Next Article in Journal
Parallelization of Modified Merge Sort Algorithm
Next Article in Special Issue
Asymmetric Equivalences in Fuzzy Logic
Previous Article in Journal
The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions

College of Science, Chongqing University of Post and Telecommunication, Chongqing 400065, China
*
Author to whom correspondence should be addressed.
Symmetry 2017, 9(9), 177; https://doi.org/10.3390/sym9090177
Original submission received: 13 July 2017 / Revised: 5 August 2017 / Accepted: 31 August 2017 / Published: 1 September 2017
(This article belongs to the Special Issue Symmetry in Fuzzy Sets and Systems)

## Abstract

:
In this paper, based on a partial order, we study the characterizations of directional derivatives and the subdifferential of fuzzy function. At the same time, we also discuss the relation between the directional derivative and the subdifferential.

## 1. Introduction

In 1965, Zadeh  introduced concepts and operations with respect to fuzzy sets, and many authors contributed to the development of fuzzy set theory and applications. Later, Zadeh proposed the fuzzy number [2,3,4] and put forward the theory of the fuzzy numerical function together with Chang . These theories and those associated with optimization theory have been extensively studied in some fields, such as economics, engineering, the stock market, greenhouse gas emissions and management science [6,7,8,9,10,11,12,13,14].
In order to solve complex optimization problems in real life, various optimization algorithms have been presented in [9,10,15,16,17,18,19,20,21,22,23]. Jia et al.  presented a new algorithm for solving the optimization problem based on the stock exchange. Afterwards, in [10,15,16], a multiple genetic algorithm and multi-objective differential evolution were used to solve multiple optimization problems efficiently. Moreover, Wah et al.  applied a genetic algorithm to optimize flow rectification efficiency of the diffuser element based on the valveless diaphragm micropump application. Precup et al.  applied the grey wolf optimizer algorithm to deal with the fuzzy optimization problem. In , in order to solve the meta-heuristics optimization problem quickly, a bio-inspired optimization algorithm based on fuzzy logic was proposed. Peraza et al.  presented a new algorithm that can solve the complex optimization problems based on uncertainty management. They  introduced a fuzzy harmony search algorithm with fuzzy logic and this algorithm was utilized to solve the fuzzy optimization problem. Amador et al. , presented a new optimization algorithm based on the fuzzy logic system.
It is well known that convexity plays a key role in fuzzy optimization theory. Therefore, the properties of convexity of fuzzy function and related problems are attached a wide range of research [24,25,26,27,28,29,30]. Subdifferentials are very important tools in convex fuzzy optimization theory. Based on a variety of different backgrounds, the derivative and differential of fuzzy function have been widely discussed. Goetschel et al. [31,32] defined the derivative of fuzzy function, which is a generalized derivative of the set-valued function. Afterwards, Buckley et al. [33,34] defined the derivatives of fuzzy function using left- and right-hand functions of its $α$-level sets and established sufficient conditions for the existence of fuzzy derivatives. Subsequently, Wang et al.  proposed the new concepts of directional derivative, differential and subdifferential of fuzzy function from $R m$ to $E 1$, and discussed the characterizations of directional derivative and differential of fuzzy function by using the directional derivative and the differential of two crisp functions that are determined.
In this paper, we investigate several characterizations of directional derivative of fuzzy function about the direction d, based on a partial order and introduce the concept of the subdifferential of fuzzy function.

