Regularization With Non-convex Separable Constraints

We consider regularization of nonlinear ill-posed problems with constraints which are non-convex. As a special case we consider separable constraints, i.e. the regularization takes place in a sequence space and the constraint acts on each sequence element with a possibly non-convex function. We derive conditions under which such a constraint provides a regularization. Moreover, we derive estimates for the error and obtain convergence rates for vanishing noise level. Our assumptions especially cover the example of regularization with a sparsity constraint in which the p-th power with 0 < p ≤ 1 is used and we present other examples as well. In particular we derive error estimates for the error measured in the quasi-norms and obtain a convergence rate of O(δ) for the error measured in ‖ · ‖p. AMS Classification: 65J20, 46A16