A characterization of strict local minimizers of order one for nonsmooth static minmax problems (Q5945931)

The authors consider the following problem: NEWLINE\[NEWLINE\text{min}\{f(x)\,| \,x\in S \}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE S:= \{x\in \mathbb R^n \,| \, g_i(x)\leq 0, \; i=1,\dots,p \}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE f(x):=\sup_{y\in Y} \phi(x,y), NEWLINE\]NEWLINE \(\phi:\mathbb R^n\times \mathbb R^m\to \mathbb R\),\, \(Y\) is a nonempty subset of \(\mathbb R^m\), and \(g_i:\mathbb R^n\to \mathbb R\).NEWLINENEWLINEA point \(x_0\in S\) is said to be a strict local minimizer of order 1 if there exist \(\epsilon > 0\) and \(\beta > 0\) such that NEWLINE\[NEWLINE f(x) \geq f(x_0) + \beta \| x-x_0\| \quad \text{for all } x\in S, \; \| x-x_0\| \leq \beta. NEWLINE\]NEWLINE Under weak assumptions on \(\phi\) and the \(g_i\), the authors derive a necessary optimality condition for a local minimizer. Moreover, under a certain constraint qualification, a necessary and sufficient condition for a strict local minimizer of order 1 is also established. The optimality conditions are multiplier rules involving Clarke's generalized gradient.