Frechet Derivative Implies Continuity N Dimention

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

Abstract

In this article, the notions of gH-directional derivative, gH-Gâteaux derivative and gH-Fréchet derivative for interval-valued functions are proposed. The existence of gH-Fréchet derivative is shown to imply the existence of gH-Gâteaux derivative and the existence of gH-Gâteaux derivative is observed to indicate the presence of gH-directional derivative. For an interval-valued gH-Lipschitz function, it is proved that the existence of gH-Gâteaux derivative implies the existence of gH-Fréchet derivative. It is observed that for an interval-valued convex function on a linear space, the gH-directional derivative exists at any point for every direction. Concepts of linear and monotonic interval-valued functions are studied in the sequel. Further, it is shown that the proposed derivatives are useful to check the convexity of an interval-valued function and to characterize efficient points of an optimization problem with interval-valued objective function. It is observed that at an efficient point of an interval-valued function, none of its gH-directional derivatives dominates zero and the gH-Gâteaux derivative must contain zero. The entire study is supported by suitable illustrative examples.

Introduction

Now-a-days, due to inherent uncertainty in many real world phenomena, the study of the functions with interval coefficients demands a significant study. Moore [26], [27] introduced the concept of interval analysis to deal with interval-valued data and the functions with interval coefficients. Thereafter, many researchers have developed the theory and applications of interval arithmetic and interval-valued functions (IVFs), for instances, see [2], [4], [10], [20], [23], [24], [34], [36], etc. and their references.

On Moore's interval arithmetic [27], many research attempts endeavoured to develop calculus and algebra with intervals. In [28], it is shown that interval systems with Moore's interval arithmetic does not form a group, because for a nonzero interval A, there does not exist any interval B such that $\mathbf{A}\oplus \mathbf{B}=\mathbf{0}$ . For the same reason, many conventional results in the calculus for real-valued functions are not true for interval-valued functions. For instance, a constant interval-valued function is not differentiable [13]. Thus, for the theoretical framework of calculus of interval-valued functions and interval analysis, Hukuhara [18] introduced a new concept of difference of two intervals, namely the Hukuhara difference. Although the concept of Hukuhara difference satisfies $\mathbf{A}{\ominus }_H\mathbf{A}=\mathbf{0}$ but the Hukuhara difference of A⊖ _H B can be calculated only when the length of A is greater than that of B. To overcome this inefficiency of the Hukuhara difference, Stefanni and Bede [32] introduced a generalized concept of the Hukuhara difference, namely gH-difference which satisfies the property $\mathbf{A}{\ominus }_{gH}\mathbf{A}=\mathbf{0}$ and the gH-difference can be calculated for any pair of intervals. Later, apart from Moore's interval arithmetic, Piegat and Landowski [30], Landowski [31] have introduced another concept of interval arithmetic, namely RDM interval arithmetic, which also assures that $\mathbf{A}-\mathbf{A}=\mathbf{0}$ for any interval A. Most of the results of RDM interval arithmetic are similar to the Moore's interval arithmetic except the substraction of an interval from itself. However, in this article we use Moore's interval arithmetic with gH-difference instead of RDM interval arithmetic (see Note 2 for the reason).

Calculus plays an important role to observe the characteristics of a function. On the calculus of IVFs, the concept of differentiability of IVFs, based on Hukuhara difference of intervals, was initially introduced by Hukuhara [18]. But this definition of Hukuhara differentiability is restrictive [10]. In general, if $\mathbf{F}\left(x\right)=\mathbf{C}\odot f\left(x\right),$ where C is a closed and bounded interval and f(x) is a real-valued function with f′(x) < 0, then F is not Hukuhara differentiable [5]. In order to refine the calculus of IVFs, the ideas of generalized Hukuhara continuity (gH-continuity), gH-differentiability and gH-gradient [13], [23], [32], [35] have been introduced with the help of Moore's interval arithmetic and gH-difference of intervals [32].

With the help of the existing calculus of IVFs, a few concepts about the problems of optimization and differential equation of IVFs have been discussed in [1], [5], [8], [19], [23], [36]. Wu [36], [37] used the concept of Hukuhara differentiability to study KKT conditions of optimization problems with IVFs. Wu [36], [37] has also illustrated the solution concept of optimization problems with IVFs by imposing a partial ordering on the set of all closed intervals and applying the existing calculus of IVFs. In 2013, the KKT conditions, based on gH-differentiability, of optimization problems with IVFs have been illustrated by Chalco-Cano and others [9]. After that, Bhurji and Panda [7] have defined interval-valued function in the parametric form and studied its properties and developed a methodology to study the existence of the efficient solution of an optimization problem with IVFs. Ghosh [13] has introduced a new definition of gH-differentiability and proposed a Newton method [13] and an updated Newton method [14] to capture the efficient solution of an optimization problem with IVFs. Recently, Ghosh et al. [16] have proposed the theory of alternatives and hence the KKT optimality condition for Interval Optimization Problems (IOPs). Importantly, it is shown in [16] that KKT optimality conditions appear with inclusion relations instead of equations.

