Alternance Form of Optimality Conditions in the Finite-Dimensional Space

V.F. Demyanov, V.N. Malozemov

Research output

1 Citation (Scopus)


In solving optimization problems, necessary and sufficient optimality conditions play an outstanding role. They allow, first of all, to check whether a point under study satisfies the conditions, and, secondly, if it does not, to find a “better” point. This is why such conditions should be “constructive” letting to solve the above-mentioned problems. For the class of directionally differentiable functions in, a necessary condition for an unconstrained minimum requires for the directional derivative to be non-negative in all directions. This condition becomes efficient for special classes of directionally differentiable functions. For example, in the case of convex and max-type functions, the necessary condition for a minimum takes the form where is a convex compact set. The problem of verifying this condition is reduced to that of finding the point of C which is the nearest to the origin. If the origin does not belong to C, we easily find the steepest descent direction and are able to construct a numerical me
Original languageEnglish
Title of host publicationSpringer Optimization and Its Applications. Vol.87: Constructive Nonsmooth Analysis and Related Topics /Editors: Demyanov, Vladimir F., Pardalos, Panos M., Batsyn, Mikhail V.
PublisherSpringer Nature
Pages265, 185-203
ISBN (Print)978-1-4614-8615-2
Publication statusPublished - 2014

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