Generalized Hukuhara Gâteaux and Fréchet derivatives of interval-valued functions and their application in optimization with interval-valued functions

Generalized Hukuhara-Clarke Derivative of Interval-valued Functions and its Properties

“…The topics of optimization [21], control theory [21], variational As the topic of this study is Clarke derivatives for IVFs, we describe below the literature on calculus of Interval-Valued Functions (IVFs). It is to be noted that there is a relatively large joint intersection of the literature survey of this paper with the recent paper by Ghosh et al [14]. However, the proposed work in this paper is completely different than that in [14].…”

“…It is to be noted that there is a relatively large joint intersection of the literature survey of this paper with the recent paper by Ghosh et al [14]. However, the proposed work in this paper is completely different than that in [14]. In [14], the properties of IVFs that are differentiable in the sense of gH-Gâteaux and gH-Fréchet derivatives have been studied.…”

“…As a linear ordering of intervals is not found so far, the development of optimization theory with interval-valued function is not a trivial extension of the conventional optimization theory. Most often [4,14,32,37], IOPs have been analyzed with respect to a partial ordering [20]. Some researchers [3,11] used ordering relations of intervals based on the parametric comparison of intervals.…”

“…Kalani et al [22] analyzed the concept of interval-valued fuzzy derivative for perfect and semiperfect interval-valued fuzzy mappings to derive a method for solving interval-valued fuzzy differential equations. Recently, Ghosh et al [14] have provided the idea of gHdirectional derivative, gH-Gâteaux derivative, and gH-Fréchet derivative of IVFs to derive the optimality conditions for IOPs.…”

“…However, the proposed work in this paper is completely different than that in [14]. In [14], the properties of IVFs that are differentiable in the sense of gH-Gâteaux and gH-Fréchet derivatives have been studied. On the other hand, we attempt to derive the properties upper gH-Clarke differentiable IVFs in this article.…”