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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: \[ \text{min}\{f(x)\,| \,x\in S \}, \] where \[ S:= \{x\in \mathbb R^n \,| \, g_i(x)\leq 0, \; i=1,\dots,p \}, \]
\[ f(x):=\sup_{y\in Y} \phi(x,y), \] \(\phi:\mathbb R^n\times \mathbb R^m\to \mathbb R\), \(Y\) is a nonempty subset of \(\mathbb R^m\), and \(g_i:\mathbb R^n\to \mathbb R\).
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 \[ f(x) \geq f(x_0) + \beta \| x-x_0\| \quad \text{for all } x\in S, \; \| x-x_0\| \leq \beta. \] 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:

90C46 Optimality conditions and duality in mathematical programming
49J35 Existence of solutions for minimax problems
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[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, Proceedings, 2nd World Congress of Nonlinear Analysts. Proceedings, 2nd World Congress of Nonlinear Analysts, Nonlinear Anal., 30 (1997), p. 5363-5367 · 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), Wiley-Interscience: Wiley-Interscience New York · Zbl 0727.90045
[10] Doležal, J., On the problem of necessary conditions for static minmax problems, Problems Control Inform. Theory, 11, 297-300 (1982) · Zbl 0505.49013
[11] Shimizu, K.; Ishizuka, Y.; Bard, J. F., Nondifferentiable and Two-Level Mathematical Programming (1997), Kluwer Academic: Kluwer Academic Boston · 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), Birkhäuser Boston: Birkhäuser Boston Cambridge · Zbl 0713.49021
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