A class of gap functions for variational inequalities

Recently Auchmuty has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. The authors develop and generalize Auchmuty’s results, and relate his class of merit functions to other works done in this field. Especially, they investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequaltiies. The authors then present new descent algorithms for variational inequalities within this framework, incuding 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 Zuhovickiℝ3i et al., and the differentiable merit function recently presented by Fukushima; 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 the authors extend those inherent in the works of Dafermos, Cohen and Wu et al.; new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty’s framework, are also given.