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11
X.: Implementation of interior point methods for large scale linear programming
 Interior Point Methods in Mathematical Programming. Kluwer Acad Pub
, 1996
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Initialization in Semidefinite Programming Via a SelfDual SkewSymmetric Embedding
, 1996
"... The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for largescale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but ..."
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Cited by 46 (12 self)
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The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for largescale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but the initialization problem remained unsolved. In this paper we show that the initialization strategy of embedding the problem in a selfdual skewsymmetric problem can also be extended to the semidefinite case. This way the initialization problem of semidefinite problems is solved. This method also provides solution for the initialization of quadratic programs and it is applicable to more general convex problems with conic formulation.
PrimalDual TargetFollowing Algorithms for Linear Programming
 ANNALS OF OPERATIONS RESEARCH
, 1993
"... In this paper we propose a method for linear programming with the property that, starting from an initial noncentral point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Al ..."
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Cited by 26 (1 self)
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In this paper we propose a method for linear programming with the property that, starting from an initial noncentral point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Along with the convergence analysis we provide a general framework which enables us to analyze various primaldual algorithms in the literature in a short and uniform way.
Duality And SelfDuality For Conic Convex Programming
, 1996
"... This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we i ..."
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Cited by 22 (8 self)
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This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primaldual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a selfdual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primaldual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, ...
Combining InteriorPoint and Pivoting Algorithms for Linear Programming
 Management Science
, 1996
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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...
LongStep PrimalDual TargetFollowing Algorithms for Linear Programming
, 1995
"... In this paper we propose a long{step target{following methodology for linear programming. This is a general framework, that enables us to analyze various long{step primal{dual algorithms in the literature in a short and uniform way. Among these are long{step central and weighted path{following metho ..."
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Cited by 8 (0 self)
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In this paper we propose a long{step target{following methodology for linear programming. This is a general framework, that enables us to analyze various long{step primal{dual algorithms in the literature in a short and uniform way. Among these are long{step central and weighted path{following 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 non{ central point, generates iterates that simultaneously get closer to optimality and closer to centrality. Key words: interiorpoint method, ane scaling method, primal{dual method, long{step method. Running title: Target{Following Methods for LP. This work is completed with the support of a research grant from SHELL. The rst author is supported by the Dutch Organization for Scientic Research (NWO), grant 611304028. The fourth author is supported by the Swiss National Foundation for Scientic Research, grant 1234002.92. ii 1...
Potential reduction algorithms
 Interior Point Methods in Mathematical Programming
, 1996
"... Potential reduction algorithms have a distinguished role in the area of interior point methods for mathematical programming. Karmarkar’s [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a ..."
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Cited by 8 (0 self)
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Potential reduction algorithms have a distinguished role in the area of interior point methods for mathematical programming. Karmarkar’s [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a
Infeasible Start Semidefinite Programming Algorithms Via SelfDual Embeddings
, 1997
"... The development of algorithms for semidefinite programming is an active research area, based on extensions of interior point methods for linear programming. As semidefinite programming duality theory is weaker than that of linear programming, only partial information can be obtained in some cases of ..."
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Cited by 5 (0 self)
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The development of algorithms for semidefinite programming is an active research area, based on extensions of interior point methods for linear programming. As semidefinite programming duality theory is weaker than that of linear programming, only partial information can be obtained in some cases of infeasibility, nonzero optimal duality gaps, etc. Infeasible start algorithms have been proposed which yield different kinds of information about the solution. In this paper a comprehensive treatment of a specific initialization strategy is presented, namely selfdual embedding, where the original primal and dual problems are embedded in a larger problem with a known interior feasible starting point. A framework for infeasible start algorithms with the best obtainable complexity bound is thus presented. The information that can be obtained in case of infeasibility, unboundedness, etc., is stated clearly. Important unresolved issues are discussed.
An Easy Way to Teach Interior Point Methods
, 1998
"... In this paper the duality theory of Linear Optimization (LO) is built up based on ideas emerged from interior point methods. All we need is elementary calculus. We will embed the LO problem and its dual in a selfdual skewsymmetric problem. Most duality results, except the existence of a strictly ..."
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Cited by 3 (0 self)
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In this paper the duality theory of Linear Optimization (LO) is built up based on ideas emerged from interior point methods. All we need is elementary calculus. We will embed the LO problem and its dual in a selfdual skewsymmetric problem. Most duality results, except the existence of a strictly complementary solution, are trivial for this embedding problem. The existence of the central path and its convergence to the analytic center of the optimal face will be proved. The proof is based on an elementary, careful analysis of a Newton step. We show also that if an almost optimal solution on the central path is known, then a simple strongly polynomial rounding procedure provides a strictly complementary optimal solution. The allone vector is feasible for the embedding problem and it is an interior point on the central path. This way an elegant solution to the initialization of IPMs is obtained as well. This approach allows to apply any interior point method to the embedding problem ...