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154
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 238 (12 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 199 (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 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 161 (19 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 121 (3 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.
On the NesterovTodd direction in semidefinite programming
 SIAM Journal on Optimization
, 1996
"... Nesterov and Todd discuss several pathfollowing and potentialreduction interiorpoint methods for certain convex programming problems. In the special case of semidefinite programming, we discuss how to compute the corresponding directions efficiently, how to view them as Newton directions, and how ..."
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Cited by 121 (23 self)
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Nesterov and Todd discuss several pathfollowing and potentialreduction interiorpoint methods for certain convex programming problems. In the special case of semidefinite programming, we discuss how to compute the corresponding directions efficiently, how to view them as Newton directions, and how to take Mehrotra predictorcorrector steps in this framework. We also provide some computational results suggesting that our algorithm is more robust than alternative methods.
SDPA (SemiDefinite Programming Algorithm) User's Manual  Version 7.0.5
, 2008
"... 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 u ..."
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Cited by 100 (30 self)
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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 69 (18 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 68 (25 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...
Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 65 (2 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
On Extending Some PrimalDual InteriorPoint Algorithms From Linear Programming to Semidefinite Programming
 SIAM Journal on Optimization
, 1998
"... This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a search direction originally proposed by HelmbergRendlVanderbeiWolkowicz [5] and KojimaShindohHara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these meth ..."
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Cited by 65 (1 self)
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This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a search direction originally proposed by HelmbergRendlVanderbeiWolkowicz [5] and KojimaShindohHara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [11] and also in [15] through different means and in different forms. In this paper, we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming (LP), producing basic relationships that have compact forms almost identical to their counterparts in LP. We also introduce a new formulation of the central path and variablemetric measures of centrality. These results provide convenient tools for deriving polynomiality results for primaldual algorithms extended from LP to SDP using the aforementioned and related search directions. We present examples...