## 2. Preliminaries

We denote by $K C$ the family of all bounded closed intervals in $R$, that is,
$K C = { [ a L , a R ] | a L , a R ∈ R a n d a L ≤ a R } .$
Given two intervals $A = [ a L , a R ]$ and $B = [ b L , b R ]$, the distance between A and B is defined by
$H ( A , B ) = max { | a L − b L | , | a R − b R | } .$
Then, $( K C , H )$ is a complete metric space .
Definition 1.
In reference , suppose that $E 1 = { u | u : R → [ 0 , 1 ] }$ satisfies the following conditions:
(1)
u is normal, that is, there exists $x 0 ∈ R$ such that $u ( x 0 ) = 1$;
(2)
u is upper semicontinuous;
(3)
u is convex, that is,
$u ( λ x + ( 1 − λ ) y ) ≥ m i n { u ( x ) , u ( y ) }$
for all $x , y ∈ R , λ ∈ [ 0 , 1 ]$;
(4)
$[ u ] 0 = { x ∈ R | u ( x ) > 0 } ¯$ is compact, where $A ¯$ denotes the closure of A.
Any $u ∈ E 1$, is called a fuzzy number. The $α$-level set of fuzzy number u is a closed and bounded interval $[ u L ( α ) , u R ( α ) ]$, where $u L ( α )$ denotes the left-hand end point of $[ u ] α$ and $u R ( α )$ denotes the right-hand end point of $[ u ] α$ .
For $u , v ∈ E 1 , k ∈ R$, the addition and scalar multiplication are defined by,
$( u + v ) ( x ) = sup s + t = x min { u ( s ) , v ( t ) } ,$
It is well known that for $u , v ∈ E 1 ,$ $k ∈ R$, then $u + v ,$ $k u ∈ E 1$, $[ u + v ] α = [ u ] α + [ v ] α$ and $[ k u ] α = k [ u ] α .$
For $x = ( x 1 , x 2 , … , x m ) ,$ $y = ( y 1 , y 2 , … , y m ) ∈ R m$, it is said that $x ≥ y$ if and only if $x i ≥ y i$ $( i = 1 , 2 , … , m )$. It is said that $x > y$ if and only if $x ≥ y$ and $x ≠ y .$
Definition 2.
In reference , for $u , v ∈ E 1$, then,
(1)
$u ⪯ v$ if and only if $[ u ] α = [ u L ( α ) ,$ $u R ( α ) ] ≤ [ v ] α = [ u L ( α ) , u R ( α ) ]$ for each $α ∈ [ 0 , 1 ] ,$ where $[ u ] α ≤ [ v ] α$ if and only if $u L ( α ) ≤ v L ( α )$ and $u R ( α ) ≤ v R ( α ) .$
(2)
$u ≺ v$ if and only if $u ⪯ v$ and there exists $α 0 ∈ [ 0 , 1 ]$ such that $u L ( α 0 ) < v L ( α 0 )$ or $u R ( α 0 ) < v R ( α 0 ) .$
(3)
if either $u ⪯ v$ or $v ⪯ u$, then u and v are comparable. Otherwise, u and v are non-comparable.
Definition 3.
In reference , if any $u , v ∈ E 1$, there exists $w ∈ E 1$ such that $u = v + w$, then the standard Hukuhara difference (H-difference) of u and v is defined by $u − ˜ v = w$.
Definition 4.
In reference , (Fuzzy function) let D be a convex set of $R$ and $F : D → E 1$ be a fuzzy function. The α-level set of F at $x ∈ D$, which is a closed and bounded interval, can be denoted by $[ F ( x ) ] α = [ F L ( x , α ) , F R ( x , α ) ] .$ Thus, F can be understood by the two functions $F L ( x , α )$ and $F R ( x , α ) ,$ which are functions from $D × [ 0 , 1 ]$ to the set of real numbers $R$, $F L ( x , α )$ is a bounded increasing function of α and $F R ( x , α )$ is a bounded decreasing function of $α .$ Moreover, $F L ( x , α ) ≤ F R ( x , α )$ for each $α ∈ [ 0 , 1 ] .$
Definition 5.
 For $u ,$ $v ∈ E 1$, the $d ∞$-distance is defined by the Hausdorff metric as,
Definition 6.
Let $X = ( a , b )$ and let $F : X → E 1$ be a fuzzy function and ${ F n ( x ) } : X → E 1$, $n ∈ N$ be a sequence of fuzzy function. If, for any $ε > 0 ,$ there exists a positive integer $M = M ( ε ) ∈ N$ such that,
$D ( F n ( x ) , F ( x ) ) < ε$
for any $x ∈ X$, all $n ≥ M$. Then the sequence ${ F n ( x ) }$ is convergent to $F ( x )$.
Definition 7.
In reference , let $F : X → E 1$ be a fuzzy function. Assume that the partial derivatives of $F L ( x , α ) , F R ( x , α )$ with respect to $x ∈ R$ for each $α ∈ [ 0 , 1 ]$ exist. The partial derivatives of $F L ( x , α )$ and $F R ( x , α )$ are denoted by $F L ′ ( x , α )$ and $F R ′ ( x , α )$, respectively. Let $Γ ( x , α ) = [ F L ′ ( x , α ) , F R ′ ( x , α ) ]$ for $x ∈ R ,$ $α ∈ [ 0 , 1 ] .$ $Γ ( x , α )$ defines the α-level set of fuzzy interval for $x ∈ R$. Then F is S-differentiable and is written as,
$[ d F ( x ) d x ] α = Γ ( x , α ) = [ F L ′ ( x , α ) , F R ′ ( x , α ) ]$
for $x ∈ R ,$ $α ∈ [ 0 , 1 ] .$

