Calculus for interval-valued functions using generalized Hukuhara derivative and applications

Chalco-Cano, Y.; Rufián-Lizana, A.; Román-Flores, Heriberto; Jiménez-Gamero, M. D.

doi:10.1016/j.fss.2012.12.004

Cited by 208 publications

(168 citation statements)

References 41 publications

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“…From ( (2) is not a linear space since an interval does not have inverse element and therefore subtraction does not have adequate properties (see [3,13]). More recently, Stefanini and Bede in [13] have introduced the following difference between two intervals.…”

Section: The Space Of Intervals and Interval-valued Functionsmentioning

confidence: 99%

“…It well known that if F is H-differentiable then it possesses the property that the diameter len(F (t)) (lenght of F (t)) is nondecreasing as t increases [2,3,13]. Thus H-derivative is a concept very restrictive.…”

Section: And Only If One Of the Following Cases Holdmentioning

confidence: 99%

“…For instance, if we consider F (x) = [1, 2] · x 3 , we have that x * = 0 is a stationary point for F but it is not a minimum.…”

Section: Sufficient Conditionsmentioning

confidence: 99%

See 2 more Smart Citations

A note on optimality conditions to interval optimization problems

Chalco-Cano

Osuna-Gómez

Hernández‐Jiménez

et al. 2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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In this article we examine to necessary and sufficient optimality conditions for interval optimization problems. We introduce a new concept of stationary point for an interval-valued function based on the gH-derivative. We show the importance the this concept from a practical and computational point of view. We introduce a new concept of invexity for gH-differentiable interval-valued function which generalizes previous concepts and we prove that it is a sufficient optimality condition. Finally, we show that the concepts of differentiability, convexity and invexity for interval-valued functions based on the differentiability, convexity and invexity of its endpoint functions are not adequate tools for interval optimization problems.

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Section: The Space Of Intervals and Interval-valued Functionsmentioning

confidence: 99%

Section: And Only If One Of the Following Cases Holdmentioning

confidence: 99%

See 1 more Smart Citation

A note on optimality conditions to interval optimization problems

Chalco-Cano

Osuna-Gómez

Hernández‐Jiménez

et al. 2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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show abstract

“…From then on, the theory of interval analysis and its application developed greatly with the joint effort of many researchers. Especially, in order to build the theory of optimization of interval value [5,7,8], theory of differential equation of interval value [1,2,6] and the differential theory of fuzzy value [3], several kinds of differentiability of interval-valued functions were invented, and the related theories were built.…”

Section: Introductionmentioning

confidence: 99%

“…The first one is to use the concepts of H-derivative of interval-valued function from nonempty subset of real number space to interval number space, and the partial H-derivative of interval-valued function from nonempty subset of n-dimensional Euclidean space to interval number space, which are given by H-difference [1,7,8]. The second one is to use the concepts of gH-derivative of interval-valued function from nonempty subset of real number space to interval number space, and the partial gHderivative of interval-valued function from nonempty subset of n-dimensional Euclidean space to interval number space, which are given by gH-difference [2,3,5]. Only the changing rate of interval-valued function in axis direction is taken into consideration, rather than in other special directions.…”

Section: Introductionmentioning

confidence: 99%

Study on differentiability problems of interval-valued functions

Bao¹,

Li²,

Bai³

2017

J. Nonlinear Sci. Appl.

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In this paper, we give the concepts of H-directional differentiability and D-directional differentiability of interval-valued functions. Then we discuss the properties of H-directional differentiable interval-valued functions and D-directional differentiable interval-valued functions. The necessary and sufficient conditions for the H-directional differentiability are given together with the sufficient conditions and the necessary and sufficient conditions for D-directional differentiability of interval-valued functions. Then we discuss the relationship between the two directional differentiability and prove these directional differentiability can be equivalent under a certain conditions.

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Uncertain Fractional Fornberg–Whitham Equations

2016

Fuzzy Arbitrary Order System

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Calculus for interval-valued functions using generalized Hukuhara derivative and applications

Cited by 208 publications

References 41 publications

A note on optimality conditions to interval optimization problems

A note on optimality conditions to interval optimization problems

Study on differentiability problems of interval-valued functions

Uncertain Fractional Fornberg–Whitham Equations

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