A brief review and correctness of the existing literature on the algebra of gH-differentiable interval-valued functions has been reported in [12]. Apart from gH-differentiability of IVFs, with the help of so-called horizontal membership functions granular derivative has been introduced in [25]. An interesting notion on the H-difference for Z-number-valued analysis has been introduced in [3]. Alongside the analysis of interval-valued functions, set-valued functions are also studied in the literature, for instances, see [17], [38], and the references therein.

An abstract local tangency property for a gH-differntiable interval-valued functions and a KKT-like optimality conditions for nondominated solutions of constrained IOPs have been developed in [33]. A KKT-optimaility condition for interval and fuzzy optimization in several variables under total and directional generalized differentiability has been derived in [11]. A few optimality conditions with inclusion relation for gH-differentiable interval-valued functions has been derived in [29]. For a detailed survey on optimality conditions for IOPs, one can refer to the references of Stefanini and Manuel [33] and Ghosh et al. [15].

In this article, we endeavor to develop the notions of generalized derivatives, such as gH-Gâteaux derivative and gH-Fréchet derivative for IVFs. The main motivation behind the study is the following two points—

•

An interval-valued function can be observed as a bunch of infinitely many real-valued functions [6], and for a vector-valued (possibly infinite dimensional) function the idea of a derivative is the Fréchet derivative.

•

To develop some numerical algorithm to capture the complete set of efficient solutions of an IOP, the idea of directional derivative and Gâteaux derivative inevitably arise.

Although in [4] the authors defined gH-directional derivative, the proposed definition of gH-directional derivative in this article is more simple, and we prove that both have identical meaning. To define gH-Gâteaux derivative and gH-Fréchet derivative and to study their properties, we propose the concepts of linear IVF and bounded linear IVF. In the sequel, we also define monotonic and bounded IVFs. Throughout the paper we use the ordering concept of intervals given in [6] which has an important role in investigating efficient solutions of an interval optimization problem [7], [13], [15], [21], [22].

Rest of the article is presented in the following sequences. The next section provides the basic terminologies and definitions on intervals and IVFs. In the Section 3, we define the gH-directional derivative of IVFs and prove that the gH-directional derivative of an interval-valued convex function always exists. Further, with the help of gH-directional derivative, we provide a necessary and sufficient condition for an efficient point of an interval optimization problem (IOP) with IVFs. The concept of linearity and the gH-Gâteaux derivative for IVFs are illustrated in Section 4. In Section 4, we also prove that at an efficient point of an interval-valued function its gH-Gâteaux derivative must contain zero. After that, the notion of gH-Fréchet derivative and its interrelation with gH-Gâteaux derivative are discussed in Section 5. It is also shown there that the chain rule holds for the gH-Fréchet derivative of an IVFs. Finally, the last section draws future directions of the study.

Section snippets

Preliminaries and terminologies

In this section, we provide basic terminologies and notations on intervals and IVFs those are used throughout the paper.

gH-Directional derivative of interval-valued functions

In this section, we define the gH-directional derivative of IVFs and prove its existence for a convex function. Further, we present an optimality condition for efficient point of an IOP with IVFs.

Definition 3.1

(gH-directional derivative). Let F be an IVF on a nonempty subset $\mathcal{X}$ of ${\mathbb{R}}^n$ . Let $\overline{x}\in \mathcal{X}$ and $h\in {\mathbb{R}}^n$ . If the limit $\underset{\lambda \to 0+}{\lim }\frac{1}{\lambda }\odot \left(\mathbf{F}(\overline{x}+\lambda h){\ominus }_{gH}\mathbf{F}\left(\overline{x}\right)\right)$ exists, then the limit is said to be gH-directional derivative of F at $\overline{x}$ in the direction h, and it is denoted by ${\mathbf{F}}_{\mathcal{D}}\left(\overline{x}\right)\left(h\right)$ .

Example 3.1

For instance, we consider the function $\mathbf{F}(x_1,x_{}$

gH-Gâteaux derivative of interval-valued functions

In this section we define the linearity concept and gH-Gâteaux derivative for IVFs.

