Pairs of Compact Convex Sets

Pallaschke, Diethard; Urbański, Ryszard

doi:10.1007/978-94-015-9920-7

Cited by 60 publications

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“…This property was first given by Pallaschke and Urbanski [11] for compact convex sets in topological vector spaces which is an useful property in investigating pairs of compact convex sets in topological vector spaces [12,13]. whereas the converse implications do not hold in general as shown in the following simple example.…”

Section: Preliminariesmentioning

confidence: 86%

(F, K, b)-vex sets and some related properties

Zang

Liu

2013

International Journal of Computer Mathematics

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Section: Preliminariesmentioning

confidence: 86%

(F, K, b)-vex sets and some related properties

Zang

Liu

2013

International Journal of Computer Mathematics

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“…However, these sets are no more weakly compact. It follows that we have to revisit the results obtained by Pallaschke and Urbanski and their co-authors ( [1,2,7,10,20,[25][26][27][28][29][30][31][32][33][34]) and the ones by Demyanov, Rubinov and their co-authors ( [8][9][10][11][12][13][14][15][16][17]). In view of the fact that we have to use operations in R, we adopt the rules devised by Moreau [24].…”

Section: Introductionmentioning

confidence: 89%

Tangentially ds functions

Caprari

Penot

2007

Optimization

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We study functions whose directional derivatives can be written as a difference of two extended real-valued sublinear function (ds functions). The pair of convex sets obtained as the support functions of these sublinear functions is provided by a familiar equivalence relation and is considered as the bi-subdifferential of the function. We study calculus rules and give application to minimization problems.

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“…Now the Pinsker formula ( [7], Proposition 3.3.6) and Theorem 2.4 imply that axiom (S5) holds for the semigroup S. Moreover it follows from Theorem 2.4 that S does not satisfy axiom (S3), because it is enough to take any two disjoint closed balls.…”

Section: Independence Of Axiomsmentioning

confidence: 96%

“…Condition (S1) is called the order cancellation law. : (a, b ) ∈ S 2 } is a commutative semigroup (see [7]) under the operation of addition defined by…”

Section: Preliminariesmentioning

confidence: 99%

On Partially Ordered Semigroups and an Abstract Set-Difference

2008

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In this paper we prove the independence of a system of five axioms (S1)-(S5), which was proposed in the book of Pallaschke and Urbański (Pairs of Compact Convex Sets, vol. 548, Kluwer Academic Publishers, Dordrecht, 2002) for partially ordered commutative semigroups. These five axioms (S1)-(S5) are stated in the introduction below. A partially ordered commutative semigroup satisfying these axioms is called a F-semigroup. By the use of a further axiom (S6) we define an abstract difference for the elements of a F-semigroup and prove some basic properties. The most interesting example of a F-semigroup are the nonempty compact convex sets of a topological vector space endowed with the Minkowski sum as operation and the inclusion as partial order. In Section 4 we apply the abstract difference to the problem of minimality of convex fractions.

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Pairs of Compact Convex Sets

Cited by 60 publications

References 0 publications

(F, K, b)-vex sets and some related properties

(F, K, b)-vex sets and some related properties

Tangentially ds functions

On Partially Ordered Semigroups and an Abstract Set-Difference

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