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52
A Cutting Plane Method from Analytic Centers for Stochastic Programming
 Mathematical Programming
, 1994
"... The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by DantzigWolfe decomposition ..."
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Cited by 49 (18 self)
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The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by DantzigWolfe decomposition the two approaches are in fact identical by duality. Here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations. The only significant difference form the aforementioned methods is that new cutting planes (or columns, by duality) will be generated not from the optimum of the linear programming relaxation, but from the analytic center of the set of localization. 1 Introduction The study of optimization problems in the presence of uncertainty still taxes the limits of methodology and software. One of the most approachable settings is that of twostaged planning under uncertainty, in which a first stage decision has to be taken bef...
High Resolution Pursuit for Feature Extraction
 Applied and Computational Harmonic Analysis
, 1995
"... Recently, adaptive approximation techniques have become popular for obtaining parsimonious representations of large classes of signals. These methods include method of frames, matching pursuit, and, most recently, basis pursuit. In this work, high resolution pursuit (HRP) is developed as an alternat ..."
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Cited by 24 (3 self)
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Recently, adaptive approximation techniques have become popular for obtaining parsimonious representations of large classes of signals. These methods include method of frames, matching pursuit, and, most recently, basis pursuit. In this work, high resolution pursuit (HRP) is developed as an alternative to existing function approximation techniques. Existing techniques do not always efficiently yield representations which are sparse and physically interpretable. HRP is an enhanced version of the matching pursuit algorithm and overcomes the shortcomings of the traditional matching pursuit algorithm by emphasizing local fit over global fit at each stage. Further, the HRP algorithm has the same order of complexity as matching pursuit. In this paper, the HRP algorithm is developed and demonstrated on 1D functions. Convergence properties of HRP are also examined. HRP is also suitable for extracting features which may then be used in recognition. 1 Introduction Recently, adaptive approxima...
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 24 (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.
Solving RealWorld Linear Ordering Problems . . .
, 1995
"... Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear prog ..."
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Cited by 22 (8 self)
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Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear programming relaxations. A point which is a good warm start for a simplexbased cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some realworld problems; the algorithm appears to be competitive with a simplexbased cutting plane algorithm.
Why a Pure Primal Newton Barrier Step May Be Infeasible
 SIAM Journal on Optimization
, 1993
"... Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when ..."
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Cited by 21 (3 self)
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Modern barrier methods for constrained optimization are sometimes portrayed conceptually as a sequence of inexact minimizations, with only a very few Newton iterations (perhaps just one) for each value of the barrier parameter. Unfortunately, this rosy image does not accurately reflect reality when the barrier parameter is reduced at a reasonable rate. We present local analysis showing why a pure Newton step in a longstep barrier method for nonlinearly constrained optimization may be seriously infeasible, even when taken from an apparently favorable point. The features described are illustrated numerically and connected to known theoretical results for convex problems satisfying selfconcordancy assumptions. We also indicate the contrasting nature of an approximate step to the desired minimizer of the barrier function. 1. Introduction 1.1. Background Interior methods, most commonly based on barrier functions, have been applied with great practical success in recent years to many con...
A polynomial primaldual Dikintype algorithm for linear programming
 FACULTY OF TECHNICAL MATHEMATICS AND COMPUTER SCIENCE, DELFT UNIVERSITY OF TECHNOLOGY
, 1993
"... In this paper we present a new primaldual affine scaling method for linear programming. The method yields a strictly complementary optimal solution pair, and also allows a polynomialtime convergence proof. The search direction is obtained by using the original idea of Dikin, namely by minimizing t ..."
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Cited by 16 (9 self)
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In this paper we present a new primaldual affine scaling method for linear programming. The method yields a strictly complementary optimal solution pair, and also allows a polynomialtime convergence proof. The search direction is obtained by using the original idea of Dikin, namely by minimizing the objective function (which is the duality gap in the primaldual case), over some suitable ellipsoid. This gives rise to completely new primaldual affine scaling directions, having no obvious relation with the search directions proposed in the literature so far. The new directions guarantee a significant decrease in the duality gap in each iteration, and at the same time they drive the iterates to the central path. In the analysis of our algorithm we use a barrier function which is the natural primaldual generalization of Karmarkar's potential function. The iteration bound is O(nL), which is a factor O(L) better than the iteration bound of an earlier primaldual affine scaling meth...
INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION
, 1995
"... Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivale ..."
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Cited by 14 (9 self)
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Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.
Polynomiality of PrimalDual Affine Scaling Algorithms for Nonlinear Complementarity Problems
, 1995
"... This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappi ..."
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Cited by 11 (4 self)
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This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primaldual affine scaling algorithms generates an approximate solution (given a precision ffl) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial of n, ln(1=ffl) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in [13].
Logarithmic Barrier Decomposition Methods for SemiInfinite Programming
, 1996
"... A computational study of some logarithmic barrier decomposition algorithms for semiinfinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmic barrier cutting plane algorithm which was presented recently by den Hertog et al., to solv ..."
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Cited by 9 (1 self)
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A computational study of some logarithmic barrier decomposition algorithms for semiinfinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmic barrier cutting plane algorithm which was presented recently by den Hertog et al., to solve semiinfinite programming problems. Usually decomposition (cutting plane methods) use cutting planes to improve the localization of the given problem. In this paper we propose an extension which uses linear cuts to solve large scale, difficult real world problems. This algorithm uses both static and (doubly) dynamic enumeration of the parameter space and allows for multiple cuts to be simultaneously added for larger/difficult problems. The algorithm is implemented both on sequential and parallel computers. Implementation issues and parallelization strategies are discussed and encouraging computational results are presented. Keywords: column generation, convex programming, cutting plane met...