On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions

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.


Introduction
In 1965, Zadeh [1] 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 [5].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. [9] 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. [17] applied a genetic algorithm to optimize flow rectification efficiency of the diffuser element based on the valveless diaphragm micropump application.Precup et al. [18] applied the grey wolf optimizer algorithm to deal with the fuzzy optimization problem.In [19], in order to solve the meta-heuristics optimization problem quickly, a bio-inspired optimization algorithm based on fuzzy logic was proposed.Peraza et al. [20] presented a new algorithm that can solve the complex optimization problems based on uncertainty management.They [22] introduced a fuzzy harmony search algorithm with fuzzy logic and this algorithm was utilized to solve the fuzzy optimization problem.Amador et al. [23], 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. [29] 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.

Preliminaries
We denote by K C the family of all bounded closed intervals in R, that is, the distance between A and B is defined by Then, (K C , H) is a complete metric space [35].

Definition 1.
In reference [27], suppose that E 1 = {u|u : R → [0, 1]} satisfies the following conditions: (1) u is normal, that is, there exists 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 [36].
For u, v ∈ E 1 , k ∈ R, the addition and scalar multiplication are defined by, 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 [29], for u, v ∈ E 1 , then, (2) u ≺ v if and only if u v and there exists (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 [37], 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 [37], (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. [35] For u, v ∈ E 1 , the d ∞ -distance is defined by the Hausdorff metric as, Definition 7. In reference [34], let F : X → E 1 be a fuzzy function.Assume that the partial derivatives of

Directional Derivative of the Fuzzy Function
Inspired by [29], 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 [36], (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 ∂ ∂x i (i = 1, 2, ..., m) stand for the partial differentiation with respect to the ith variable x i .For each α ∈ [0, 1], F L (x, α) and F R (x, α) have continuous partial derivatives so that ∂F L (x,α) 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 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 [38], let D be a convex set of R m and F : D → E 1 be a fuzzy function. ( (3) for any x 1 , x 2 ∈ D with x 1 = x 2 and each λ ∈ (0, 1).
Theorem 1.In reference [36], 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], 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 , Proof.F is a convex fuzzy function on D. According to Definition 9, we obtain that: 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 Now combining ( 8) and ( 9) imply that and Taking limits for λ → 0 + , we get Conversely, since F is a S-differentiable fuzzy function, and there exist x, y ∈ D with x > y such that That is, Let x = x 2 and y = (1 That is, Now combining (14) × (1 − λ) and (17) × λ, we have Similarly, The equations ( 19) and (20) imply 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 , Proof.The proof is similar to the proof of Theorem 2.

Theorem 4.
In reference [39], 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 + λd ∈ D for any λ > 0 and sufficiently small.If h(λ) : (0, +∞) → (−∞, +∞] is defined by Definition 10.In reference [36], (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 + λd 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 Proof.Since F : D → E 1 is a convex and S-differentiable fuzzy function.From Theorem 2, for That is, Since F is a convex fuzzy function.From Definition 10 and Theorem 4, we conclude that and Now combining ( 23), ( 24), ( 25) and ( 26), we have 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 + λd ∈ 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, For any t > 0 and let τ = tλ, we obtain 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 + λd ∈ 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 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 + λd ∈ 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 By Theorem 6, we obtain that Now combining (33) and (34), it follows that Therefore, F (x, d) is subadditive about the direction d.

Definition 11.
In reference [40], 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 + λd ∈ 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 + λd ∈ U( x, δ) D (|λd| < δ) and for all n ∈ N.That is, for each α ∈ [0, 1].From Definition 10, we conclude that 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).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 + λd ∈ 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.
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 + λd ∈ 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 (45) for any d ∈ R m , all n ≥ M. Hence, by Definition 5, we obtain that sup Therefore, for any ε > 0, there exists a positive integer  That is, l(d) F ( x, d).
Thus, l(d) is a subgradient.That is, the subdifferential ∂F( x) is closed.

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. [41], Buckley [42]).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.

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).

4 .
Subdifferential of Fuzzy FunctionDefinition 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 + λd ∈ 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 or all 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 or all d ∈ R m }