  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.