Definition 4.1

(Linear IVF). Let $\mathcal{Y}$ be a linear subspace of ${\mathbb{R}}^n$ . The function $\mathbf{F}:\mathcal{Y}\to I\left(\mathbb{R}\right)$ is said to be linear if

(i)

$\mathbf{F}\left(\lambda x\right)=\lambda \odot \mathbf{F}\left(x\right)\text{for}\text{all}x\in \mathcal{Y}\text{and}\text{for}\text{all}\lambda \in \mathbb{R}$ and

(ii)

for all $x,y\in \mathcal{Y},$ $\text{either}\mathbf{F}\left(x\right)\oplus \mathbf{F}\left(y\right)=\mathbf{F}(x+y)\text{or}\text{none}\text{of}\mathbf{F}\left(x\right)\oplus \mathbf{F}\left(y\right)\text{and}\mathbf{F}(x+y)\text{dominates}\text{the}\text{other.}$

Note 5

Let $\mathbf{F}:\mathcal{Y}\to I\left(\mathbb{R}\right)$ be a linear IVF. If $\mathbf{F}\left(x\right)=[\underline{f}\left(x\right),\overline{f}\left(x\right)],$ then $\underline{f}\left(\lambda x\right)=\lambda \overline{f}\left(x\right)$ and $\overline{f}\left(\lambda x\right)=\lambda \underline{f}\left(x\right)$ for λ < 0.

Lemma 4.1

If an IVF $\mathbf{F}:\mathcal{Y}\to I\left(\mathbb{R}\right)$ on a linear subspace $\mathcal{Y}$ of ${\mathbb{R}}^n$ is linear, then $\mathbf{F}\left(x\right)\nprec \mathbf{0}⟹\mathbf{0}\nprec \mathbf{F}(-x).$

Proof

Let F(x)⊀0. Then, $\begin{array}{ccc}& & [\underline{f}\left(x\right),\overline{f}\left(x\right)\hfill \end{array}$

gH-Fréchet derivatives of interval-valued functions

It is noteworthy that gH-Gâteaux derivative does not imply the gH-continuity of interval-valued function. For instance, consider the following example.

Example 5.1

Let $\mathbf{C}=[a,b]$ and $\mathbf{F}:{\mathbb{R}}^2\to I\left(\mathbb{R}\right)$ be defined by $\mathbf{F}(x_1,x_2)=\{\begin{array}{cc}\left(\frac{x_1^8{x}_2^2}{x_1^{16}+x_2^4}\right)\odot \mathbf{C}& \text{if}(x_1,x_2)\ne (0,0)\\ \mathbf{0}& \text{otherwise}.\end{array}$ Then gH-directional derivative of F at $(0,0)\in {\mathbb{R}}^2$ in the direction $(h_1,h_2)\in {\mathbb{R}}^2$ is given by ${\mathbf{F}}_{\mathcal{D}}(0,0)\left(h\right)=\underset{\lambda \to 0+}{\lim }\frac{1}{\lambda }\odot \left(\mathbf{F}((0,0)\oplus \lambda (h_1,h_2)){\ominus }_{gH}\mathbf{F}(0,0)\right)=\underset{\lambda \to 0+}{\lim }\frac{1}{\lambda }\odot \mathbf{F}\left(\lambda h\right)=\mathbf{0}.$ Clearly ${\mathbf{F}}_{\mathcal{D}}(0,0)$ is gH-continuous and linear in h. Therefore, F has gH-Gâteaux derivative at (0, 0).

Conclusion and future directions

In this article, majorly four concepts on interval-valued functions have been studied—gH-directional derivative (Definition 3.1), gH-Gatéaux derivative (Definition 4.3), gH-Fréchet derivative (Definition 5.1) and linear IVFs (Definition 4.1). It has been shown that the Gatéaux and Fréchet derivative values are helpful to characterize convexity of an IVF (Theorem 4.1) and to characterize the efficient solutions of an IOP (Theorems 3.2 and 4.2) with interval-valued objective functions. One can

Declaration of Competing Interest

None.

Acknowledgement

The authors put a sincere thanks to the reviewers and editors for their valuable comments to enhance the paper. Authors are also grateful to Prof. H.-C. Wu, Department of Mathematics, National Kaohsiung Normal University, Taiwan for his comments and suggestions on an initial version of the manuscript. The first and fourth authors gratefully acknowledge the financial support through the Early Career Research Award (ECR/2015/000467), Science & Engineering Research Board, Government of India. The

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Source: https://www.sciencedirect.com/science/article/abs/pii/S0020025519308746