Solving variational inequality problems via smoothing-nonsmooth reformulations

D. Sun, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

29 Citations (Scopus)


It has long been known that variational inequality problems can be reformulated as nonsmooth equations. Recently, locally high-order convergent Newton methods for nonsmooth equations have been well established via the concept of semismoothness. When the constraint set of the variational inequality problem is a rectangle, several locally convergent Newton methods for the reformulated nonsmooth equations can also be globalized. In this paper, our main aim is to provide globally and locally high-order convergent Newton methods for solving variational inequality problems with general constraints. To achieve this, we first prove via convolution that these nonsmooth equations can be well approximated by smooth equations, which have desirable properties for the design of Newton methods. We then reformulate the variational inequality problems as equivalent smoothing-nonsmooth equations and apply Newton-type methods to solve the latter systems, and so the variational inequality problems. Stronger convergence results have been obtained.
Original languageEnglish
Pages (from-to)37-62
Number of pages26
JournalJournal of Computational and Applied Mathematics
Issue number1-2
Publication statusPublished - 1 Apr 2001
Externally publishedYes


  • Reformulation
  • Smoothing
  • Variational inequalities

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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