On Gap Functions for Quasi‐Variational Inequalities

For variational inequalities, various merit functions, such as the gap function, the regularized gap function, the D‐gap function and so on, have been proposed. These functions lead to equivalent optimization formulations and are used to optimization‐based methods for solving variational inequalities. In this paper, we extend the regularized gap function and the D‐gap functions for a quasi‐variational inequality, which is a generalization of the variational inequality and is used to formulate generalized equilibrium problems. These extensions are shown to formulate equivalent optimization problems for quasi‐variational inequalities and are shown to be continuous and directionally differentiable.