint methods for large scale SDPs (semidefinite programs) has been that the primal positive semidefinite variable matrix becomes fully dense in general even when all data matrices are sparse. Based on some fundamenta ..."
Abstract - Cited by 102 (31 self) - Add to MetaCart
A critical disadvantage of primal-dual interior-point methods against dual interiorpoint methods for large scale SDPs (semidefinite programs) has been that the primal positive semidefinite variable matrix becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite matrix completion, this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage. Our method is used in two ways. One is a conversion of a sparse SDP having a large scale positive semidefinite variable matrix into an SDP having multiple but smaller size positive semidefinite variable matrices to which we can effectively apply any interior-point method for SDPs employing a standard block-diagonal matrix data structure. The other way is an incorporation of our method into primal-dual interior-point methods which we can apply directly to a given SDP. In Part II of this article, we wi...
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Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results

by Kazuhide Nakata, Katsuki Fujisawa, Mituhiro Fukuda, Masakazu Kojima, Kazuo Murota
"... In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about po ..."
Abstract - Cited by 50 (18 self) - Add to MetaCart
In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two di#erent ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over di#erent classes of SDPs show that these methods can be very e#cient for some problems. Keywords: Semidefinite programming; Primal-dual interior-point method; Matrix completion problem; Clique tree; Numerical results. # Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8565 Japan (nakata@zzz.t.u-tokyo.ac.jp ). + Department of Architecture and Architectural Systems, Kyoto University, Kyoto 606-8501 Japan (fujisawa@is-mj.archi.kyoto-u.ac.jp). # Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 OhOkayama, Meguro-ku, Tokyo 152-8552 Japan (mituhiro@is.titech.ac.jp). The author was supported by The Ministry of Education, Culture, Sports, Science and Technology of Japan. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 OhOkayama, Meguro-ku, Toky...
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Consistent bipartite graph co-partitioning for star-structured high-order heterogeneous data co-clustering

by Bin Gao, Tie-yan Liu, Xin Zheng, Qian-sheng Cheng, Wei-ying Ma - KDD , 2005
"... Heterogeneous data co-clustering has attracted more and more attention in recent years due to its high impact on various applications. While the co-clustering algorithms for two types of heterogeneous data (denoted by pair-wise co-clustering), such as documents and terms, have been well studied in t ..."
Abstract - Cited by 50 (2 self) - Add to MetaCart
Heterogeneous data co-clustering has attracted more and more attention in recent years due to its high impact on various applications. While the co-clustering algorithms for two types of heterogeneous data (denoted by pair-wise co-clustering), such as documents and terms, have been well studied in the literature, the work on more types of heterogeneous data (denoted by high-order co-clustering) is still very limited. As an attempt in this direction, in this paper, we worked on a specific case of high-order coclustering in which there is a central type of objects that connects the other types so as to form a star structure of the interrelationships. Actually, this case could be a very good abstract for many real-world applications, such as the co-clustering of categories, documents and terms in text mining. In our philosophy, we treated such kind of problems as the fusion of multiple pairwise co-clustering sub-problems with the constraint of the star structure. Accordingly, we proposed the concept of consistent bipartite graph co-partitioning, and developed an algorithm based on semi-definite programming (SDP) for efficient computation of the clustering results. Experiments on toy problems and real data both verified the effectiveness of our proposed method.
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Randomized Heuristics for the Max-Cut Problem

by P. Festa, P. M. Pardalos, M. G. C. Resende, C. C. Ribeiro - 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. ..."
Abstract - Cited by 41 (16 self) - Add to MetaCart
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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An Independent Benchmarking of SDP and SOCP Solvers

by H. D. Mittelmann - MATHEMATICAL PROGRAMMING
"... ..."
Abstract - Cited by 41 (3 self) - Add to MetaCart
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Implementation of interior point methods for mixed semidefinite and second order cone optimization problems

by Jos F. Sturm - Optimization Methods and Software
"... There is a large number of implementational choices to be made for the primal-dual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different resear ..."
Abstract - Cited by 41 (0 self) - Add to MetaCart
There is a large number of implementational choices to be made for the primal-dual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide an elaborate discussion of the implementation in SeDuMi.
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Implementation and Evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0

by Makoto Yamashita, Katsuki Fujisawa, Masakazu Kojima , 2002
"... Abstract. The SDPA (SemiDefinite Programming Algorithm) is a software package for solving general SDPs (SemiDefinite Programs). It is written in C++ with the help of LAPACK for numerical linear algebra for dense matrix computation. The purpose of this paper is to present a brief description of the l ..."
Abstract - Cited by 36 (12 self) - Add to MetaCart
Abstract. The SDPA (SemiDefinite Programming Algorithm) is a software package for solving general SDPs (SemiDefinite Programs). It is written in C++ with the help of LAPACK for numerical linear algebra for dense matrix computation. The purpose of this paper is to present a brief description of the latest version of the SDPA and its high performance for large scale problems through numerical experiment and comparison with some other major software packages for general SDPs. Key words.
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Semi-Definite Programming for Topology Optimization of Trusses under Multiple Eigenvalue Constraints

by M. Ohsaki, K. Fujisawa, N. Katoh, Y. Kanno - Computer Methods in Applied Mechanics and Engineering , 1999
"... Topology optimization problem of trusses for specified eigenvalue of vibration is formulated as Semi-Definite Programming (SDP), and an algorithm is presented based on the Semi-Definite Programming Algorithm (SDPA) which utilizes extensively the sparseness of the matrices. Since the sensitivity coef ..."
Abstract - Cited by 27 (16 self) - Add to MetaCart
Topology optimization problem of trusses for specified eigenvalue of vibration is formulated as Semi-Definite Programming (SDP), and an algorithm is presented based on the Semi-Definite Programming Algorithm (SDPA) which utilizes extensively the sparseness of the matrices. Since the sensitivity coefficients of the eigenvalues with respect to the design variables are not needed, the SDPA is especially useful for the case where the optimal design has multiple fundamental eigenvalue. Global and local modes are defined and a procedure is presented for generating optimal topology from the practical point of view. It is shown in the examples, that SDPA has advantage over existing methods in view of computational efficiency and accuracy of the solutions, and an optimal topology with five-fold fundamental eigenvalue is found without any difficulty. 1 Introduction The eigenvalues of free vibration as well as the linear buckling load factor are important performance measures of the structures. ...
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