On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces

This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the (primal-dual) solution set and give a sufficient condition for such an error bound to hold. In the second part of the paper, we consider an algorithm of augmented Lagrangian type for the solution of such variational problems. We give some global convergence properties of the method and then use the error bound theory to provide estimates for the rate of convergence and to deduce boundedness of the sequence of penalty parameters. Finally, numerical results for optimal control, Nash equilibrium problems, and elliptic parameter estimation problems are presented.Comment: 27 page

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

This paper deals with quasi-variational inequality problems (QVIs) in a generic Banach space setting. We provide a theoretical framework for the analysis of such problems which is based on two key properties: the pseudomonotonicity (in the sense of Brezis) of the variational operator and a Mosco-type continuity of the feasible set mapping. We show that these assumptions can be used to establish the existence of solutions and their computability via suitable approximation techniques. In addition, we provide a practical and easily verifiable sufficient condition for the Mosco-type continuity property in terms of suitable constraint qualifications. Based on the theoretical framework, we construct an algorithm of augmented Lagrangian type which reduces the QVI to a sequence of standard variational inequalities. A full convergence analysis is provided which includes the existence of solutions of the subproblems as well as the attainment of feasibility and optimality. Applications and numerical results are included to demonstrate the practical viability of the method.Comment: 27 pages, 3 figure