ON GENERAL AUGMENTED LAGRANGIANS AND A MODIFIED SUBGRADIENT ALGORITHM
Nonsmooth Optimization; General Augmented Lagrangians; Modified Subgradient Algorithm; Banach spaces
In this thesis we study a modified subgradient algorithm applied to the dual problem generated by augmented Lagrangians. We consider an optimization problem with equality constraints and study an exact version of the algorithm with a sharp Lagrangian in finite dimensional spaces. An inexact version of the algorithm is extended to infinite dimensional spaces and we apply it to a dual problem of an extended real-valued optimization problem. The dual problem is constructed via augmented Lagrangians which include sharp Lagrangian as a particular case. The sequences generated by these algorithms converge to a dual solution when the dual optimal solution set is nonempty. They have the property that all accumulation points of a primal sequence, obtained without extra cost, are primal solutions. We relate the convergence properties of these modified subgradient algorithms to differentiability of the dual function at a dual solution, and exact penalty property of these augmented Lagrangians. In the second part of this thesis, we propose and analyze a general augmented Lagrangian function, which includes several augmented Lagrangians considered in the literature. In this more general setting, we study a zero duality gap property, exact penalization and convergence of a sub-optimal path related to the dual problem.