Generalized Invexity and Generalized Invariant Monotonicity

X. M. Yang

Department of Mathematics, Chongqing Normal University, Chongqing 400047, China.

X. Q. Yang and K. L. Teo

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.

During the last thirty years, an increase of research activities in generalized convexity field has been witnessed. A concept closely related to the convexity of a real-valued function is the monotonicity of a vector-valued function. It is well known that the convexity of a real-valued function is equivalent to the monotonicity of the corresponding gradient function. It is worth noting that monotonicity has played a very important role in for study of the existence and solution methods of variational inequality problems, complementarity problems and more generally, in equilibrium problems. An important breakthrough generalization of this relation was given for various pseudoconvexities/quasiconvexities and pseudomonotonicities/quasimonotonicities (See Karamardian and Schaible (Journal of Optimization Theory and Applications, Vol. 66(1990), pp.37-46). Subsequently, there is an increasing interest in the study of monotonicity and generalized monotonicity and their relationships to convexity and generalized convexity. On the other hand, some relationships between generalized invexity and invarant generalized monotonicity were given by Pini and Singh (Journal of Information and Optimization Sciences, Vol. 20, pp.215-233, 1999) under certain conditions.

In this paper, we study generalized invariant monotonicity and its relationships with generalized invariant convexity. Several types of generalized invariant monotonicity which are generalizations of the (strict) monotonicity, (strict) pseudomonotonicity, and quasimonotonicity are introduced by Karamardian and Schaible (Journal of Optimization Theory and Applications, Vol. 66(1990), pp.37-46). The relations among generalized invariant monotonicities and generalized invexities are also established. Noted that the conditions assumed in this paper are different from those assumed in Pini and Singh’s paper. Several examples are given to show that these generalized invariant monotonicities are proper generalization of the corresponding generalized monotonicities. Moreover, some examples are also presented to illustrate the properly inclusive relations among the generalized invariant monotonicities.