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

Lei Guo, Xiaojun Chen

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)1-31
Number of pages31
JournalMathematical Programming
DOIs
Publication statusAccepted/In press - 24 Sep 2019

Keywords

  • Approximation method
  • Mathematical program with complementarity constraints
  • Non-Lipschitz continuity
  • Optimality condition
  • Sparse solution

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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