Augmented Lagrangian methods for optimal control problems governed by mixed quasi‐equilibrium problems with applications

Abstract

The main goal of this article is to study a general augmented Lagrangian method for optimal control problems governed by mixed (quasi-)equilibrium problems.We establish zero duality gap properties between a primal problem and its augmented Lagrangian dual problem. As a consequence, we give some existence results for optimal control problem governed by evolutionary quasi-variational inequalities. An application to optimal control of obstacle problems described by quasi-hemivariational inequalities is studied. The results obtained in this article are new and improves considerably many recent results in literature. K E Y W O R D Saugmented Lagrangian methods, mixed quasi-equilibrium problems, Mosco convergence, optimal control problems, quasi-hemivariational inequalities, quasi-variational inequalities