## 3. Directional Derivative of the Fuzzy Function

Inspired by , we discuss some relations among the gradient and directional derivative of fuzzy function. Moreover, several characterizations of the directional derivative of fuzzy function about the direction d are investigated, based on the partial order ⪯.
Definition 8.
In reference , (Gradient of a fuzzy function) let D be a convex set of $R m$ and $F : D → E 1$ be a fuzzy function. For $x ∈ D$ and stand for the partial differentiation with respect to the $i t h$ variable $x i$. For each $α ∈ [ 0 , 1 ]$, $F L ( x , α )$ and $F R ( x , α )$ have continuous partial derivatives so that and are continuous about x. Define
$[ ∂ F ( x ) ∂ x i ] α = [ ∂ F L ( x , α ) ∂ x i , ∂ F R ( x , α ) ∂ x i ]$
for each $i = 1 , 2 , … , m ,$ $α ∈ [ 0 , 1 ]$. If for each $i = 1 , 2 , … , m$, (1) defines the α-level set of fuzzy number, then F is S-differential at x. The gradient of the fuzzy function $F ( x )$ at x, denoted by $∇ ∼ F x$, is defined as:
Remark 1.
For the gradient of fuzzy function, we use the symbol $∇ ∼$, whereas for the gradient of a real valued function, we use the symbol .
Definition 9.
In reference , let D be a convex set of $R m$ and $F : D → E 1$ be a fuzzy function.
(1)
F is convex on D if
$F ( λ x 1 + ( 1 − λ ) x 2 ) ⪯ λ F ( x 1 ) + ( 1 − λ ) F ( x 2 )$
for any $x 1 , x 2 ∈ D$ and each $λ ∈ [ 0 , 1 ]$.
(2)
F is strictly convex on D if
$F ( λ x 1 + ( 1 − λ ) x 2 ) ≺ λ F ( x 1 ) + ( 1 − λ ) F ( x 2 )$
for any $x 1 , x 2 ∈ D$ with $x 1 ≠ x 2$ and each $λ ∈ ( 0 , 1 )$.
Theorem 1.
In reference , let D be a convex set of $R m$ and $F : D → E 1$ be a fuzzy function, F is convex on D, if and only if for each $α ∈ [ 0 , 1 ]$, $F L ( x , α )$ and $F R ( x , α )$ are convex on D, that is, for each $λ ∈ [ 0 , 1 ] ,$ $x 1 ,$$x 2 ∈ D$, and each $α ∈ [ 0 , 1 ]$,
$F L ( ( λ x 1 + ( 1 − λ ) x 2 ) , α ) ≤ λ F L ( x 1 , α ) + ( 1 − λ ) F L ( x 2 , α )$
and
$F R ( ( λ x 1 + ( 1 − λ ) x 2 ) , α ) ≤ λ F R ( x 1 , α ) + ( 1 − λ ) F R ( x 2 , α ) .$
Theorem 2.
Let D be a convex set of $R m$ and $F : D → E 1$ be a S-differentiable fuzzy function. Then F is a convex fuzzy function on D if and only if, for any $x 1 , x 2 ∈ D$ with $( x 1 > x 2 )$ such that
Proof.
F is a convex fuzzy function on D. According to Definition 9, we obtain that:
$F ( λ x 1 + ( 1 − λ ) x 2 ) ⪯ λ F ( x 1 ) + ( 1 − λ ) F ( x 2 )$
for any $x 1 , x 2 ∈ D$ with $x 1 > x 2$, and each $λ ∈ [ 0 , 1 ]$. By Theorem 1, for each $λ ∈ [ 0 , 1 ]$ we have that
$F L ( ( λ x 1 + ( 1 − λ ) x 2 ) , α ) ≤ λ F L ( x 1 , α ) + ( 1 − λ ) F L ( x 2 , α )$
and
$F R ( ( λ x 1 + ( 1 − λ ) x 2 ) , α ) ≤ λ F R ( x 1 , α ) + ( 1 − λ ) F R ( x 2 , α ) .$
Now combining (8) and (9) imply that
and
Taking limits for $λ → 0 +$, we get
$∇ F L ( x 2 , α ) T ( x 1 − x 2 ) ≤ F L ( x 1 , α ) − F L ( x 2 , α )$
and
$∇ F R ( x 2 , α ) T ( x 1 − x 2 ) ≤ F R ( x 1 , α ) − F R ( x 2 , α ) .$
That is,
Conversely, since F is a S-differentiable fuzzy function, and there exist $x , y ∈ D$ with $x > y$ such that
For any $x 1 , x 2 ∈ D$ and each $λ ∈ [ 0 , 1 ]$. Suppose that $x = x 1$ and $y = ( 1 − λ ) x 1 + λ x 2$. It follows that
That is,
