Adjoint coexhausters in nonsmooth analysis and extremality conditions
type: Item
instance of: scholarly article
title: Adjoint coexhausters in nonsmooth analysis and extremality conditions
zbMATH Open document ID: 1280.90112
author: V. F. Demyanov
author: Majid E. Abbasov
published in: Journal of Optimization Theory and Applications
publication date: 2013-05-08T00:00:00Z
review text: The article of M. E. Abbasov and V. F. Demyanov is a valuable contribution to the field of nondifferentiable or nonsmooth calculus and, especially, optimization. This investigation takes place in a wide analytic setting, in the tradition of the quasidifferential which it extends and employs. This work is well structured, well written and, e.g., by figures, well visualized. Classically, in ``smooth'' mathematical calculus and optimization, a differentiable function is studied by means of the derivative or, multidimensionally, the gradient. In the case of nondifferentiable functions, nonsmooth analysis gets used. In convex analysis and minimax theory, the corresponding classes of functions are investigated by the subdifferential, quasidifferentiable functions are evaluated by the quasidifferential (being is a pair of sets). Any directionally differentiable function is treated in terms of upper and lower exhausters (families of convex sets). Here, conditions for a minimizer are given with an upper exhauster; while conditions for a maximizer are represented by a lower exhauster. Hence, upper exhausters are called proper ones for the minimization whereas a lower exhauster is referred to as a proper one for the maximization task. The directional derivatives and so, the exhausters permit first-order approximations of the increment of the regarded function. As functions of the direction, these approximations are positively homogeneous. They permit the stating of optimality conditions, the finding of steepest ascent and descent directions and the constructing of numerical procedures. If, however, e.g., the maximizer of the function has to be found, but one has an upper exhauster (being not proper for the maximization), it is required to employ a lower exhauster. One could try to express conditions for a maximum in expressions of an upper exhauster (which is an adjoint one for the maximization). First research on this was done by Roshchina, and then Abbasov. Generally, exhaus…
reviewed by: Gerhard-Wilhelm Weber
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C46
zbMATH DE Number: 6161532
zbMATH Keywords: nonsmooth analysis
zbMATH Keywords: directional derivatives
zbMATH Keywords: proper and adjoint exhausters and coexhausters
zbMATH Keywords: extremality conditions
MaRDI profile type: Publication
full work available at URL: s10957-012-0144-0
OpenAlex ID: W1971509874
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DOI: 10.1007/S10957-012-0144-0
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description: scientific article; zbMATH DE number 6161532