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 the vanishing noise level. Our assumptions especially cover the example of regularization with a sparsity constraint in which the pth 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(delta) for the error measured in parallel to center dot parallel to(p).