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Generalized subdifferentials and exhausters in nonsmooth analysis. (English. Russian original) Zbl 1151.49015
Dokl. Math. 76, No. 2, 652-655 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 416, No. 1, 18-21 (2007).
Introduction: Nonsmooth analysis, as an independent part and a natural extension of classical (“smooth”) calculus, was formed in the 1960–1970s. Historically, the first classes of nonsmooth functions to be extensively studied were those of convex and minimax functions.
As a result, convex analysis and minimax theory came into being. It turns out that the main tool for investigation of the mentioned classes of functions is subdifferential. Using subdifferential, we can, in particular, formulate conditions for a minimum and find steepest descent directions. The properties of the subdifferentials of convex and minimax functions mentioned above have triggered numerous attempts to find a similar tool in the nonsmooth case. Various generalizations of the concept of subdifferential were proposed and studied. In the present paper, for some most popular subdifferentials in finite-dimensional spaces, calculus rules are constructed. This is done by means of exhausters [see V. F. Demyanov, in: Demyanov, V. (ed.) et al., Quasidifferentiability and related topics. Dedicated to Prof. Franco Giannessi on his 65th birthday and to Prof. Diethard Pallaschke on his 60th birthday. Dordrecht: Kluwer Academic Publishers. Nonconvex Optim. Appl. 43, 85–137 (2000; Zbl 1138.49301)].

MSC:
49J52 Nonsmooth analysis
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