Abstract
We consider a parametrized constrained optimization problem, which can be represented as the Moreau-type infimal convolution of the norm and some (nonconvex in general) function f. This problem arises particularly in optimal control and approximation theory. We assume that the admissible set A is weakly convex and function f is Lipschitz continuous and weakly convex on the convex hull of A. We show that the problem is Tykhonov well-posed and the solution of the problem is unique and Lipschitz continuous in some neighbourhood of A. Exact estimates for the size of the neighbourhood and for the Lipschitz constant are obtained. Based on these results we prove lower regularity of the optimal value (marginal) function of this problem in some neighbourhood of A.
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