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Mathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation

We consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessar... Full description

Journal Title: Mathematical programming 2019-09-24, Vol.185 (1-2), p.455-485
Main Author: Guo, Lei
Other Authors: Chen, Xiaojun
Format: Electronic Article Electronic Article
Language: English
Subjects:
Approximation
Business schools
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Nonlinear programming
Numerical Analysis
Pricing
Qualifications
Road use pricing
Theoretical
Publisher: Berlin/Heidelberg: Springer Berlin Heidelberg
ID: ISSN: 0025-5610
Zum Text:
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recordid: cdi_crossref_primary_10_1007_s10107_019_01435_7
title: Mathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation
format: Article
creator:
  • Guo, Lei
  • Chen, Xiaojun
subjects:
  • Approximation
  • Business schools
  • Calculus of Variations and Optimal Control
  • Optimization
  • Combinatorics
  • Full Length Paper
  • Mathematical and Computational Physics
  • Mathematical Methods in Physics
  • Mathematical programming
  • Mathematics
  • Mathematics and Statistics
  • Mathematics of Computing
  • Nonlinear programming
  • Numerical Analysis
  • Pricing
  • Qualifications
  • Road use pricing
  • Theoretical
ispartof: Mathematical programming, 2019-09-24, Vol.185 (1-2), p.455-485
description: We consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.
language: eng
source:
identifier: ISSN: 0025-5610
fulltext: no_fulltext
issn:
  • 0025-5610
  • 1436-4646
url: Link


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titleMathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation
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descriptionWe consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.
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subjectApproximation ; Business schools ; Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Full Length Paper ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematical programming ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Nonlinear programming ; Numerical Analysis ; Pricing ; Qualifications ; Road use pricing ; Theoretical
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descriptionWe consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.
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abstractWe consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.
copBerlin/Heidelberg
pubSpringer Berlin Heidelberg
doi10.1007/s10107-019-01435-7
orcididhttps://orcid.org/0000-0001-8053-0121