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An interior-point piecewise linear penalty method for nonlinear programming

We present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an ℓ 2 -penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barr... Full description

Journal Title: Mathematical programming 2009-07-14, Vol.128 (1-2), p.73-122
Main Author: Chen, Lifeng
Other Authors: Goldfarb, Donald
Format: Electronic Article Electronic Article
Language: English
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Publisher: Berlin/Heidelberg: Springer-Verlag
ID: ISSN: 0025-5610
Link: http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=24208784
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title: An interior-point piecewise linear penalty method for nonlinear programming
format: Article
creator:
  • Chen, Lifeng
  • Goldfarb, Donald
subjects:
  • Applied sciences
  • Barriers
  • Calculus of variations and optimal control
  • Calculus of Variations and Optimal Control
  • Optimization
  • Combinatorics
  • Convergence
  • Eigen values
  • Exact sciences and technology
  • Experimental design
  • Full Length Paper
  • Iterative methods
  • Mathematical analysis
  • Mathematical and Computational Physics
  • Mathematical Methods in Physics
  • Mathematical models
  • Mathematical programming
  • Mathematics
  • Mathematics and Statistics
  • Mathematics of Computing
  • Methods
  • Nonlinear programming
  • Nonlinearity
  • Numerical Analysis
  • Operational research and scientific management
  • Operational research. Management science
  • Penalty function
  • Probability and statistics
  • Sciences and techniques of general use
  • Statistics
  • Studies
  • Theoretical
ispartof: Mathematical programming, 2009-07-14, Vol.128 (1-2), p.73-122
description: We present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an ℓ 2 -penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barrier function and the infeasibility measure at a subset of previous iterates and this set is updated at every iteration. The ℓ 2 -penalty function is a traditional penalty function defined by a single penalty parameter. At every iteration the step direction is computed from a regularized Newton system of the first-order equations of the barrier problem proposed in Chen and Goldfarb (Math Program 108:1–36, 2006). Iterates are updated using a line search. In particular, a trial point is accepted if it provides a sufficient reduction in either of the penalty functions. We show that the proposed method has the same strong global convergence properties as those established in Chen and Goldfarb (Math Program 108:1–36, 2006). Moreover, our method enjoys fast local convergence. Specifically, for each fixed small barrier parameter  μ , iterates in a small neighborhood (roughly within o ( μ )) of the minimizer of the barrier problem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear.
language: eng
source:
identifier: ISSN: 0025-5610
fulltext: no_fulltext
issn:
  • 0025-5610
  • 1436-4646
url: Link


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descriptionWe present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an ℓ 2 -penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barrier function and the infeasibility measure at a subset of previous iterates and this set is updated at every iteration. The ℓ 2 -penalty function is a traditional penalty function defined by a single penalty parameter. At every iteration the step direction is computed from a regularized Newton system of the first-order equations of the barrier problem proposed in Chen and Goldfarb (Math Program 108:1–36, 2006). Iterates are updated using a line search. In particular, a trial point is accepted if it provides a sufficient reduction in either of the penalty functions. We show that the proposed method has the same strong global convergence properties as those established in Chen and Goldfarb (Math Program 108:1–36, 2006). Moreover, our method enjoys fast local convergence. Specifically, for each fixed small barrier parameter  μ , iterates in a small neighborhood (roughly within o ( μ )) of the minimizer of the barrier problem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear.
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subjectApplied sciences ; Barriers ; Calculus of variations and optimal control ; Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Convergence ; Eigen values ; Exact sciences and technology ; Experimental design ; Full Length Paper ; Iterative methods ; Mathematical analysis ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematical models ; Mathematical programming ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Methods ; Nonlinear programming ; Nonlinearity ; Numerical Analysis ; Operational research and scientific management ; Operational research. Management science ; Penalty function ; Probability and statistics ; Sciences and techniques of general use ; Statistics ; Studies ; Theoretical
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descriptionWe present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an ℓ 2 -penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barrier function and the infeasibility measure at a subset of previous iterates and this set is updated at every iteration. The ℓ 2 -penalty function is a traditional penalty function defined by a single penalty parameter. At every iteration the step direction is computed from a regularized Newton system of the first-order equations of the barrier problem proposed in Chen and Goldfarb (Math Program 108:1–36, 2006). Iterates are updated using a line search. In particular, a trial point is accepted if it provides a sufficient reduction in either of the penalty functions. We show that the proposed method has the same strong global convergence properties as those established in Chen and Goldfarb (Math Program 108:1–36, 2006). Moreover, our method enjoys fast local convergence. Specifically, for each fixed small barrier parameter  μ , iterates in a small neighborhood (roughly within o ( μ )) of the minimizer of the barrier problem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear.
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4Combinatorics
5Convergence
6Eigen values
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abstractWe present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an ℓ 2 -penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barrier function and the infeasibility measure at a subset of previous iterates and this set is updated at every iteration. The ℓ 2 -penalty function is a traditional penalty function defined by a single penalty parameter. At every iteration the step direction is computed from a regularized Newton system of the first-order equations of the barrier problem proposed in Chen and Goldfarb (Math Program 108:1–36, 2006). Iterates are updated using a line search. In particular, a trial point is accepted if it provides a sufficient reduction in either of the penalty functions. We show that the proposed method has the same strong global convergence properties as those established in Chen and Goldfarb (Math Program 108:1–36, 2006). Moreover, our method enjoys fast local convergence. Specifically, for each fixed small barrier parameter  μ , iterates in a small neighborhood (roughly within o ( μ )) of the minimizer of the barrier problem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear.
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