  A learning approach to selecting regularization parameters in multi-penalty
Tikhonov regularization is investigated. It leads to a bilevel optimization
problem, where the lower level problem is a Tikhonov regularized problem
parameterized in the regularization parameters. Conditions which ensure the
existence of solutions to the bilevel optimization problem of interest are
derived, and these conditions are verified for two relevant examples.
Difficulties arising from the possible lack of convexity of the lower level
problems are discussed. Optimality conditions are given provided that a
reasonable constraint qualification holds. Finally, results from numerical
experiments used to test the developed theory are presented.
