Quaisidifferentials in Kantorovich Spaces

Basaeva, Elena K.; Kusraev, A. G.; Kutateladze, S. S.

doi:10.1007/s10957-016-0899-9

Cited by 8 publications

(4 citation statements)

References 5 publications

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“…Optimality conditions were formulated in terms of quasidifferentials as well as the procedures of finding directions of a steepest descent and ascent when these conditions are not satisfied [16]. Thereafter the theory of quasidifferentials progressed rapidly due to the many significant studies in the area [17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Converting exhausters and coexhausters

Abbasov¹

2021

Preprint

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Exhausters and coexhausters are notions of constructive nonsmooth analysis which are used to study extremal properties of functions. An upper exhauster (coexhauster) is used to get an approximation of a considered function in the neighborhood of a point in the form of min max of linear (affine) functions. A lower exhauster (coexhauster) is used to represent the approximation in the form of max min of linear (affine) functions. Conditions for a minimum in a most simple way are expressed by means of upper exhausters and coexhausters, while conditions for a maximum are described in terms of lower exhausters and coexhausters. Thus the problem of obtaining an upper exhauster or coexhauster when the lower one is given and vice verse arises. We study this problem in the paper and propose new method for its solution which allows one to pass easily between min max and max min representations.

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Section: Introductionmentioning

confidence: 99%

Converting exhausters and coexhausters

Abbasov¹

2021

Preprint

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“…Inclusion-minimal pairs were studied by Bauer [5], Scholtes [26,34], Pallaschke [15,16,27,28] and by the authors [12,13,19,20] in connection with quasidifferential calculus. Quasidifferential calculus was developed by Demyanov and Rubinov [8] and studied by many authors including Zhang, Xia, Gao and Wang [38] Basaeva, Kusraev and Kutateladze [4], Antczak [2], Abbasov [1], Dolgopolik [10] and others.…”

Section: Introductionmentioning

confidence: 99%

Minimal pairs of convex sets which share a recession cone

Grzybowski¹,

Urbański²

2020

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Robinson introduced a quotient space of pairs of convex sets which share their recession cone. In this paper minimal pairs of unbounded convex sets, i.e. minimal representations of elements of Robinson's spaces are investigated. The fact that a minimal pair having property of translation is reduced is proved. In the case of pairs of two-dimensional sets a formula for an equivalent minimal pair is given, a criterion of minimality of a pair of sets is presented and reducibility of all minimal pairs is proved. Shephard-Weil-Schneider's criterion for polytopal summand of a compact convex set is generalized to unbounded convex sets. An application of minimal pairs of unbounded convex sets to Hartman's minimal representation of dc-functions is shown. Examples of minimal pairs of three-dimensional sets are given.

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“…Since then, several collections of papers [8,14] and monographs [13,15,16] were devoted to quasidifferential calculus and its applications in the finite dimensional case. Infinite dimensional extensions of quasidifferential calculus were analysed in [5,12,21,44,50,63]. A generalization of the concept of quasidifferentiability called ε-quasidifferentiability was proposed by Gorokhovik [29][30][31].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, the problem of when necessary optimality conditions for quasidifferentiable problems become sufficient was analyzed in [2,27] under generalized invexity assumptions, while optimality conditions for vector quasidifferentiable optimization problems were studied by Glover et al [28], Basaeva [3,4] (see also [5,44]), and Antczak [1].…”

Section: Introductionmentioning

confidence: 99%

A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

Dolgopolik

2019

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This paper is devoted to an analysis of optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials. To this end, we obtain a novel description of convex subcones of the contingent cone to a set defined by quasidifferentiable equality and inequality constraints. With the use of this description we derive optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials based on a new constraint qualification. The main feature of these optimality conditions and constraint qualification is the fact that they depend on individual elements of quasidifferentials of constraints. To illustrate the theoretical results, we present two simple examples in which the optimality conditions obtained in this paper are not satisfied at a given point, while optimality conditions in terms of various subdifferentials (in fact, any outer semicontinuous/limiting subdifferential) fail to disqualify this point as nonoptimal.

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Quaisidifferentials in Kantorovich Spaces

Cited by 8 publications

References 5 publications

Converting exhausters and coexhausters

Converting exhausters and coexhausters

Minimal pairs of convex sets which share a recession cone

A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

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