Robust Constrained Best Approximation in Banach Spaces - Salkhordeh, S.; Mohebi, H. (Set-valued and variational analysis, 2025)
The theory of constrained best approximation in Banach spaces has been systematically developed well over a decade and effective characterizations of best approximations have been known under some qualifications on the constraints. The existing theory yet does not explain how to characterize a best approximation in the face of data uncertainty in the constraints. In this paper, we explain when the best approximation over uncertain nonconvex constraints in Asplund aces is immunized against data uncertainty. This study is done by characterizing the best approximation of the robust counterpart of the uncertain constrained best approximation problem where the constraints are enforced for all possible uncertainties within the prescribed uncertainty sets. We show that under the robust nondegeneracy constraint qualification the same kind of effective characterizations of constrained best approximation hold for the robust best approximation that is immunized against data uncertainty. Moreover, we present Lagrange multipliers characterizations of the robust constrained best approximation that is immunized against data uncertainty under the robust nondegeneracy constraint qualification by following the robust optimization approach. Some examples are given to illustrate the obtained results.