Paper detail

Stability of the optimal values under small perturbations of the constraint set

This paper discusses a general and useful stability principle which, roughly speaking, says that given a uniformly continuous function defined on an arbitrary metric space, if the function is bounded on the constraint set and we slightly change this set, then its optimal (extreme) values on this set vary slightly, and, moreover, they are actually uniformly continuous as a function of the constraint set. The principle holds in a much more general setting than a metric space, since the distance function may be asymmetric, may attain negative and even infinite values, and so on. This stability principle leads to applications in parametric optimization, mixed linear-nonlinear programming and analysis of Lipschitz continuity, as well as to a general scheme for tackling a wide class of non-convex and non-smooth optimization problems. We also discuss the issue of stability when the objective function is merely continuous. As a byproduct of our analysis we obtain a significant generalization of the concept of a generalized inverse of a linear operator and a very general variant of the so-called "Hoffman's Lemma".