$∇ F L ( y , α ) T ( x 1 − y ) ≤ F L ( x 1 , α ) − F L ( y , α )$
and
$∇ F R ( y , α ) T ( x 1 − y ) ≤ F R ( x 1 , α ) − F R ( y , α ) .$
Let $x = x 2$ and $y = ( 1 − λ ) x 1 + λ x 2$, we get
That is,
$∇ F L ( y , α ) T ( x 2 − y ) ≤ F L ( x 2 , α ) − F L ( y , α )$
and
$∇ F R ( y , α ) T ( x 2 − y ) ≤ F R ( x 2 , α ) − F R ( y , α ) .$
Now combining $( 14 ) × ( 1 − λ )$ and $( 17 ) × λ$, we have
$∇ F L ( y , α ) T ( ( 1 − λ ) x 1 + λ x 2 − y ) ≤ ( 1 − λ ) F L ( x 1 , α ) + λ F L ( x 2 , α ) − F L ( y , α ) .$
Similarly,
$∇ F R ( y , α ) T ( ( 1 − λ ) x 1 + λ x 2 − y ) ≤ ( 1 − λ ) F R ( x 1 , α ) + λ F R ( x 2 , α ) − F R ( y , α ) .$
The equations (19) and (20) imply
$F ( λ x 1 + ( 1 − λ ) x 2 ) ⪯ λ F ( x 1 ) + ( 1 − λ ) F ( x 2 ) .$
Therefore, F is a convex fuzzy function on D. ☐
Theorem 3.
Let D be a convex set of $R m$ and $F : D → E 1$ be a S-differentiable fuzzy function. Then F is a strictly convex fuzzy function on D if and only if, for any $x 1 , x 2 ∈ D$ with $( x 1 > x 2 )$ such that
Proof.
The proof is similar to the proof of Theorem 2. ☐
Theorem 4.
In reference , let D be a convex set of $R m$ and $f : D → ( − ∞ , + ∞ ]$ be a convex real valued function. For $x ∈ D ,$ let $d ∈ R m$ such that $x + λ d ∈ D$ for any $λ > 0$ and sufficiently small. If $h ( λ ) : ( 0 , + ∞ ) → ( − ∞ , + ∞ ]$ is defined by
$h ( λ ) = f ( x + λ d ) − f ( x ) λ ,$
then $h ( λ )$ is a nondecreasing function. Moreover, if f is differential at x, then
$lim λ → 0 + h ( λ ) = lim λ → 0 + f ( x + λ d ) − f ( x ) λ = ∇ f ( x ) T d .$
Definition 10.
In reference , (directional derivative of a fuzzy function) Let D be a convex set of $R m$ and $F : D → E 1$ be a fuzzy function. For $x ∈ D$, let $d ∈ R m$ such that $x + λ d$ for any $λ > 0$ and sufficiently small. The directional derivative of F at x along the vector d (if it exists) is a fuzzy number denoted by $F ′ ( x , d )$ whose α-level set is defined as:
where
and
Theorem 5.
Let D be a convex set of $R m$ and $F : D → E 1$ be a convex and S-differentiable fuzzy function. For $x ∈ D$, let $d ∈ R m$, $d = ( d 1 , d 2 , … , d m )$, $d i > 0 ,$ $i = 1 , 2 , … , m ,$ for $m ∈ N$. The directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$. Then
$F ′ ( x , d ) ⪯ F ( x + d ) − ˜ F ( x ) .$
Proof.
Since $F : D → E 1$ is a convex and S-differentiable fuzzy function. From Theorem 2, for $x ∈ D ,$ $d ∈ R m$, we obtain
$∇ ˜ F ( x ) T d ⪯ F ( x + d ) − ˜ F ( x ) .$
That is,
$∇ F L ( x , α ) T d ≤ F L ( x + d , α ) − F L ( x , α )$
and
$∇ F R ( x , α ) T d ≤ F R ( x + d , α ) − F R ( x , α ) .$
Since F is a convex fuzzy function. From Definition 10 and Theorem 4, we conclude that
$F L ′ ( ( x , d ) , α ) = lim λ → 0 + F L ( x + λ d , α ) − F L ( x , α ) λ = ∇ F L ( x , α ) T d$
and
$F R ′ ( ( x , d ) , α ) = lim λ → 0 + F R ( x + λ d , α ) − F R ( x , α ) λ = ∇ F R ( x , α ) T d$
Now combining (23), (24), (25) and (26), we have
$F L ′ ( ( x , d ) , α ) = ∇ F L ( x , α ) T d ≤ F L ( x + d , α ) − F L ( x , α )$
and
$F R ′ ( ( x , d ) , α ) = ∇ F R ( x , α ) T d ≤ F R ( x + d , α ) − F R ( x , α ) .$
The Equations (27) and (28) imply $F ′ ( x , d ) ⪯ F ( x + d ) − ˜ F ( x ) .$ ☐
Theorem 6.
