Calm Minima in Parameterized Finite-Dimensional Optimization

Abstract

Calmness is a restricted form of local Lipschitz continuity where one point of comparison is fixed. We study the calmness of solutions to parameterized optimization problems of the form $$ \min \{f(x,w)\} \hbox{ over all } x \in\reals^{n}, $$ where the extended real-valued objective function f is continuously prox-regular in x with compatible parameterization in w. This model covers most finite-dimensional optimization problems, though we focus particular attention here on the case of parameterized nonlinear programming. We give a second-order sufficient condition for there to exist unique optimal solutions that are calm with respect to the parameter. We also characterize a slightly stronger stability property in terms of the same second-order condition, thus clarifying the gap between our sufficient condition and the calmness property. In the case of nonlinear programming, our results complement a long study of the stability properties of optimal solutions: for instance, one consequence of our results is that the Mangasarian--Fromovitz constraint qualification when paired with a new (and relatively weak) second-order condition ensures the calmness of solutions to parameterized nonlinear programs.