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36
Monotonicity of primaldual interiorpoint algorithms for semidefinite programming problems
, 1998
"... We present primaldual interiorpoint 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 ..."
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Cited by 216 (35 self)
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We present primaldual interiorpoint 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 LargeScale Sparse Semidefinite Programs for Combinatorial Optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1998
"... We present a dualscaling interiorpoint 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 dualscaling interiorpoint 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 interiorpoint algorithms for approximating the maximum cut semidefinite programs with dimension upto 3000.
Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework
 SIAM JOURNAL ON OPTIMIZATION
, 1999
"... A critical disadvantage of primaldual interiorpoint 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 ..."
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Cited by 102 (31 self)
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A critical disadvantage of primaldual interiorpoint 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 interiorpoint method for SDPs employing a standard blockdiagonal matrix data structure. The other way is an incorporation of our method into primaldual interiorpoint methods which we can apply directly to a given SDP. In Part II of this article, we wi...
Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results
"... 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 primaldual interiorpoint methods. This framework is based on some results about po ..."
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Cited by 50 (18 self)
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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 primaldual interiorpoint 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 primaldual interiorpoint 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; Primaldual interiorpoint method; Matrix completion problem; Clique tree; Numerical results. # Department of Applied Physics, The University of Tokyo, 731 Hongo, Bunkyoku, Tokyo 1138565 Japan (nakata@zzz.t.utokyo.ac.jp ). + Department of Architecture and Architectural Systems, Kyoto University, Kyoto 6068501 Japan (fujisawa@ismj.archi.kyotou.ac.jp). # Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 1528552 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, 2121 OhOkayama, Meguroku, Toky...
Consistent bipartite graph copartitioning for starstructured highorder heterogeneous data coclustering
 KDD
, 2005
"... Heterogeneous data coclustering has attracted more and more attention in recent years due to its high impact on various applications. While the coclustering algorithms for two types of heterogeneous data (denoted by pairwise coclustering), such as documents and terms, have been well studied in t ..."
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Cited by 50 (2 self)
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Heterogeneous data coclustering has attracted more and more attention in recent years due to its high impact on various applications. While the coclustering algorithms for two types of heterogeneous data (denoted by pairwise coclustering), such as documents and terms, have been well studied in the literature, the work on more types of heterogeneous data (denoted by highorder coclustering) is still very limited. As an attempt in this direction, in this paper, we worked on a specific case of highorder 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 realworld applications, such as the coclustering of categories, documents and terms in text mining. In our philosophy, we treated such kind of problems as the fusion of multiple pairwise coclustering subproblems with the constraint of the star structure. Accordingly, we proposed the concept of consistent bipartite graph copartitioning, and developed an algorithm based on semidefinite programming (SDP) for efficient computation of the clustering results. Experiments on toy problems and real data both verified the effectiveness of our proposed method.
Randomized Heuristics for the MaxCut Problem
 Optimization Methods and Software
, 2002
"... Given an undirected graph with edge weights, the MAXCUT 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 MAXCUT 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.
Implementation of interior point methods for mixed semidefinite and second order cone optimization problems
 Optimization Methods and Software
"... There is a large number of implementational choices to be made for the primaldual 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 ..."
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Cited by 41 (0 self)
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There is a large number of implementational choices to be made for the primaldual 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.
Implementation and Evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0
, 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 ..."
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Cited by 36 (12 self)
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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.
SemiDefinite Programming for Topology Optimization of Trusses under Multiple Eigenvalue Constraints
 Computer Methods in Applied Mechanics and Engineering
, 1999
"... Topology optimization problem of trusses for specified eigenvalue of vibration is formulated as SemiDefinite Programming (SDP), and an algorithm is presented based on the SemiDefinite Programming Algorithm (SDPA) which utilizes extensively the sparseness of the matrices. Since the sensitivity coef ..."
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Cited by 27 (16 self)
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Topology optimization problem of trusses for specified eigenvalue of vibration is formulated as SemiDefinite Programming (SDP), and an algorithm is presented based on the SemiDefinite 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 fivefold 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. ...