Let D be a convex set of $R m$ and $F : D → E 1$ be a S-differentiable fuzzy function. For $x ∈ D$, let $d ∈ R m$ such that $x + λ d ∈ D$ for any $λ > 0$ and sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$. Then $F ′ ( x , d )$ is a strictly positive homogeneous fuzzy function.
Proof.
Since the directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$. By Definition 10, we have,
and
For any $t > 0$ and let $τ = t λ$, we obtain
and
Now combining (29) and (30), we have that
Therefore, $F ′ ( x , d )$ is a strictly positive homogeneous fuzzy function. ☐
Theorem 7.
Let D be a convex set of $R m$ and $F : D → E 1$ be a convex and S-differential fuzzy function. For $x ∈ D$, let $d ∈ R m$ such that $x + λ d ∈ D$ for any $λ > 0$ sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$. Then $F ′ ( x , d )$ is a convex fuzzy function about the direction d.
Proof.
For any $λ 1 , λ 2 ∈ ( 0 , 1 )$, any $d 1 , d 2 ∈ R m$, let $λ 1 = 1 − λ 2 .$ By Definition 10 and Theorem 1, we get that
and
Hence, by Theorem 1, $F ′ ( x , d )$ is a convex fuzzy function about the direction d. ☐
Theorem 8.
Let D be a convex set of $R m$ and $F : D → E 1$ be a convex and S-differential fuzzy function. For $x ∈ D$, let $d ∈ R m$ such that $x + λ d ∈ D$ for any $λ > 0$ sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$. Then $F ′ ( x , d )$ is subadditive about the direction d.
Proof.
F is a S-differential fuzzy function on D, and the directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$. For arbitrary $d 1 , d 2 ∈ R m$. By Theorem 7, we have that
$F ′ ( x , 1 2 d 1 + 1 2 d 2 ) ⪯ 1 2 F ′ ( x , d 1 ) + 1 2 F ′ ( x , d 2 ) .$
By Theorem 6, we obtain that
$F ′ ( x , 1 2 d 1 + 1 2 d 2 ) = 1 2 F ′ ( x , d 1 + d 2 ) .$
Now combining (33) and (34), it follows that
$F ′ ( x , d 1 + d 2 ) ⪯ F ′ ( x , d 1 ) + F ′ ( x , d 2 ) .$
Therefore, $F ′ ( x , d )$ is subadditive about the direction d. ☐
Definition 11.
In reference , let D be a convex set of $R m$ and $F : D → E 1$ be a fuzzy function. Let $x ¯ ∈ D$, if there exists $δ 0 > 0$ and no $x ∈ U ( x ¯ , δ 0 ) ⋂ D$ such that $F ( x ) ⪯ F ( x ¯ )$, then $x ¯$ is a local minimum solution of $F ( x )$.
Theorem 9.
Let D be a convex set of $R m$ and $F : D → E 1$ be a fuzzy function. For $x ¯ ∈ D$, let $d ∈ R m$ such that $x ¯ + λ d ∈ D$ for any $λ > 0$ sufficiently small. The directional derivative of F at $x ¯$ along the vector d is a fuzzy number denoted by $F ′ ( x ¯ , d )$ if $0 ∈$ inter $[ F ′ ( x ¯ , d ) ] 0$, where inter A denotes the interior of the set $A .$ Then, $x ¯$ is a local minimum solution of $F ( x )$.
Proof.
Suppose that $x ¯$ is not a local minimum solution of $F ( x )$. Hence, there exists a sequence ${ x n } n = 1 ∞$ and any $δ > 0$ such that $x n = x ¯ + λ d ∈ U ( x ¯ , δ ) ⋂ D$ ($| λ d | < δ$) and
$F ( x ¯ + λ d ) = F ( x n ) ⪯ F ( x ¯ )$
for all $n ∈ N$. That is,
$F L ( x ¯ + λ d , α ) ≤ F L ( x ¯ , α )$
and
$F R ( x ¯ + λ d , α ) ≤ F R ( x ¯ , α ) .$
for each $α ∈ [ 0 , 1 ] .$ From Definition 10, we conclude that
and
for each $α ∈ [ 0 , 1 ] .$ Hence, we have $0 ∉$ inter $[ F ′ ( x ¯ , d ) ] 0$. This is a contradiction with the hypothesis. Then $x ¯$ is a local minimum solution of $F ( x )$. ☐

