Reduction of finite exhausters

Grzybowski, Jerzy; Pallaschke, Diethard; Urbański, Ryszard

doi:10.1007/s10898-009-9444-9

Cited by 17 publications

(10 citation statements)

References 19 publications

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“…After applying the developed procedure we usually get a family with many redundant sets. These sets can be discarded via various reduction techniques and methods presented in [38][39][40][41].…”

Section: Discussionmentioning

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: Discussionmentioning

confidence: 99%

Converting exhausters and coexhausters

Abbasov¹

2021

Preprint

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“…Neither the essentially primal graphical derivatives [16] nor dual coderivative objects [14] allow for well-defined dual characterisations. The exhauster approach is not without drawbacks: such constructions inherently lack uniqueness, and whilst some works are dedicated to finding minimal objects [17], it is shown that minimal exhausters do not exist in some cases [9]. The conjecture that we are studying in this paper is in a similar vein: we want to establish the uniqueness of a dual characterisation of a function by establishing a steady 2-cycle in the relevant dynamical system defined by the conversion operator.…”

Section: Introductionmentioning

confidence: 97%

On the Conjecture by Demyanov–Ryabova in Converting Finite Exhausters

Sang

2017

J Optim Theory Appl

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In this paper, we prove the conjecture of Demyanov and Ryabova on the length of cycles in converting exhausters in an affinely independent setting and obtain a combinatorial reformulation of the conjecture.Given a finite collection of polyhedra, we can obtain its "dual" collection by forming another collection of polyhedra, which are obtained as the convex hull of all support faces of all polyhedra for a given direction in space. If we keep applying this process, we will eventually cycle due to the finiteness of the problem. Demyanov and Ryabova claim that this cycle will eventually reach a length of at most two.We prove that the conjecture is true in the special case, that is, when we have affinely independent number of vertices in the given space. We also obtain an equivalent combinatorial reformulation for the problem, which should advance insight for the future work on this problem.

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“…Since the supremum of convex sets sup i∈I k∈I \{i} A k is the convex hull i∈I k∈I \{i} A k and the infimum inf i∈I A i is the intersection i∈I A i , the equivalence (c) ⇔ (e) follows immediately. The equivalence (b) ⇔ (c) follows from Theorem 5.2 in[15]. It should be mentioned, however, that in[15] the property (c) is called a shadowing property.The equivalence (e) ⇔ (f) is obvious.…”

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confidence: 93%

“…The equivalence (b) ⇔ (c) follows from Theorem 5.2 in[15]. It should be mentioned, however, that in[15] the property (c) is called a shadowing property.The equivalence (e) ⇔ (f) is obvious. By summand, we understand a summand with respect to Minkowski addition.…”

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confidence: 93%

On minimal representations by a family of sublinear functions

Grzybowski

Küçük

et al. 2014

J Glob Optim

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This paper is a continuation of Grzybowski et al. (J Glob Optim 46:589-601, 2010) and is motivated by the study of exhausters i.e. families of closed convex sets. By Minkowski duality closed convex sets correspond to sublinear functions. Here we study the criteria of reducing representations of pointwise infimum of an infinite family of sublinear functions. A family { f i } i∈I of sublinear functions is by definition an exhaustive family of upper convex approximations of its pointwise infimum inf i∈I f i. A family of closed convex sets is by definition an exhauster of a pointwise infimum of a family of support functions of these convex sets. We establish codependence between infimum of a subset of Minkowski-Rådström-Hörmander cone and translation property of intersection of an exhauster (Sect. 4), between reducing an exhauster to single convex set and shadowing property of intersection of an exhauster (Theorem 5.1) and among all four of these properties (Theorem 6.1). In Grzybowski et al. (2010) the first example of two different minimal upper exhausters of the same function was presented. Here we give an example of infinitely many minimal exhausters of the same ph-function (Example 7.2). In Sect. 8, we give a criterion of minimality of an exhauster with the help of polars of sets belonging to the exhauster. We illustrate this criterion with two interesting examples (Example 8.3).

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Reduction of finite exhausters

Abstract: Minkowski–Rådström–Hörmander spaces, Exhausters, Pairs of closed bounded convex sets, 46B20, 52A05, 54B20,

Cited by 17 publications

References 19 publications

Converting exhausters and coexhausters

Converting exhausters and coexhausters

On the Conjecture by Demyanov–Ryabova in Converting Finite Exhausters

On minimal representations by a family of sublinear functions

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