Regularized Gap Functions For Nonsmooth Quasi-variational Inequalities

In this paper, we use regularized gap functions to reformulate quasi-variational inequalities into minimization problems. In optimization prob-lems, it is significant to study the directional derivatives and subdifferentials of objective function, for example, we can use the directional derivatives and sub-differentials of objective function to study the necessary optimality conditions and algorithms for optimization problems. We discuss the directional differentia-bility of marginal function and estimate the upper and lower bounds of a class of marginal functions in parametric convex programs. Then we employ these results to study the directional differentiability and the upper bound of limit-subdifferentials of the regularized gap functions for generalized quasi-variational inequalities and Clarke directional derivatives of the regularized gap functions and D-gap functions for nonsmooth quasi-variational inequalities.