Calm minima in parameterized finite-dimensional optimization

Calmness is a restricted form of local Lipschitz continuity where one point of comparison is fixed. We study the calmness of solutions to parameterize d optimization problems of the form min {f (x, w)} over all x epsilon R-n, where the extended real-valued objective function f is continuously prox-re gular in x with compatible parameterization in w. This model covers most fi nite-dimensional optimization problems, though w 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 st ronger stability property in terms of the same second-order condition, thus clarifying the gap between our sufficient condition and the calmness prope rty. In the case of nonlinear programming, our results complement a long st udy of the stability properties of optimal solutions: for instance, one con sequence of our results is that the Mangasarian-Fromovitz constraint qualif ication when paired with a new ( and relatively weak) second-order conditio n ensures the calmness of solutions to parameterized nonlinear programs.