Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions

Abstract In this paper, we present the concepts of generalized derivative, directional generalized derivative, subdifferential and conjugate for n-dimensional fuzzy-number-valued functions and discuss the characterizations of generalized derivative and directional generalized derivative by, respectively, using the derivative and directional derivative of crisp functions that are determined by the fuzzy mapping. Furthermore, the relations among generalized derivative, directional generalized derivative, subdifferential and convexity for n-dimensional fuzzy-number-valued functions are investigated. Finally, under two kinds of partial orderings defined on the set of all n-dimensional fuzzy numbers, the duality theorems and saddle point optimality criteria in fuzzy optimization problems with constraints are discussed.


Introduction
In 1972, Chang and Zadeh [1] introduced the concept of fuzzy numbers with the consideration of the properties of probability functions. Since then the fuzzy numbers have been extensively studied by many authors. Fuzzy numbers are a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. As part of the development of theories about fuzzy numbers and its applications, researchers began to study the differentiability and integrability of fuzzy mappings. Initially, Puri and Ralescu [2] defined the g-derivative of fuzzy mappings from an open subset of a normed space into the n-dimension fuzzy number space E n using the Hukuhara difference. In 1987, Kaleve [3] investigated fuzzy differential equations based on the g-derivative. In 2010, Farajzadeh et al. [4] proposed a numerical method for the solution of the fuzzy heat equation. Furthermore, Wang and Wu [5] defined the directional derivative of fuzzy mappings from R n into E 1 . However, the Hukuhara difference between two fuzzy numbers exists only under very restrictive conditions and the H-difference of two fuzzy numbers does not always exist. The g-difference proposed in [6,7] overcomes these shortcomings of the above discussed concepts, and the g-difference of two fuzzy numbers always exists. Based on the generalizations of the Hukuhara difference for fuzzy sets, Bede and Stefanini [8] introduced and studied a new generalized differentiability concept for fuzzy-valued functions from R into E 1 in 2013. Using the fuzzy g-difference introduced in Stefanini [7], in 2016, Hai et al. [9] defined and studied generalized differentiability for n-dimensional fuzzy-number-valued functions on [ ] a b , . In this paper, we generalize the concepts of generalized derivative and support-function-wise derivative for n-dimensional fuzzy-number-valued to ⊆ M R m . Furthermore, the directional generalized derivative, subdifferential and conjugate for fuzzy-number-valued functions are investigated, and we give characteristic theorems for the generalized derivative and directional generalized derivative for fuzzy-number-valued functions.
Recently, convexity has been increasingly important in the study of extremum problems in many areas of applied mathematics. In fact, convex analysis [10] is an important branch of mathematics, and it also has wide application in convex optimization. If the values of the objective function that is sought optimum solution are crisp real numbers, the optimization is a general crisp optimization [11]. But in reality, sometimes, the values of the objective function only are estimated values, so it is more suitable that the values are expressed with fuzzy numbers, and the optimization is a fuzzy optimization. In 1992, Nanda and Kar [12] introduced and discussed the concepts of convex fuzzy mappings from a vector space over the field R into E 1 , established criteria for convex fuzzy mappings. In 2005, Zhang et al. [13] discussed the convex fuzzy mappings and discussed the duality theory in fuzzy mathematical programming problems with fuzzy coefficients based on the ordering of two fuzzy numbers proposed in [12]. Under a general setting of partial ordering defined on the set of all fuzzy numbers, Wu [14] investigated the duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on two solution concepts for primal problem and three solution concepts for dual problem in 2007. A well-known fact in mathematical programming is that variational inequality problems have a close relation with the optimization problems. Similarly, the fuzzy variational inequality (inclusions) problems also have a close relation with fuzzy optimization problems. In 2009, Ahmad and Farajzadeh [15] investigated random variational inclusions with random fuzzy mappings and defined an iterative algorithm to compute the approximate solutions of random variational inclusion problem. However, very few studies have investigated the convexity and duality in fuzzy optimization of n-dimensional fuzzy-number-valued functions. The main reason is that there is almost no related research about the ordering and the difference of n-dimensional fuzzy numbers. In 2016, Gong and Hai introduced the concept of a convex fuzzy-numbervalued function based on a new ordering ≼ c of n-dimensional fuzzy numbers [16] and investigated differentiability for n-dimensional fuzzy-number-valued functions on [ ] a b , and Karush-Kuhn-Tucker (KKT) conditions in fuzzy optimization problems based on the ordering ≼ s in [9]. In 2019, Xie and Gong [17] investigated variational-like inequalities for n-dimensional fuzzy-vector-valued functions and obtained optimality conditions for fuzzy multiobjective optimization problems. In this paper, under the two kinds of partial orderings defined on the set of all n-dimensional fuzzy numbers, the duality theorems and saddle point optimality criteria in fuzzy optimization problems are discussed.
