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A characterization of strict local minimizers of order one for nonsmooth static minmax problems. (English) Zbl 1121.90123
The authors consider the following problem: $$\text{min}\{f(x)\,\vert \,x\in S \}, $$ where $$ S:= \{x\in \Bbb R^n \,\vert \, g_i(x)\leq 0, \; i=1,\dots,p \}, $$ $$ f(x):=\sup_{y\in Y} \phi(x,y), $$ $\phi:\Bbb R^n\times \Bbb R^m\to \Bbb R$,  $Y$ is a nonempty subset of $\Bbb R^m$, and $g_i:\Bbb R^n\to \Bbb R$. A 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 $$ f(x) \geq f(x_0) + \beta \Vert x-x_0\Vert \quad \text{for all } x\in S, \; \Vert x-x_0\Vert \leq \beta. $$ 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.

MSC:
90C46Optimality conditions, duality
49J35Minimax problems (existence)
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Full Text: DOI
References:
[1] Cromme, L.: Strong uniqueness: A far-reaching criterion for the convergence of iterative procedures. Numer. math. 29, 179-193 (1978) · Zbl 0352.65012
[2] Studniarski, M.: Sufficient conditions for the stability of local minimum points in nonsmooth optimization. Optimization 20, 27-35 (1989) · Zbl 0679.90072
[3] Klatte, D.: Stable local minimizers in semi-infinite optimization: regularity and second-order conditions. J. comput. Appl. math. 56, 137-157 (1994) · Zbl 0823.90121
[4] Ward, D. E.: Characterizations of strict local minima and necessary conditions for weak sharp minima. J. optim. Theory appl. 80, 551-571 (1994) · Zbl 0797.90101
[5] Still, G.; Streng, M.: Optimality conditions in smooth nonlinear programming. J. optim. Theory appl. 90, 483-515 (1996) · Zbl 0866.90117
[6] Studniarski, M.: Characterizations of strict local minima for some nonlinear programming problems. Nonlinear anal. 30 (1997) · Zbl 0914.90243
[7] Clarke, F. H.: Generalized gradients and applications. Trans. amer. Math. soc. 205, 247-262 (1975) · Zbl 0307.26012
[8] Schmitendorf, W. E.: Necessary conditions and sufficient conditions for static minmax problems. J. math. Anal. appl. 57, 683-693 (1977) · Zbl 0355.90066
[9] Clarke, F. H.: Optimization and nonsmooth analysis. (1983) · Zbl 0582.49001
[10] Doležal, J.: On the problem of necessary conditions for static minmax problems. Problems control inform. Theory 11, 297-300 (1982)
[11] Shimizu, K.; Ishizuka, Y.; Bard, J. F.: Nondifferentiable and two-level mathematical programming. (1997) · Zbl 0878.90088
[12] Sutti, C.: Monotone generalized differentiability in nonsmooth optimization. Riv. mat. Sci. econom. Social. 18, 83-89 (1995) · Zbl 0868.90115
[13] Luu, D. V.; Oettli, W.: Necessary optimality conditions for non-smooth minimax problems. Z. anal. Anwendungen 12, 709-721 (1993) · Zbl 0806.49020
[14] Aubin, J. -P.; Frankowska, H.: Set-valued analysis. (1990) · Zbl 0713.49021