Smoothing by mollifiers. I: Semi-infinite optimization
The authors study the standard semi-infinite programming problem where the index set is described by finitely many inequality constraints and all functions involved are twice continuously differentiable. They begin with a summary of the essential definitions and properties respectively of the feasible set, the lower level problem, regularity, stationarity, the Reduction Ansatz, and so-called mollifiers. Then they prove that, under some regularity assumptions, the nonempty compact feasible set of the problem can be approximated arbitrary well by a level set of a single smooth function which is constructed as the mollification of the optimal value function of the lower level problem. Next correspondences are shown between the Karush-Kuhn-Tucker points of the original and the smooth finite optimization problem defined via the approximated feasible set, and a result concerning the associated Morse indices is presented. The latter result, which is proved in part II of the paper (appearing in J. Glob. Optimiz.), finally enables the prove of the connectedness of the so-called min-max digraph for semi-infinite programming problems.
scientific article; zbMATH DE number 5306197
scientific article