Comparison Between Quasidifferentials and Exhausters

Abbasov, Majid E.

doi:10.1007/s10957-017-1167-3

Cited by 14 publications

(8 citation statements)

References 20 publications

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“…This happens due to the fact that quasidifferentiable set-valued mapping is not continuous in Hausdorff metric. Similar results have been reported with exhausters [8][9][10][11][12][13][14][15][16] which can be viewed as a generalization of quasidifferentials [17].…”

Section: Introductionsupporting

confidence: 87%

Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

Abbasov¹

2021

Preprint

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Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to develop simultaneously. Moreover, codifferentials and coexhausters are strongly connected with DC functions. In this paper we trace analogies between all these objects, and prove the equivalence of the boundedness and optimality conditions described in terms of these notions. This allows one to extend the results derived in terms of one object to the problems stated via the other one. Another contribution of this paper is the study of connection between nonhomogeneous approximations and directional derivatives and formulate optimality conditions in terms of nonhomogeneous approximations.

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Section: Introductionsupporting

confidence: 87%

Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

Abbasov¹

2021

Preprint

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“…Inequality ( 19) is equivalent to the fact that g ∈ K + ( C), where K + ( C) is a conjugate cone of cone( C). Thus ( 18) and ( 19) implies (17).…”

Section: Optimality Conditionsmentioning

confidence: 87%

“…This notion brought attention of many researchers [19][20][21][22][23][24]. It turned out that unconstrained optimality conditions for the minimum most organically can be expressed in terms of upper exhausters (see [11,17,18]). Therefore an upper exhauster was called proper for the minimization problem and adjoint for the maximization one.…”

Section: Dini and Hadamard Directional Derivatives Exhaustersmentioning

confidence: 99%

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Optimality conditions for an exhausterable function on an exhausterable set

Abbasov

2019

J Glob Optim

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Exhausters are families of convex compact sets that allow one to represent directional derivative of the studied function at the considered point in the form of InfMax or SupMin of linear functions. Functions for which such a representation is valid we call exhausterable. The class of these functions is quite wide and contains many nonsmooth ones. The set which is given by exhausterable function is also called exhausterable.In the present paper we describe optimality conditions for an exhausterable function on an exhausterable set. These conditions can be used for solving of many nondifferentiable optimization problems. An example that illustrate obtained results is provided.

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“…The order cancellation law or Rådström cancellation theorem [22,24], investigated also in [14,17,19], enables an embedding of a semigroup of convex sets into a quotient vector space called a Minkowski-Rådström-Hörmander space [11,27]. This space plays a crucial role in differentiation of nonsmooth functions (quasidifferential calculus of Demyanov and Rubinov [1,6,7,10,13,20]) and in integration and differentiation in the theory of multifunctions [4,5]. The cancellation by unbounded sets is especially challenging.…”

Section: Introductionmentioning

confidence: 99%

Order cancellation law in the family of bounded convex sets

Grzybowski

Urbański

2019

J Glob Optim

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In this paper generalize Robinson's version of an order cancellation law for subsets of vector spaces in which we cancel by unbounded sets. We introduce the notion of weakly narrow sets in normed spaces, study their properties and prove the order cancellation law where the canceled set is weakly narrow. Also we prove the order cancellation law for closed convex subsets of topological vector space where the canceled set has bounded Hausdorff-like distance from its recession cone. We topologically embed the semigroup of closed convex sets sharing a recession cone having bounded Hausdorff-like distance from it into a topological vector space. This result extends Bielawski and Tabor's generalization of Rådström theorem.

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Comparison Between Quasidifferentials and Exhausters

Cited by 14 publications

References 20 publications

Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

Optimality conditions for an exhausterable function on an exhausterable set

Order cancellation law in the family of bounded convex sets

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