The augmented Lagrangian methods and applications

Abstract

The purpose of this thesis is to study a general augmented Lagrangian scheme for optimization and optimal control problems. We establish zero duality gap and exact penalty properties between a primal optimization problem and its augmented Lagrangian dual problem, and characterize local and global solutions for a class of non-Lipschitz penalty problems. We also obtain the existence of an optimal control for an optimal control problem governed by a variational inequality with monotone type mappings, and establish zero duality gap between this optimal control problem and its nonlinear Lagrangian dual problem. Under the assumptions that the augmenting function satisfies the level-coercive condition and the perturbation function satisfies a growth condition, a necessary and sufficient condition for a vector to support an exact penalty representation of the problem of minimizing an extended real function is established. Moreover, in general Banach spaces, under the assumption that the augmenting function satisfies a valley at zero condition and the perturbation function satisfies a growth condition, a necessary and sufficient condition for a zero duality gap property between the primal problem and its augmented Lagrangian dual problem is established. We show that under some conditions the inequality and equality constrained optimization problem and its augmented Lagrangian problem both have optimal solutions. On the other hand it is shown that every weak limit point of a sequence of optimal solutions generated by its augmented Lagrangian problem is a solution of the original constrained optimization problem. Sufficient conditions for the existence of an exact penalization representation and an asymptotically minimizing sequence for a constrained optimization problem are established. It is shown that the second order sufficient condition implies a strict local minimum of a class of non-Lipschitz penalty problems with any positive penalty parameter. The generalized representation condition and the second order sufficient condition imply a global minimum of these penalty problems. We apply our results to quadratic programming and linear fractional programming problems. We study an optimal control problem where the state system is defined by a variational inequality problem with monotone type mappings. We first study a variational inequality problem for monotone type mappings. Under some general coercive assumption, we establish existence results of a solution of variational inequality problems with generalized pseudomonotone mappings, generalized pseudo-monotone perturbation and T-pseudomonotone perturbation of maximal monotone mappings respectively. We obtain several existence results of an optimal control of the optimal control problem governed by a variational inequality with monotone type mappings. Moreover, as an application, we get several existence results of an optimal control for the optimal control problem where the system is defined by a quasilinear elliptic variational inequality problem with an obstacle. By using nonlinear Lagrangian methods, we obtain one necessary condition and several sufficient conditions for the zero duality gap property between the optimal control problem where the state of the system is defined by a variational inequality problem for monotone type mappings and its nonlinear Lagrangian dual problem. We also apply our results to an example where the variational inequality problem leads to a linear elliptic obstacle problem. The study of this thesis has used tools from nonlinear functional analysis, non-linear programming, nonsmooth analysis and numerical linear algebra.