To make our analysis possible, we present the preliminary terminology used throughout this paper in Section 2. In Section 3, we present the concept of generalized derivative, directional generalized derivative, subdifferential and conjugate for fuzzy-number-valued functions and obtain characteristic theorems of the generalized derivative and directional generalized derivative for fuzzy-number-valued functions. Furthermore, the relations among generalized derivative, directional generalized derivative, subdifferential and convexity for n-dimensional fuzzy-number-valued functions are discussed. In Sections 4 and 5, the Lagrange duality theorem and the optimality conditions, including the KKT conditions and the saddle point optimality criteria, in fuzzy optimization problems with constraints for n-dimensional fuzzy-number-valued functions are derived, respectively, under the partial orderings ≼ c and ≼ s defined on the set of all n-dimensional fuzzy numbers. Section 6 concludes this paper.

Preliminaries
Throughout this paper, R n denotes the n-dimensional Euclidean space, n and C n denote the spaces of nonempty compact and compact convex sets of R n , respectively. Let ( ) R n be the set of all fuzzy subsets on R n . A fuzzy set u on R n is a mapping We use E n to denote the fuzzy number space [18,19]. When = n 1, u is called a one-dimensional fuzzy number, and the fuzzy number space denoted by E 1 or E.
It is clear that each ∈ u R n can be considered as a fuzzy number u defined by In particular, the fuzzy number 0 is defined as x y x y r , : 1 A special kind of n-dimension fuzzy number is the fuzzy n-cell number proposed in [20]. : For any ∈ x R n , the addition and scalar multiplication can be defined, respectively, as It is well known that for any ∈ u v E , n and ∈ k R, be the unit sphere of R n and 〈⋅ ⋅〉 , be the inner product in R , n i.e., The generalized Hukuhara difference has been extended to the fuzzy case in [7]. For any ∈ u v E , , n the generalized Hukuhara difference (gH-difference for short) is the fuzzy number w, if it exists, such that It is possible that the gH-difference of two fuzzy numbers does not exist. To solve this shortcoming, in [8] a new difference between fuzzy numbers was proposed. Using the convex hull (conv) the new difference was defined as follows.
Definition 2.6. [6,8] The generalized difference (g-difference for short) of two fuzzy numbers ∈ u v E , n is given by its level set as we have Optimization problems for n-dimensional fuzzy-number-valued functions  1455 n n between u and v is defined by the equation: where d is the Hausdorff metric given by : , i n f , n is a complete metric space and satisfies ( + , for any ∈ u v w E , , n and ∈ k R. We , which coincides with the definition of ordering of u v , proposed by Goetschel and Voxman [10].
Similarly, we can define the lower bound and the greatest lower bound of a subset of E 1 .

We call
n a n-dimensional fuzzy-number-valued function. We denote its lower u-level set as , and the strict lower u-level set as , and the epigraph as n . F is said to be lower semicontinuous at x 0 if for any ε, a neighborhood U of x 0 exists when ∈ x U 0 , and we have ( if it is both lower semicontinuous and upper semicontinuous at x 0 , and that it is continuous if and only if it is continuous at every point of M. , , then we have by (2.20) 1 2 , for an arbitrary positive fuzzy-number ε, and ≠ C R n , that is, n be quasi-convex. The normal operator and strict normal operator of F at x are set-valued functions, which are defined as the normal cones to ( ) are convex sets.
is called the conjugate of the fuzzy-number-valued function F.
and ≠ C R n . It follows that ) a n-dimensional fuzzy vector, denoted it as ( … ) u u u , , , n 1 2 , and call the collection of all n-dimensional fuzzy vectors (i.e., the Cartesian product × × ⋯× E E E n n n n       ) n-dimensional fuzzy vector space, and denote it as ( ) E n n .
For any n-dimensional fuzzy vectors = ( 0, 1 . We use the following convention for equalities and inequalities: 3 Directional g-derivative and subdifferential for fuzzy-numbervalued functions In this section, we generalize the concepts of g-derivative proposed by Hai et al.
to ⊆ M R m , and the directional g-derivative and subdifferential for fuzzy-number-valued functions are investigated. In addition, we give some kinds of definitions of convexity for fuzzy-number-valued functions, so that we can more conveniently discuss convex fuzzy mappings and convex fuzzy programming. Let n be the n-dimensional fuzzy-number-valued function (fuzzy-number-valued functions for short). The fuzzy-number-valued functions in the following arguments are assumed to be comparable.  then we say that F has the jth partial generalized derivative (g-derivative for short) at x 0 , denoted by . If all the partial g-derivatives at x 0 exist, then we say F is generalized differentiable (g-differentiable for short) at x 0 , denoted by ′( ) F x g 0 . Here the limit is taken in the metric space such that 0 0 then we say F is g-differentiable at x 0 , and such u is called the gradient of F at x 0 , denoted by ∇ ( ) F x g 0 .
