A Class of Gap Functions for Variational Inequalities

Torbjörn Larsson and Michael Patriksson 
Department of Mathematics 
Linköping Institute of Technology
S-581 83 Linköping 
Sweden

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