Results 1  10
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50
LOQO: An interior point code for quadratic programming
, 1994
"... ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex ..."
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Cited by 156 (9 self)
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ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1.
Implementation of Interior Point Methods for Large Scale Linear Programming
 in Interior Point Methods in Mathematical Programming
, 1996
"... In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on bot ..."
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Cited by 70 (22 self)
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In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on both sides. The significant difference between interior point and simplex based methods is reflected not only in the theoretical background but also in the practical implementation. In this paper we give an overview of the most important characteristics of advanced implementations of interior point methods. First, we present the infeasibleprimaldual algorithm which is widely considered the most efficient general purpose IPM. Our discussion includes various algorithmic enhancements of the basic algorithm. The only shortcoming of the "traditional" infeasibleprimaldual algorithm is to detect a possible primal or dual infeasibility of the linear program. We discuss how this problem can be solve...
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 60 (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 simplified homogeneous and selfdual linear programming algorithm and its implementation
 Annals of Operations Research
, 1996
"... 1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x ..."
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Cited by 56 (5 self)
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1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x
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 55 (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...
On Extending PrimalDual InteriorPoint Algorithms from Linear Programming to Semidefinite Programming
, 1995
"... This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a linearized complementarity equation originally proposed by Kojima, Shindoh and Hara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these methods, a number of ba ..."
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Cited by 47 (0 self)
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This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a linearized complementarity equation originally proposed by Kojima, Shindoh and Hara [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 very short derivation of the key equalities and inequalities along the exact line used in linear programming (LP), producing basic relationships that have highly compact forms almost identical to their counterparts in LP. We also introduce a new definition of the central path and variablemetric measures of centrality. These results provide convenient tools for extending existing polynomiality results for many, if not most, algorithms from LP to SDP with little complication. We present examples of such extensions, including the longstep infeasible...
A QMRbased interiorpoint algorithm for solving linear programs
 Math. Programming
, 1994
"... A new approach for the implementation of interiorpoint methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2\Theta2block systems of linear equations that arise within the interiorpoint algorithm. These linear systems ..."
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Cited by 39 (4 self)
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A new approach for the implementation of interiorpoint methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2\Theta2block systems of linear equations that arise within the interiorpoint algorithm. These linear systems are solved by a symmetric variant of the quasiminimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3 \Theta 3block systems to symmetric 2 \Theta 2block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interiorpoint algorithm is proposed. Some indefini...
A Truncated PrimalInfeasible DualFeasible Network Interior Point Method
, 1994
"... . In this paper we introduce the truncated primalinfeasible dualfeasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is ..."
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Cited by 29 (3 self)
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. In this paper we introduce the truncated primalinfeasible dualfeasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria of the iterative solver employed for the solution of the system. In the implementation, a preconditioned conjugate gradient method is used as the iterative solver. The details of the implementation are described and the code, pdnet, is tested on a large set of standard minimum cost network flow test problems. Computational results indicate that the implementation is competitive with stateoftheart network flow codes. Key Words. Interior point method, linear programming, network flows, primalinfeasible dualfeasible, truncated Newton method, conjugate gradient, maximum flow, experimental test...
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 and compleme ..."
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Cited by 24 (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...
InfeasibleInteriorPoint PrimalDual PotentialReduction Algorithms For Linear Programming
 SIAM Journal on Optimization
, 1995
"... . In this paper, we propose primaldual potentialreduction algorithms which can start from an infeasible interior point. We first describe two such algorithms and show that both are polynomialtime bounded. One of the algorithms decreases the TanabeToddYe primaldual potential function by a const ..."
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Cited by 20 (4 self)
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. In this paper, we propose primaldual potentialreduction algorithms which can start from an infeasible interior point. We first describe two such algorithms and show that both are polynomialtime bounded. One of the algorithms decreases the TanabeToddYe primaldual potential function by a constant at each iteration under the condition that the duality gap decreases by at most the same ratio as the infeasibility. The other reduces a new potential function, which has one more term in the TanabeToddYe potential function, by a fixed constant at each iteration without any other conditions on the step size. Finally, we describe modifications of these methods (incorporating centering steps) which dramatically decrease their computational complexity. Our algorithms also extend to the case of monotone linear complementarity problems. Key words. Polynomial Time, Linear Programming, PrimalDual, InfeasibleInteriorPoint Algorithm, Potential Function. AMS subject classifications. 90C05, ...