Sub-Differential Characterizations of Lower Semi-Continuous Quasi-Convex Functions on Infinite-Dimensional Spaces and Optimality Conditions Using Variational Inequalities

摘要

We study optimizations under a weak condition of convexity, called quasi-convexity in infinite dimensional spaces. Although many theorems involving the characterizations of quasi-convex functions and optimizations in finite dimensional spaces appear in the literature, very few results exist on the characterizations of quasi-convex functions in infinite dimensional spaces which involve a generalized derivatives of quasi-convex functions. Although the condition for , is known to be necessary optimality condition for existence of a minimizer in quasi-convex programming for some sub-differentials, it is not a sufficient condition. We extend the study of subdifferential characterization of quasi-convex functions in infinite dimensional spaces by using some variational inequalities approach to obtain a necessary and sufficient condition for to be either a local minimum or a global minimum.