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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:
Publisher: Berlin/Heidelberg: Springer Berlin Heidelberg
ID: ISSN: 0025-5610
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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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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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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.
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pubSpringer Berlin Heidelberg
doi10.1007/s10107-019-01435-7
orcididhttps://orcid.org/0000-0001-8053-0121