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25
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 180 (30 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
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.
Interior methods for nonlinear optimization
 SIAM Review
, 2002
"... Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their ..."
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Cited by 76 (4 self)
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Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar’s widely publicized announcement in 1984 of a fast polynomialtime interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.
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...
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...
On the implementation of an algorithm for largescale equality constrained optimization
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques ..."
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Cited by 38 (11 self)
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Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques for solving the subproblems occurring in the algorithm. Second derivative information can be used, but when it is not available, limited memory quasiNewton approximations are made. The performance of the code is studied using a set of difficult test problems from the CUTE collection.
Presolve Analysis of Linear Programs Prior to Applying an Interior Point Method
 INFORMS Journal on Computing
, 1994
"... Several issues concerning an analysis of large and sparse linear programming problems prior to solving them with an interior point based optimizer are addressed in this paper. Three types of presolve procedures are distinguished. Routines from the first class repeatedly analyze an LP problem formula ..."
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Cited by 34 (6 self)
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Several issues concerning an analysis of large and sparse linear programming problems prior to solving them with an interior point based optimizer are addressed in this paper. Three types of presolve procedures are distinguished. Routines from the first class repeatedly analyze an LP problem formulation: eliminate empty or singleton rows and columns, look for primal and dual forcing or dominated constraints, tighten bounds for variables and shadow prices or just the opposite, relax them to find implied free variables. The second type of analysis aims at reducing a fillin of the Cholesky factor of the normal equations matrix used to compute orthogonal projections and includes a heuristic for increasing the sparsity of the LP constraint matrix and a technique of splitting dense columns in it. Finally, routines from the third class detect, and remove, different linear dependecies of rows and columns in a constraint matrix. Computational results on problems from the Netlib collection, inc...
Regularized Symmetric Indefinite Systems in Interior Point Methods for Linear and Quadratic Optimization
 Optimization Methods and Software
, 1998
"... This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefi ..."
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Cited by 30 (11 self)
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This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidefinite systems known to be strongly factorizable to a form of Choleskylike factorization. Two different regularization techniques, primal and dual, are very well suited to the (infeasible) primaldual interior point algorithm. This particular algorithm, with an extension of multiple centrality correctors, is implemented in our solver HOPDM. Computational results are given to illustrate the potential advantages of the approach when applied to the solution of very large linear and convex quadratic programs. Keywords: Linear programming, convex quadratic programming, symmetric quasidefinite systems, primaldual regularization, pri...
APPLYING NEW OPTIMIZATION ALGORITHMS TO MODEL PREDICTIVE CONTROL
"... The connections between optimization and control theory have been explored by many researchers, and optimization algorithms have been applied with success to optimal control. The rapid pace of developments in model predictive control has given rise to a host of new problems to which optimization has ..."
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Cited by 27 (1 self)
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The connections between optimization and control theory have been explored by many researchers, and optimization algorithms have been applied with success to optimal control. The rapid pace of developments in model predictive control has given rise to a host of new problems to which optimization has yet to be applied. Concurrently, developments in optimization, and especially in interiorpoint methods, have produced a new set of algorithms that may be especially helpful in this context. In this paper, we reexamine the relatively simple problem of control of linear processes subject to quadratic objectives and general linear constraints. We show how new algorithms for quadratic programming can be applied efficiently to this problem. The approach extends to several more general problems in straightforward ways.
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...