Smoothing by mollifiers. Part I: semi-infinite optimization

We show that a compact feasible set of a standard semi-infinite optimization problem can be approximated arbitrarily well by a level set of a single smooth function with certain regularity properties. This function is constructed as the mollification of the lower level optimal value function. Moreover, we use correspondences between Karush–Kuhn–Tucker points of the original and the smoothed problem, and between their associated Morse indices, to prove the connectedness of the so-called min–max digraph for semi-infinite problems.