## 4. Subdifferential of Fuzzy Function

Definition 12.
(Subdifferential of a fuzzy function) Let D be a convex set of $R m$ and $F : D → E 1$ be a fuzzy number-valued function. For $x ¯ ∈ D$, let $d ∈ R m$ such that $x ¯ + λ d ∈ D$ for any $λ > 0$ and sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$.
(1)
A fuzzy function $l ( d ) : R m → E 1$ with
$l ( d ) ⪯ F ′ ( x ¯ , d ) f o r a l l d ∈ R m .$
Then $l ( d )$ is called a subgradient of F at $x ¯ .$
(2)
Define the set
$∂ F ( x ¯ ) = { l ( d ) | l ( d ) ⪯ F ′ ( x ¯ , d ) f o r a l l d ∈ R m }$
The set $∂ F ( x ¯ )$ of all subgradients of F at $x ¯$ is called the subdifferential of F at $x ¯$.
Now, we present some basic properties of subdifferential of fuzzy function.
Theorem 10.
Let D be a convex set of $R m$ and $F : D → E 1$ be a S-differential fuzzy function. For $x ¯ ∈ D$, let $d ∈ R m$ such that $x ¯ + λ d ∈ D$ for any $λ > 0$ and is sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$. Then, $∂ F ( x ¯ )$ is convex.
Proof.
Take any $l 1 ( d ) ,$ $l 2 ( d ) ∈ ∂ F ( x ¯ )$ and $λ ∈ [ 0 , 1 ]$. By Definition 12, it follows that
$l 1 ( d ) ⪯ F ′ ( x ¯ , d )$
and
$l 2 ( d ) ⪯ F ′ ( x ¯ , d )$
that is,
$l L 1 ( d , α ) ≤ F L ′ ( ( x ¯ , d ) , α )$
$l R 1 ( d , α ) ≤ F R ′ ( ( x ¯ , d ) , α ) ,$
and
$l L 2 ( d , α ) ≤ F L ′ ( ( x ¯ , d ) , α )$
$l R 2 ( d , α ) ≤ F R ′ ( ( x ¯ , d ) , α ) .$
Now combining $λ × ( 39 )$ and $( 1 − λ ) × ( 41 )$, we have that
Similarly, we obtain
The Equations (43) and (44) imply
$λ × l 1 ( d ) + ( 1 − λ ) l 2 ( d ) ⪯ F ′ ( x ¯ , d ) .$
By Definition 12, we obtain
$λ × l 1 ( d ) + ( 1 − λ ) l 2 ( d ) ∈ ∂ F ( x ¯ ) .$
Then $∂ F ( x ¯ )$ is convex. ☐
Theorem 11.
Let D be a convex set of $R m$ and $F : D → E 1$ be a fuzzy function. For $x ¯ ∈ D$, let $d ∈ R m$ such that $x ¯ + λ d ∈ D$ for any $λ > 0$ and sufficiently small. The directional derivative of F at x along the vector d is a fuzzy number denoted by $F ′ ( x , d )$. Then $∂ F ( x ¯ )$ is closed.
Proof.
Take an arbitrary sequence of fuzzy functions ${ l n ( d ) } n ∈ N$ of subgradient is convergent to fuzzy functions $l ( d )$. By Definition 6, for any $ε > 0 ,$ there exists a positive integer $M = M ( ε ) ∈ N$ such that
$D ( l n ( x ) , l ( x ) ) < ε$
for any $d ∈ R m$, all $n ≥ M .$ Hence, by Definition 5, we obtain that
$sup α ∈ [ 0 , 1 ] max { | l n . L ( d , α ) − l L ( d , α ) | , | l n . R ( d , α ) − l R ( d , α ) | } < ε .$
Therefore, for any $ε > 0 ,$ there exists a positive integer $M = M ( ε ) ∈ N$ such that $| l n . L ( d , α ) − l L ( d , α ) | < ε$ and $| l n . R ( d , α ) − l R ( d , α ) | < ε$ for any $d ∈ R m$, all $n ≥ M$, and each $α ∈ [ 0 , 1 ] .$
That is,
$lim n → ∞ l n . L ( d , α ) = l L ( d , α )$
and
$lim n → ∞ l n . R ( d , α ) = l R ( d , α )$
In view of Definition 12, we get
$l n ( d ) ⪯ F ′ ( x ¯ , d ) .$
That is,
$l n . L ( d , α ) ≤ F L ′ ( ( x ¯ , d ) , α )$
and
$l n . R ( d , α ) ≤ F R ′ ( ( x ¯ , d ) , α ) .$
Now combining (46), (47), (48) and (49), we obtain
$l L ( d , α ) ≤ F L ′ ( ( x ¯ , d ) , α )$
and
$l R ( d , α ) ≤ F R ′ ( ( x ¯ , d ) , α ) .$
That is,
$l ( d ) ⪯ F ′ ( x ¯ , d ) .$
Thus, $l ( d )$ is a subgradient. That is, the subdifferential $∂ F ( x ¯ )$ is closed. ☐

## 5. Conclusions

We have investigated several characterizations of directional derivative of fuzzy function about the direction, based on the partial order. For example, we present strictly positive homogeneity, convexity and subadditivity of directional derivative of fuzzy functions. And we also propose the sufficient optimality condition for fuzzy optimization problems. Afterwards, we introduce the concept of the subdifferential of convex fuzzy function. And we present some basic characterizations of subdifferential of fuzzy function and application in the convex fuzzy programming. Thus, we will apply the subdifferentiablity of fuzzy function to deal with the multiobjective fuzzy optimization problem in the future. Constrained optimization problems involving fuzzy functions are an interesting field for future study. For example, finance represents a good field to implement models for sensitive analysis through fuzzy mathematics. Several authors are working hard to shape sources of uncertainty: prices, interest rates, volatilities, etc. (see Guerra et al. , Buckley ). Therefore, fuzzy optimization problems based on parameter uncertainty sources are a topic of interest in many applications. Inspired by [20,22], directional derivatives and subdifferentials of fuzzy functions will be extensively applied in some fields, such as economics, engineering, stock market greenhouse gas emission and interest rates.

## Acknowledgments

This work was supported by The National Natural Science Foundations of China (Grant Nos. 11671001 and 61472056).

## Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

## Conflicts of Interest

The authors declare that they have no competing interests.

