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arXiv · 2210.02852

Generalized Hukuhara Hadamard Derivative of Interval-valued Functions and Its Applications to Interval Optimization

Ram Surat Chauhan·Debdas Ghosh·Qamrul Hasan Ansari
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Abstract

In this article, we study the notion of gH-Hadamard derivative for interval-valued functions (IVFs) and its applications to interval optimization problems (IOPs). It is shown that the existence of gH-Hadamard derivative implies the existence of gH-Frechet derivative and vise-versa. Further, it is proved that the existence of gH-Hadamard derivative implies the existence of gH-continuity of IVFs. We found that the composition of a Hadamard differentiable real-valued function and a gH-Hadamard differentiable IVF is gHHadamard differentiable. Further, for finite comparable IVF, we prove that the gH-Hadamard derivative of the maximum of all finite comparable IVFs is the maximum of their gH-Hadamard derivative. The proposed derivative is observed to be useful to check the convexity of an IVF and to characterize efficient points of an optimization problem with IVF. For a convex IVF, we prove that if at a point the gH-Hadamard derivative does not dominate to zero, then the point is an efficient point. Further, it is proved that at an efficient point, the gH-Hadamard derivative does not dominate zero and also contains zero. For constraint IOPs, we prove an extended Karush-Kuhn-Tucker condition by using the proposed derivative. The entire study is supported by suitable examples.

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Ram Surat Chauhan, Debdas Ghosh, Qamrul Hasan Ansari. 2022-10-04. Generalized Hukuhara Hadamard Derivative of Interval-valued Functions and Its Applications to Interval Optimization. https://arxiv.org/abs/2210.02852

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