Mathematical Programming

, Volume 64, Issue 1–3, pp 53–79 | Cite as

A class of gap functions for variational inequalities

  • Torbjörn Larsson
  • Michael Patriksson
Article
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Abstract

Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.

Key words

Finite-dimensional variational inequalities Merit functions Variational principles Successive approximation algorithms Descent algorithms 
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Copyright information

© The Mathematical Programming Society, Inc 1994

Authors and Affiliations

  • Torbjörn Larsson
    • 1
  • Michael Patriksson
    • 1
  1. 1.Department of MathematicsLinköping Institute of TechnologyLinköpingSweden

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