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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 63 (5 self)
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1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x
Multiple centrality corrections in a primaldual method for linear programming
 Computational Optimization and Applications
, 1996
"... ..."
A Computational View of InteriorPoint Methods for Linear Programming
 IN: ADVANCES IN LINEAR AND INTEGER PROGRAMMING
, 1994
"... Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primaldual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing te ..."
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Cited by 16 (10 self)
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Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primaldual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing techniques, the initialization approaches, the methods of computing search directions (and lying behind them linear algebra techniques), centering strategies and methods of stepsize selection. Several reasons for the manifestations of numerical difficulties like e.g.: the primal degeneracy of optimal solutions or the lack of feasible solutions are explained in a comprehensive way. A motivation for obtaining an optimal basis is given and a practicable algorithm to perform this task is presented. Advantages of different methods to perform postoptimal analysis (applicable to interior point optimal solutions) are discussed. Important questions that still remain open in the implementations of i...
A Short Survey on Ten Years Interior Point Methods
, 1995
"... The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in 80s emphasis was on developing and implementing efficient variants of interior point methods for LP, the nineties have shown applicability ..."
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Cited by 4 (0 self)
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The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in 80s emphasis was on developing and implementing efficient variants of interior point methods for LP, the nineties have shown applicability to certain structured nonlinear programming and combinatorial problems. We will give a historical account of the developments and outline the major contributions to the field in the last decade. An important class of problems to which interior point methods are applicable is semidefinite optimization, which has recently gained much attention. It has a lot of applications in various fields (like control and system theory, combinatorial optimization, algebra, statistics, structural design) and can be efficiently solved with interior point methods.
LongStep PrimalDual TargetFollowing Algorithms for Linear Programming
, 1996
"... In this paper we propose a longstep targetfollowing methodology for linear programming. This is a general framework, that enables us to analyze various longstep primaldual algorithms in the literature in a short and uniform way. Among these are longstep central and weighted pathfollowing metho ..."
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In this paper we propose a longstep targetfollowing methodology for linear programming. This is a general framework, that enables us to analyze various longstep primaldual algorithms in the literature in a short and uniform way. Among these are longstep central and weighted pathfollowing methods and algorithms to compute a central point or a weighted center. Moreover, we use it to analyze a method with the property that it, starting from an initial noncentral point, generates iterates that simultaneously get closer to optimality and closer to centrality.
Experimental Investigations in Combining Primal Dual Interior Point Method and Simplex Based LP Solvers
, 1993
"... 2. Terminal criteria and complementarity in the PD solution ..."
Preconditioned Conjugate Gradients in an Interior Point Method for Twostage Stochastic Programming ∗
, 1994
"... We develop a variant of an interior point method for solving twostage stochastic linear programming problems. The problems are solved in a deterministic equivalent form in which the first stage variables appear as dense columns. To avoid their degrading influence on the adjacency structure AA T (an ..."
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We develop a variant of an interior point method for solving twostage stochastic linear programming problems. The problems are solved in a deterministic equivalent form in which the first stage variables appear as dense columns. To avoid their degrading influence on the adjacency structure AA T (and the Cholesky factor) an iterative method is applied to compute orthogonal projections. Conjugate gradient algorithm with a structureexploiting preconditioner is used. The method has been applied to solve real–life stochastic optimization problems. Preliminary computational results show the feasibility of the approach for problems with up to 80 independent scenarios (a deterministic equivalent linear program has 14001 constraints and 63690 variables). Key words: interior point method, twostage stochastic programs, conjugate gradient algorithm.