Accuracy estimates of regularization methods and the well-posedness of nonlinear constrained optimization problems

We consider the problem of minimizing a nonlinear functional on a closed set in a Hilbert space. The functional to be minimized and the admissible set may be specified with errors. It is established that a necessary and sufficient condition for the existence of regularization procedures with an accuracy estimate uniform across different classes of functionals and admissible sets is the uniform well-posedness of these classes of minimization problems. A necessary and sufficient condition for the existence of a regularizing operator that does not use information about the error level of the input data is obtained. The proofs partially rely on the variational principles of Ekeland and Borwein–Preiss. Similar results were previously known for regularization procedures for ill-posed inverse problems, as well as for unconstrained optimization problems.