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

We investigate the nonlinear minimization problem on a convex closed set in a Hilbert space. It is shown that the uniform conditional well-posedness of a class of problems with weakly lower semicontinuous functionals is the necessary and sufficient condition for existence of regularization procedures with accuracy estimates uniform on this class. We also establish a necessary and sufficient condition for the existence of regularizing operators which do not use information on the error level in input data. Similar results were previously known for regularization procedures of solving ill-posed inverse problems.