## References

1. Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
2. Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci. 1975, 8, 199–249. [Google Scholar] [CrossRef]
3. Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning-II. Inf. Sci. 1975, 8, 301–357. [Google Scholar] [CrossRef]
4. Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning-III. Inf. Sci. 1975, 9, 43–80. [Google Scholar] [CrossRef]
5. Chang, S.S.L.; Zadeh, L.A. On fuzzy mappings and control. IEEE Trans. Syst. Man Cybern 1972, 2, 30–34. [Google Scholar] [CrossRef]
6. Jelušič, P.; Žlender, B. Discrete optimization with fuzzy constraints. Symmetry 2017, 9, 87. [Google Scholar] [CrossRef]
7. Qiu, D.; Xing, Y.; Chen, S. Solving multi-Objective Matrix Games with Fuzzy Payoffs through the Lower Limit of the Possibility Degree. Symmetry 2017, 9, 130. [Google Scholar] [CrossRef]
8. Xiao, Y.; Zhang, T.; Ding, Z.; Li, C. The Study of Fuzzy Proportional Integral Controllers Based on Improved Particle Swarm Optimization for Permanent Magnet Direct Drive Wind Turbine Converters. Energies 2016, 9, 343. [Google Scholar] [CrossRef]
9. Jia, J.; Zhao, A.; Guan, S. Forecasting Based on High-Order Fuzzy-Fluctuation Trends and Particle Swarm Optimization Machine Learning. Symmetry 2017, 9, 124. [Google Scholar] [CrossRef]
10. Li, K.; Pan, L.; Xue, W.; Jiang, H.; Mao, H. Multi-Objective Optimization for Energy Performance Improvement of Residential Buildings: A Comparative Study. Energies 2017, 10, 245. [Google Scholar] [CrossRef]
11. Xu, J.; Wang, Z.; Wang, J.; Tan, C.; Zhang, L.; Liu, X. Acoustic-Based Cutting Pattern Recognition for Shearer through Fuzzy C-Means and a Hybrid Optimization Algorithm. Appl. Sci. 2016, 6, 294. [Google Scholar] [CrossRef]
12. Huang, C.-W.; Lin, Y.-P.; Ding, T.-S.; Anthony, J. Developing a Cell-Based Spatial Optimization Model for Land-Use Patterns Planning. Sustainability 2014, 6, 9139–9158. [Google Scholar] [CrossRef]
13. Wang, X.; Ma, J.-J.; Wang, S.; Bi, D.-W. Distributed Particle Swarm Optimization and Simulated Annealing for Energy-efficient Coverage in Wireless Sensor Networks. Sensors 2007, 7, 628–648. [Google Scholar] [CrossRef]
14. Cai, X.; Zhu, J.; Pan, P.; Gu, R. Structural Optimization Design of Horizontal-Axis Wind Turbine Blades Using a Particle Swarm Optimization Algorithm and Finite Element Method. Energies 2012, 5, 4683–4696. [Google Scholar] [CrossRef]
15. Sato, H. MOEA/D using constant-distance based neighbors designed for many-objective optimization. In Proceedings of the 2015 IEEE Congress on Evolutionary Computation, Sendai, Japan, 25–28 May 2015; pp. 2867–2874. [Google Scholar]
16. Feng, S.; Yang, Z.; Huang, M. Hybridizing Adaptive Biogeography-Based Optimization with Differential Evolution for Multi-Objective Optimization Problems. Information 2017, 8, 83. [Google Scholar] [CrossRef]
17. Wah, L.; Abdul, A.I.H. Neuro-Genetic Optimization of the Diffuser Elements for Applications in a Valveless Diaphragm Micropumps System. Sensors 2009, 9, 7481–7497. [Google Scholar]
18. Precup, R.E.; David, R.C.; Szedlak-Stinean, A.I.; Petriu, E.M.; Dragan, F. An Easily Understandable Grey Wolf Optimizer and Its Application to Fuzzy Controller Tuning. Algorithms 2017, 10, 68. [Google Scholar] [CrossRef]
19. Caraveo, C.; Valdez, F.; Castillo, O. A New Meta-Heuristics of Optimization with Dynamic Adaptation of Parameters Using Type-2 Fuzzy Logic for Trajectory Control of a Mobile Robot. Algorithms 2017, 10, 85. [Google Scholar] [CrossRef]
20. Peraza, C.; Valdez, F.; Melin, P. Optimization of Intelligent Controllers Using a Type-1 and Interval Type-2 Fuzzy Harmony Search Algorithm. Algorithms 2017, 10, 82. [Google Scholar] [CrossRef]
21. Bernal, E.; Castillo, O.; Soria, J.; Valdez, F. Imperialist Competitive Algorithm with Dynamic Parameter Adaptation Using Fuzzy Logic Applied to the Optimization of Mathematical Functions. Algorithms 2017, 10, 18. [Google Scholar] [CrossRef]
