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94
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
 SIAM Journal on Optimization
, 1993
"... We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to S ..."
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Cited by 471 (11 self)
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We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to SDP. Next we present an interior point algorithm which converges to the optimal solution in polynomial time. The approach is a direct extension of Ye's projective method for linear programming. We also argue that most known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity also carrying over in a similar fashion. Finally we study the significance of these results in a variety of combinatorial optimization problems including the general 01 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k partite subgraph problem in graphs, and va...
SDPT3  a MATLAB software package for semidefinite programming
 OPTIMIZATION METHODS AND SOFTWARE
, 1999
"... This software package is a Matlab implementation of infeasible pathfollowing algorithms for solving standard semidefinite programming (SDP) problems. Mehrotratype predictorcorrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP problem are also imp ..."
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Cited by 214 (11 self)
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This software package is a Matlab implementation of infeasible pathfollowing algorithms for solving standard semidefinite programming (SDP) problems. Mehrotratype predictorcorrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP problem are also implemented. Four types of search directions are available, namely, the AHO, HKM, NT, and GT directions. A few classes of SDP problems are included as well. Numerical results for these classes show that our algorithms are fairly efficient and robust on problems with dimensions of the order of a few hundreds.
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 181 (34 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 semidefinitequadraticlinear programs using SDPT3
 MATHEMATICAL PROGRAMMING
, 2003
"... This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primaldual pathfollowing algorithm ..."
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Cited by 139 (18 self)
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This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primaldual pathfollowing algorithms. The software developed by the authors uses Mehrotratype predictorcorrector variants of interiorpoint methods and two types of search directions: the HKM and NT directions. A discussion of implementation details is provided and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported.
Semidefinite optimization
 Acta Numerica
, 2001
"... Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the ..."
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Cited by 103 (2 self)
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Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or ” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.
SDPA (Semidefinite Programming Algorithm)  User's Manual
, 1995
"... Abstract. The SDPA (SemiDefinite Programming Algorithm) [5] is a software package for solving semidefinite programs (SDPs). It is based on a Mehrotratype predictorcorrector infeasible primaldual interiorpoint method. The SDPA handles the standard form SDP and its dual. It is implemented in C++ l ..."
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Cited by 94 (28 self)
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Abstract. The SDPA (SemiDefinite Programming Algorithm) [5] is a software package for solving semidefinite programs (SDPs). It is based on a Mehrotratype predictorcorrector infeasible primaldual interiorpoint method. The SDPA handles the standard form SDP and its dual. It is implemented in C++ language utilizing the LAPACK [1] for matrix computations. The SDPA version 7.0.5 enjoys the following features: • Efficient method for computing the search directions when the SDP to be solved is large scale and sparse [4]. • Block diagonal matrix structure and sparse matrix structure are supported for data matrices. • Sparse or dense Cholesky factorization for the Schur matrix is automatically selected. • An initial point can be specified. • Some information on infeasibility of the SDP is provided. This manual and the SDPA can be downloaded from the WWW site
Exploiting Sparsity in PrimalDual InteriorPoint Methods for Semidefinite Programming
 Mathematical Programming
, 1997
"... Abstract. The HelmbergRendlVanderbeiWolkowicz/KojimaShindohHara/Monteiro and the NesterovTodd search directions have been used in many primaldual interiorpoint methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when a semidefinite prog ..."
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Cited by 66 (19 self)
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Abstract. The HelmbergRendlVanderbeiWolkowicz/KojimaShindohHara/Monteiro and the NesterovTodd search directions have been used in many primaldual interiorpoint methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when a semidefinite program to be solved is large scale and sparse.
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 61 (27 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...
A superlinearly convergent predictorcorrector method for degenerate LCP in a wide neighborhood of the central path with O (√n L)iteration complexity
, 2006
"... ..."
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 30 (11 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...