Generalized Monotonicity — Concepts and Uses

Abstract

The role of monotonicity of the operator in Complementarity Problems and Variational Inequality Problems corresponds to the role of convexity of the objective function in Mathematical Programming Problems. Various concepts of generalized monotonicity are presented which are related to generalized convexity concepts. Special characterizations in case of differentiable operators can be obtained. Using some of these concepts, two sets of existence results are given for generalized monotone Variational Inequality Problems in Banach spaces. In the paper with Hadjisavvas recent existence results by Cottle and Yao for pseudomonotone Complementarity Problems in Hilbert spaces are extended to the general case of quasimonotone Variational Inequality Problems in Banach spaces. In the work with Yao the equivalence of strictly monotone Complementarity Problems, least element problems and other related problems in Banach lattices, studied by Riddel, is extended to the strictly pseudomonotone case.