On weak sharp minima for a special class of nonsmooth functions

We present a characterization of weak sharp local minimizers of order one for a function f: ℝⁿ → ℝ defined by ${\displaystyle f\left(x\right):=max\{f_i\left(x\right)\mid i=1,...,p\}}$, where the functions ${\displaystyle f_i}$ are strictly differentiable. It is given in terms of the gradients of ${\displaystyle f_i}$ and the Mordukhovich normal cone to a given set on which f is constant. Then we apply this result to a smooth nonlinear programming problem with constraints.