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Zbl 1012.49014
Studniarski, Marcin
On weak sharp minima for a special class of nonsmooth functions.
(English)
[J] Discuss. Math., Differ. Incl. Control Optim. 20, No.2, 195-207 (2000). ISSN 1509-9407; ISSN 2084-0365/e

Let $f: \bbfR^n\to [-\infty,+\infty]$ be a function which is finite and constant on a set $S\subseteq \bbfR^n$. 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 $$f(x)- f(\overline x)\ge \beta(\text{dist}(x, S))^m\quad\text{for all }x\in B(\overline x,\varepsilon),$$ 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.
MSC 2000:
*49J52 Nonsmooth analysis (other weak concepts of optimality)
49K35 Minimax problems (necessity and sufficiency)

Keywords: weak sharp local minimizers; strictly differentiable functions; Mordukhovich normal cone

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