n be a fuzzy-number-valued function, ∈ x M. The one-sided directional g-derivative of F at x with respect to a vector ∈ y R m is defined to be the limit We say the one-sided directional g-derivative ′( ) F x y , g is two-sided if and only if ′( − ) F x y , g exists and We also say F is g-differentiable in the direction y at x. Here the limit is taken in the metric space n be a fuzzy-number-valued function. If F is g-differentiable at x, then the directional g-derivatives ′( ) F x y , g are two-sided, and Proof. Since F is g-differentiable at x, then for any ∈ y R m and ≠ y 0, , sup max , , , .     lim sup max  inf min  , ,  , , ,  , , , , ,  ,   , ,  , , ,  , , , , ,  ,   sup max  , ,  , lim sup sup max , , ,   i n fm i n  , ,  ,  ,  infmin  , ,  ,   ,        n be e-convex, then F is convex.
:  x r  x  i  n  ,  , , ,  , ,  , , ,  ,  ,  1 , The proof is similar to the proof of Theorem 6.1.2 in the study of Mangasarian [24].
The set of all subgradients of F at x is called the subdifferential of F at x and is denoted by is called the subdifferential of F. If ∂ ( ) F x is not empty, F is said to be subdifferentiable at x.  Proof. If ∂ ( ) F x is empty, the theorem is trivial. If ∂ ( ) F x is not empty, then suppose ∈ ∂ ( ) ξ ξ F x , 1 2 , for all ∈ [ ] λ 0, 1 , we have Therefore, Since F is lower semicontinuous at x, then we obtain ( ) ≽ ( ) n be convex and x be a vector in M. Then for all ≥ λ 0, .
for each ∈ x S, then ★ x is said to be an optimal solution to (FOP1), and we denote the optimal value in (FOP1) by ★ p , i.e., = ( ) ★ ★ p F x . We assume is nonempty. In the following, we denote ( ) = ( ( ) ( ) … ( )) x n associated with problem 4.1 is defined as . We refer to α k as the Lagrange multiplier associated with the kth inequality constraint ( ) ≼ g x 0 i.e., for ∈ + α R l , ∈ β R t , When the Lagrangian is unbounded below in x, the dual function takes on the value −∞.
Proposition 4.2. The dual function yields lower bounds on the optimal value ★ p of problem 4.1, i.e., for any ≥ α 0 and any β, we have H˜0.

T T c
Since each term in the first sum is nonpositive, and each term in the second sum is zero, then by (2.20) holds for every feasible point x, inequality 37 follows. This completes the proof. □ For each pair ( ) α β , with ≥ α 0, the Lagrange dual function gives us a lower bound on the optimal value ★ p of the fuzzy optimization problem 4.1. Thus, we have a lower bound that depends on some parameters α β , . A natural question is what is the best lower bound that can be obtained from the Lagrange dual function. This leads to the optimization problem This problem is called the Lagrange dual problem associated (DFOP1) with the problem (FOP1). The original problem (FOP1) is also called the primal problem.
, is said to be the dual feasible set of the primal problem (FOP1), that is, it is the feasible set of the dual problem (DFOP1).
, , then we refer to the pair ( ) ★ ★ α β , as the dual optimal solution or optimal Lagrange multipliers, and the optimal value of the Lagrange dual problem denoted by ★ d , i.e., = ( ) , are feasible solutions to the primal problem (FOP1) and the Lagrange dual problem (DFOP1), respectively, then weak duality holds: Proof. By definition, we have are the optimal solutions to the primal problem (FOP1) and the Lagrange dual problem (DFOP1), respectively, then . . , .
c c (4.7) We say that strong duality holds if a dual optimal solution with strong duality. Then the complementary slackness condition holds: Proof. By definition, we have k k Now we investigate optimality conditions, which are called the KKT conditions, for the solutions to be primal and dual optimal, when the primal problem is e-convex. , the constraint conditions are equivalent to . Obviously, the feasible set of (FOP1′) is equivalent to the feasible set of (FOP1). Let We refer to ( ) = ( ( ) ( ) … ( )) ∈ + α r α r α r α r R , , , p T as the Lagrange multiplier vectors containing parameter.
