On weak sharp minima for a special class of nonsmooth functions (Q2717933)

Let \(f: \mathbb{R}^n\to [-\infty,+\infty]\) be a function which is finite and constant on a set \(S\subseteq \mathbb{R}^n\). A point \(\overline x\in S\) is called weak sharp local minimizer of order \(m\) for \(f\) if there exist \(\beta> 0\) and \(\varepsilon> 0\) such that NEWLINE\[NEWLINEf(x)- f(\overline x)\geq \beta(\text{dist}(x, S))^m\quad\text{for all }x\in B(\overline x,\varepsilon),NEWLINE\]NEWLINE where \(\text{dist}(x,S)\) is the distance of \(x\) from \(S\). The main result of the paper is a characterization of weak sharp local minimizers of order 1 for a function \(f\) which is a finite maximum of strictly differentiable functions. The characterization makes use of the Mordukhovich normal cone to \(S\). The result is then applied to a smooth constrained optimization problem.