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This paper discusses a general and useful stability principle which, roughlyspeaking, says that given a uniformly continuous function defined on anarbitrary metric space, if the function is bounded on the constraint set and weslightly change this set, then its optimal (extreme) values on this set varyslightly, and, moreover, they are actually uniformly continuous as a functionof the constraint set. The principle holds in a much more general setting thana metric space, since the distance function may be asymmetric, may attainnegative and even infinite values, and so on. This stability principle leads toapplications in parametric optimization, mixed linear-nonlinear programming andanalysis of Lipschitz continuity, as well as to a general scheme for tackling awide class of non-convex and non-smooth optimization problems. We also discussthe issue of stability when the objective function is merely continuous. As abyproduct of our analysis we obtain a significant generalization of the conceptof a generalized inverse of a linear operator and a very general variant of theso-called "Hoffman's Lemma".

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