  We present a method to solve a special class of parameter identification
problems for an elliptic optimal control problem to global optimality. The
bilevel problem is reformulated via the optimal-value function of the
lower-level problem. The reformulated problem is nonconvex and standard
regularity conditions like Robinson's CQ are violated. Via a relaxation of the
constraints, the problem can be decomposed into a family of convex problems and
this is the basis for a solution algorithm. The convergence properties are
analyzed. It is shown that a penalty method can be employed to solve this
family of problems while maintaining convergence speed. For an example problem,
the use of the identity as penalty function allows for the solution by a
semismooth Newton method. Numerical results are presented. Difficulties and
limitations of our approach to solve a nonconvex problem to global optimality
are discussed.
