Appl Math Optim 21:223-241 (1990) Applied Mathematics and Optimization © 1990 Springer-Verlag New York Inc. An Augmented Lagrangian Technique for Variational Inequalities* K. Ito 1 and K. Kunisch2 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA 2 Institut fiir Mathematik, Technische Universit~it Graz, Kopernikusgasse 24, A-8010 Graz, Austria Communicated by J. Stoer Abstract. A general framework for the treatment of a class of elliptic variational inequalities by an augmented Lagrangian method, when inequalities with infinite-dimensional image space are augmented, is developed. Applications to the obstacle problem, the elastoplastic torsion problem, and the Signorini problem are given. 1. Introduction In this paper we develop a general framework for the treatment of a class of variational inequalities by an augmented Lagrangian method. The variational inequalities are formulated as optimization problems with inequality constraints whose image space is infinite dimensional. While the augmented Lagrangian method has been developed exhaustively for equality constraints, see, for instance, [Be], IH], [FG], and [G1], much less is known for the case when inequality constraints are present. If the image space of the inequality constraint is finite dimensional, inequality constraints can be treated essentially like equality constraints, provided that the strict complemen- tarity assumption holds, see, e.g., [Be]. In [IK1] we developed a theory which * The research of the first author was supported in part by the Air Force Office of Scientific Research under Grants AFOSR-84-0398 and AFOSR-85-0303, by the National Aeronautics and Space Administration under Grant NAG-l-1517, and by NSF under Grant No. UINT-8521208. The second author's research was supported in part by the Fonds zur F/Srderung der wissenschaftlichen Forschung under $3206 and P6005.