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108
Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones
, 1998
"... SeDuMi is an addon for MATLAB, that lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This pape ..."
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Cited by 1334 (4 self)
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SeDuMi is an addon for MATLAB, that lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.
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 362 (17 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.
A Rank Minimization Heuristic with Application to Minimum Order System Approximation
, 2001
"... Several problems arising in control system analysis and design, such as reduced order controller synthesis, involve minimizing the rank of a matrix variable subject to linear matrix inequality (LMI) constraints. Except in some special cases, solving this rank minimization probiem (globally) is ve ..."
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Cited by 269 (10 self)
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Several problems arising in control system analysis and design, such as reduced order controller synthesis, involve minimizing the rank of a matrix variable subject to linear matrix inequality (LMI) constraints. Except in some special cases, solving this rank minimization probiem (globally) is very difficult. One simple and surprisingly effective heuristic, applicable when the matrix variable is symmetric and positive semidefinite, is to minimize its trace in place of its rank. This results in a semidefinite program (SDP) which can be efficiently solved. In this paper we describe a generalization of the trace heuristic that applies to general nonsymmetric, even nonsquare, matrices, and reduces to the trace heuristic when the matrix is positive selinidefinite. The heuristic is to replace the (nonconvex) rank objective with the sum of the singular values of the matrix, which is the dual of the spectral norm. We show that this problem can be reduced to an SDP, hence efficiently solved. To motivate the heuristic, we show that the dual spectral norm is ^ the convex envelope of the rank on the set of matrices with norm less than one. We demonstrate the method on the problem of minimum order system approximation.
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 233 (22 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.
CSDP, a C library for semidefinite programming.
, 1997
"... this paper is organized as follows. First, we discuss the formulation of the semidefinite programming problem used by CSDP. We then describe the predictor corrector algorithm used by CSDP to solve the SDP. We discuss the storage requirements of the algorithm as well as its computational complexity. ..."
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Cited by 209 (1 self)
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this paper is organized as follows. First, we discuss the formulation of the semidefinite programming problem used by CSDP. We then describe the predictor corrector algorithm used by CSDP to solve the SDP. We discuss the storage requirements of the algorithm as well as its computational complexity. Finally, we present results from the solution of a number of test problems. 2 The SDP Problem We consider semidefinite programming problems of the form max tr (CX)
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 104 (30 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 89 (3 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
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 80 (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.
The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem
 SIAM REVIEW
, 2006
"... We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ2 of the Laplacian of the weighted graph. I ..."
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Cited by 67 (5 self)
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We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ2 of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize λ2 subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, i.e., the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for “unfolding ” highdimensional data that lies on a lowdimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.
Local Convergence of PredictorCorrector InfeasibleInteriorPoint Algorithms for SDPs and SDLCPs
 Mathematical Programming
, 1997
"... . An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the MizunoToddYe type predictorcorrector primaldual interiorpoint method for LPs (linear programs) to SDPs, and suggests that we need to force the genera ..."
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Cited by 60 (4 self)
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. An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the MizunoToddYe type predictorcorrector primaldual interiorpoint method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A MizunoToddYe type predictorcorrector infeasibleinteriorpoint algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. Key words. Semidefinite Programming, InfeasibleInteriorPoint Method, PredictorCorrectorMethod, Superlinear Convergence, PrimalDual Nondegeneracy Abbreviated Title. InteriorPoint Algorithms for SDPs y Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 152, Japa...