22. Peraza, C.; Valdez, F.; Garcia, M.; Melin, P.; Castillo, O. A New Fuzzy Harmony Search Algorithm Using Fuzzy Logic for Dynamic Parameter Adaptation. Algorithms 2016, 9, 69. [Google Scholar] [CrossRef]
23. Amador-Angulo, L.; Mendoza, O.; Castro, J.R.; Rodrguez-Diaz, A.; Melin, P.; Castillo, O. Fuzzy Sets in Dynamic Adaptation of Parameters of a Bee Colony Optimization for Controlling the Trajectory of an Autonomous Mobile Robot. Sensors 2016, 16, 1458. [Google Scholar] [CrossRef] [PubMed]
24. Ammar, E.E. On convex fuzzy mapping. J. Fuzzy Math. 2006, 14, 501–512. [Google Scholar]
25. Ammar, E.E. On fuzzy convexity and parametric fuzzy optimization. Fuzzy Sets Syst. 1992, 49, 135–141. [Google Scholar] [CrossRef]
26. Nanda, S.; Kar, K. Convex fuzzy mappings. Fuzzy Sets Syst. 1992, 48, 129–132. [Google Scholar] [CrossRef]
27. Rockafellar, R.T. Convex Analysis; Princeton University Press: Princeton, NJ, USA, 2015; Volume 17, pp. 5–101. [Google Scholar]
28. Syau, Y.R. Differentiability and convexity of fuzzy mappings. Comput. Math. Appl. 2001, 41, 73–81. [Google Scholar] [CrossRef]
29. Wang, G.X.; Wu, C.X. Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming. Fuzzy Sets Syst. 2003, 138, 559–591. [Google Scholar] [CrossRef]
30. Yang, X.M.; Teo, K.L.; Yang, X.Q. A characterization of convex function. Appl. Math. Lett. 2000, 13, 27–30. [Google Scholar] [CrossRef]
31. Goetschel, R.; Voxman, W. Elementary fuzzy calculus. Fuzzy Sets Syst. 1986, 18, 31–43. [Google Scholar] [CrossRef]
32. Puri, M.L.; Ralescu, D.A. Differentials of fuzzy functions. J. Math. Anal. Appl. 1983, 91, 552–558. [Google Scholar] [CrossRef]
33. Buckley, J.J.; Feuring, T. Introduction to fuzzy partial differential equations. Fuzzy Sets Syst. 1999, 105, 241–248. [Google Scholar] [CrossRef]
34. Buckley, J.J.; Eslami, E.; Feuring, T. Fuzzy differential equations. Fuzzy Sets Syst. 2000, 110, 43–54. [Google Scholar] [CrossRef]
35. Diamond, P.; Kloeden, P. Metric spaces of fuzzy sets. Fuzzy Sets Syst. 1990, 35, 241–249. [Google Scholar] [CrossRef]
36. Panigrahi, M.; Panda, G.; Nanda, S. Convex fuzzy mapping with differentiability and its application in fuzzy optimization. Eur. J. Oper. Res. 2000, 185, 47–62. [Google Scholar] [CrossRef]
37. Li, L.; Liu, S.Y.; Zhang, J.K. On fuzzy generalized convex mappings and optimality conditions for fuzzy weakly univex mappings. Fuzzy Sets Syst. 2015, 280, 107–132. [Google Scholar] [CrossRef]
38. Gong, Z.T.; Hai, S.X. Convexity of n-dimensional fuzzy functions and its applications. Fuzzy Sets Syst. 2016, 295, 19–36. [Google Scholar] [CrossRef]
39. Lin, G.H. The Basis of Nonlinear Optimization; Science Publishing Company: Beijing, China, 2010. [Google Scholar]
40. Osuna-Góomez, R.; Chalco-Cano, Y.; Rufiáan-Lizana, A.; Hernáandez-Jiméenez, B. Necessary and sufficient conditions for fuzzy optimality problem. Fuzzy Sets Syst. 2016, 296, 112–123. [Google Scholar] [CrossRef]
41. Guerra, M.L.; Sorini, L.; Stefanini, L. Options priece sensitives through fuzzy numbers. Comput. Math. Appl. 2011, 61, 515–526. [Google Scholar] [CrossRef]
42. Buckley, J.J. The fuzzy mathematics of finance. Fuzzy Sets Syst. 1987, 21, 257–273. [Google Scholar] [CrossRef]

## Share and Cite

MDPI and ACS Style

Zhang, W.; Xing, Y.; Qiu, D. On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions. Symmetry 2017, 9, 177. https://doi.org/10.3390/sym9090177

AMA Style

Zhang W, Xing Y, Qiu D. On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions. Symmetry. 2017; 9(9):177. https://doi.org/10.3390/sym9090177

Chicago/Turabian Style

Zhang, Wei, Yumei Xing, and Dong Qiu. 2017. "On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions" Symmetry 9, no. 9: 177. https://doi.org/10.3390/sym9090177

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.