Nonlinear Sharp Minimum and the Stability of a Local Minimum on Metric Spaces

We put the emphasis on nonlinear (weak) sharp minimizers with respect to a gauge function. This concept plays an important role in both theoretical and numerical aspects of optimization. In the last part of the contribution we study the stability (in some appropriate sense) of local/global minimizers of an objective function f perturbed to f + g by a function g belonging to a suitable class of Lipschitz functions defined on metric spaces.