DOI: 10.1007/s002459910017 Appl Math Optim 41:343–364 (2000) © 2000 Springer-Verlag New York Inc. Optimal Control of Elliptic Variational Inequalities ∗ K. Ito 1 and K. Kunisch 2 1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA 2 Institut f¨ ur Mathematik, Karl-Franzens-Universit¨ at Graz, A-8010 Graz, Austria Communicated by J. Stoer Abstract. Optimality systems for optimal control problems governed by elliptic variational inequalities are derived. Existence of appropriately defined Lagrange multipliers is proved. A primal–dual active set method is proposed to solve the optimality systems numerically. Examples with and without lack of strict comple- mentarity are included. Key Words. Optimal control, Variational inequalities, Augmented Lagrangians, Lagrange multipliers, Active set methods, Strict complementarity. AMS Classification. 49J24, 49J40, 49K24, 90C30. 1. Introduction In this paper we continue our line of investigation of augmented Lagrangian techniques as an efficient tool for both analysis and numerical treatment of optimization problems ∗ The research of Kazufumi Ito was supported in part by AFSOR under Contracts F-49620-95-1-0437 and F-49620-95-1-0447, and that of Karl Kunisch was supported in part by the Fonds zur F¨ orderung der wissenschaftlichen Forschung, SFB “Optimization and Control.”