Boundedness and regularity properties of semismooth reformulations of variational inequalities

Liqun Qi

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1 Citation (Scopus)


The Karush-Kuhn-Tucker (KKT) system of the variational inequality problem over a set defined by inequality and equality constraints can be reformulated as a system of semismooth equations via an nonlinear complementarity problem (NCP) function. We give a sufficient condition for boundedness of the level sets of the norm function of this system of semismooth equations when the NCP function is metrically equivalent to the minimum function; and a sufficient and necessary condition when the NCP function is the minimum function. Nonsingularity properties identified by Facchinei, Fischer and Kanzow, 1998, SIAM J. Optim. 8, 850-869, for the semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function, which is an irrational regular pseudo-smooth NCP function, hold for the reformulation based on other regular pseudo-smooth NCP functions. We propose a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent to the minimum NCP function. When it is used to the generalized Newton method for solving the variational inequality problem, an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers only.
Original languageEnglish
Pages (from-to)343-366
Number of pages24
JournalJournal of Global Optimization
Issue number2
Publication statusPublished - 1 Jun 2006


  • Generalized Newton method
  • Global convergence
  • Lagrangian multipliers
  • Quadratic convergence
  • Variational inequalities

ASJC Scopus subject areas

  • Applied Mathematics
  • Control and Optimization
  • Management Science and Operations Research
  • Global and Planetary Change

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