On the Nonlinear Complementarity Problem

Variational inequality theory besides being elegant, exciting and rich, also provides us a unified and natural framework to study a large class of linear and nonlinear problems arising in mathematical and engineering sciences. Equally important is the area of operations research known as complementarity theory, which has received much attention during the last twenty years. It has been shown by Karamardian [ 11, that if the convex set involved in a variational inequality problem and complementarity problem is a convex cone, then both problems are equivalent. In fact, variational inequality problems are more general than the complementarity problems, and incluce them as special cases. Recently various extensions and generalizations of the variational inequality and complementarity problems have been introduced and analyzed. An important and useful generalization of the variational inequality problems is the mildly nonlinear variational inequality problem introduced and studied by Noor [2,3]. On the other hand, the complementarity problem has also been extended by Karamardian [l] and Dolcetta [4] to the generalized complementarity problem and the quasi (implict) complementarity problem. For related work and applications, see Ahn [S], Crank [6], Mangasarian [7], and Noor [S].