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

This paper presents a general and useful stability principle which, roughly speaking, says that given a uniformly continuous function defined on an arbitrary metric space, if we slightly change the constraint set over which the optimal (extreme) values of the function are sought, then these values vary slightly. Actually, this apparently new 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. Our 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, and the stability of the sets of minimizers and maximizers.