Upper Hölder Continuity of Minimal Points

We derive criteria for upper Lipschitz/Hölder continuity of the set of minimal points of a given subset $A\subset Y$ of a normed space Y when A is subjected to perturbations. To this aim we introduce the rate of containment of A, a real-valued function of one real variable, which measures the depart from minimality as a function of the distance from the minimal point set. The main requirement we impose is that for small arguments the rate of containment is a sufficiently fast growing function. The obtained results are applied to parametric vector optimization problems to derive conditions for upper Hölder continuity of the performance multifunction.