We now assume that the feasible set of (FOP1′)   x i T T Now we prove the necessity. Since ★ x is an optimal solution to (FOP1′), ★ x minimizes ( ) () ★ ★ L x α β r , , over x, it follows that its gradient must vanish at ★ x , we obtain (4.16) and equivalently have (4.12). Since strong duality holds, by (4.9), it is not difficult to obtain (4.13). Conditions (4.14) and (4.15) hold since they are constraint conditions of (FOP1′) and (DFOP1′), respectively.
Conversely, since ★ ★ ★ x α β , , satisfy the KKT conditions (4. 12-4.15), then ★ ★ ★ x α β , , also satisfy the KKT conditions (4. 13-4.16) with the objective function is the real-valued function ( ) F x r , i , thus, ★ x is an optimal solution to the optimization problem and ( ) is an optimal solution to its Lagrange dual problem, that is, ∀ ∈ x int M and ∀( ) ∈ ( ) , . (4.18) By reductio ad absurdum, suppose that ★ x is not an optimal solution of (FOP1′), then there exists ′ ∈ , which is in contradiction to (4.17). Therefore, ★ x is an optimal solution to (FOP1′). Equation where , the convex optimization problem (OP1) satisfies Slater's condition, therefore, for (OP1) and its Lagrange dual problem, the strong duality holds [25]. Since the feasible set of (FOP1) 4.1 is equivalent to the feasible set of (OP1) 4.19, then for (FOP1) and its Lagrange dual problem, the strong duality holds. □ In other words, . If x is not a feasible solution, then there exists , o t h e r w i s e .
Therefore, we can express the optimal value of the primal problem as If ★ x is a primal optimal solution, then we obtain Since strong duality holds, then = ★ ★ p d , thus, we have , where ∈ + α R l . Therefore, ,¯,¯inf¯.
x c x c It follows that ,¯,¯¯. Thus, x is a primal optimal solution, ( ) α β,¯is a dual optimal solution, and strong duality holds. □ Now, we give an example cited from [16] to illustrate the fuzzy optimization.
Example 4.10. A company operates a training program for all new employees and without loss of generality. During the training program, we have to characterize the working state of one person. It is well known that the working state of one person changes with time. It is not appropriate to characterize the working efficiency of one person based only on their production speed because we also need to consider the quality of their products. If we denote the production speed and the qualification rate by x 1 and x 2 , respectively, then the working state of the person can be characterized by a two-dimensional quantity ( ) x x , 1 2 . However, the quantity is only an estimated quantity, then using a two-dimensional fuzzy numbervalued function ( ) u x x , 1 2 to express the quantity is more appropriate than using a crisp two-dimensional quantity. Suppose that one person's production speed and qualification rate are about 100 and 0.95, respectively, then the person's working state can be expressed by the two-dimensional fuzzy numbervalued function ( ) u x x , 1 2 , which is defined as follows: . Consider the following fuzzy optimization problem , satisfy the condition of Theorem 4.6, therefore, = t 1 is an optimal solution to the above fuzzy optimization problem.
uniformly for ∈ [ ] r 0, 1 and ∈ − x S , n 1 then F is said to be support-function-wise convex (s-convex) on M. Conversely, if x α β,˜,˜are any points that satisfy the KKT conditions (5.4)-(5.7), then x and ( ) α β,˜are primal solution and dual optimal solution, and strong duality holds.
Proof. If x is an optimal solution to (FOP2) and ( ) α β,˜is an optimal solution to (DFOP2) with strong duality, since under the ordering ≼ s , the support function of a fuzzy number is a real-valued function, then analogous to Theorem 4.6, it is not difficult to prove that x α β,˜,˜satisfy conditions (5.4)-(5.7).
Conversely, for any ∈ x S, we have strong duality holds (see [25]). Since the feasible set of (FOP2) (5.2) is equivalent to the feasible set of (OP2) (5.9), then for (FOP2) and its Lagrange dual (DOP2) (5.3), the strong duality holds. □ x α β , , forms a saddle point for the Lagrangian. Conversely, if ( ) x α β,¯,¯is a saddle point of the Lagrangian, then x is a primal optimal solution, ( ) α β,¯is a dual optimal solution, and strong duality holds.

Conclusion
We present the concepts of generalized derivative, directional generalized derivative, subdifferential and conjugate for n-dimensional fuzzy-number-valued functions from R m to E n and give the characteristic theorems of generalized derivative and directional generalized. We examine the relations among generalized derivative, directional generalized derivative, subdifferential and convexity for n-dimensional fuzzynumber-valued functions. Additionally, under two kinds of partial orderings defined on the set of all n-dimensional fuzzy numbers, we discuss the duality theorems and saddle point optimality criteria in fuzzy optimization problems with constraints. The next step for the continuation of the research direction proposed here is to investigate the fuzzy optimization problems under non-differentiable case.