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57
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 221 (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.
Linear Programming: Foundations and Extensions
, 1996
"... under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. ISBN 0000000000 The text for this book was formated in Time ..."
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Cited by 148 (0 self)
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under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. ISBN 0000000000 The text for this book was formated in TimesRoman and the mathematics was formated in Michael Spivak’s Mathtimes using AMSL ATEX(which is a macro package for Leslie Lamport’s L ATEX, which itself is a macro package for Donald Knuth’s TEXtext formatting system) and converted from deviceindependent to postscript format using DVIPS. The figures were produced using SHOWCASE on a Silicon Graphics, Inc. workstation and were incorporated into the text as encapsulated postscript files with the macro package called PSFIG.TEX. To my parents, Howard and Marilyn, my dear wife, Krisadee, and the babes, Marisa and Diana Contents
Implementation of interior point methods for large scale linear programming
 Interior point methods in mathematical programming
, 1996
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Solving LargeScale Linear Programs by InteriorPoint Methods Under the MATLAB Environment
 Optimization Methods and Software
, 1996
"... In this paper, we describe our implementation of a primaldual infeasibleinteriorpoint algorithm for largescale linear programming under the MATLAB 1 environment. The resulting software is called LIPSOL  Linearprogramming InteriorPoint SOLvers. LIPSOL is designed to take the advantages of M ..."
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Cited by 62 (2 self)
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In this paper, we describe our implementation of a primaldual infeasibleinteriorpoint algorithm for largescale linear programming under the MATLAB 1 environment. The resulting software is called LIPSOL  Linearprogramming InteriorPoint SOLvers. LIPSOL is designed to take the advantages of MATLAB's sparsematrix functions and external interface facilities, and of existing Fortran sparse Cholesky codes. Under the MATLAB environment, LIPSOL inherits a high degree of simplicity and versatility in comparison to its counterparts in Fortran or C language. More importantly, our extensive computational results demonstrate that LIPSOL also attains an impressive performance comparable with that of efficient Fortran or C codes in solving largescale problems. In addition, we discuss in detail a technique for overcoming numerical instability in Cholesky factorization at the endstage of iterations in interiorpoint algorithms. Keywords: Linear programming, PrimalDual infeasibleinteriorp...
A QMRbased interiorpoint algorithm for solving linear programs”, AT&T Bell Laboratories and Institute für Angewandte Mathematik und Statistik
, 1995
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On implementing a primaldual interiorpoint method for conic quadratic optimization
 MATHEMATICAL PROGRAMMING SER. B
, 2000
"... Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linea ..."
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Cited by 36 (4 self)
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Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case. Conic quadratic optimization problems can in theory be solved efficiently using interiorpoint methods. In particular it has been shown by Nesterov and Todd that primaldual interiorpoint methods developed for linear optimization can be generalized to the conic quadratic case while maintaining their efficiency. Therefore, based on the work of Nesterov and Todd, we discuss an implementation of a primaldual interiorpoint method for solution of largescale sparse conic quadratic optimization problems. The main features of the implementation are it is based on a homogeneous and selfdual model, handles the rotated quadratic cone directly, employs a Mehrotra type predictorcorrector
Presolve analysis of linear programs prior to applying an interior point method
 INFORMS Journal on Computing
, 1997
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Homogeneous InteriorPoint Algorithms for Semidefinite Programming
 Department of Mathematics, The University of Iowa
, 1995
"... A simple homogeneous primaldual feasibility model is proposed for semidefinite programming (SDP) problems. Two infeasibleinteriorpoint algorithms are applied to the homogeneous formulation. The algorithms do not need big M initialization. If the original SDP problem has a solution, then both algo ..."
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Cited by 34 (8 self)
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A simple homogeneous primaldual feasibility model is proposed for semidefinite programming (SDP) problems. Two infeasibleinteriorpoint algorithms are applied to the homogeneous formulation. The algorithms do not need big M initialization. If the original SDP problem has a solution, then both algorithms find an fflapproximate solution (i.e., a solution with residual error less than or equal to ffl) in at most O( p n ln(ae ffl 0 =ffl)) steps, where ae is the trace norm of a solution and ffl 0 is the residual error at the (normalized) starting point. A simple way of monitoring possible infeasibility of the original SDP problem is provided such that in at most O( p n ln(aeffl 0 =ffl)) steps either an fflapproximate solution is obtained, or it is determined that there is no solution with trace norm less than or equal to a given number ae ? 0. Key Words: semidefinite programming, homogeneous interiorpoint algorithm, polynomial complexity. Abbreviated Title: Homogeneous al...
On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility an ..."
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Cited by 26 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
Modified Cholesky Factorizations In InteriorPoint Algorithms For Linear Programming
 SIAM Journal on Optimization
"... . We investigate a modified Cholesky algorithm typical of those used in most interiorpoint codes for linear programming. Choleskybased interiorpoint codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky algorithms (allowing us to take f ..."
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Cited by 25 (3 self)
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. We investigate a modified Cholesky algorithm typical of those used in most interiorpoint codes for linear programming. Choleskybased interiorpoint codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky algorithms (allowing us to take full advantage of software written by specialists in that area); they tend to be more efficient than competing approaches that use alternative factorizations; and they perform robustly on most practical problems, yielding good interiorpoint steps even when the coefficient matrix of the main linear system to be solved for the step components is illconditioned. We investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis, illustrating our results with computational experiments. Finally, we point out the potential limitations of this approach. Key words. interiorpoint algorithms and software, Cholesky factorization, matrix p...