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

  • Ting Xie , Zengtai Gong EMAIL logo and Dapeng Li
Published/Copyright: December 7, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

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.

Keywords: fuzzy n-cell number; n-dimensional fuzzy-number-valued function; directional generalized derivative; subdifferential; fuzzy optimization
MSC 2010: 26E50

1 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 functions from [ a , b ] ⊆ R 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-number-valued 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.

2 Preliminaries

Throughout this paper, R n denotes the n-dimensional Euclidean space, K n and K C n denote the spaces of nonempty compact and compact convex sets of R n , respectively. Let ℱ ( R n ) be the set of all fuzzy subsets on R n . A fuzzy set u on R n is a mapping u : R n → [ 0 , 1 ] , and u ( x ) is the degree of membership of the element x in the fuzzy set u. For each fuzzy set u, we denote its r-level set as [ u ] r = { x ∈ R n : u ( x ) ≥ r } for any r ∈ ( 0 , 1 ] , and in some references also denoted by u r for short. The support of u we denote by supp u where supp u = { x ∈ R n : u ( x ) > 0 } . The closure of supp u defines the 0-level of u, i.e., [ u ] 0 = c l ( supp u ) . Here c l ( M ) denotes the closure of set M. Fuzzy set u ∈ ℱ ( R n ) is called a fuzzy number if

  1. u is a normal fuzzy set, i.e., there exists an x 0 ∈ R n such that u ( x 0 ) = 1 ,

  2. u is a convex fuzzy set, i.e., u ( λ x + ( 1 − λ ) y ) ≥ min { u ( x ) , u ( y ) } for any x , y ∈ R n and λ ∈ [ 0 , 1 ] ,

  3. u is upper semicontinuous and

  4. [ u ] 0 = c l ( supp u ) = c l ⋃ r ∈ ( 0 , 1 ] [ u ] r is compact.

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

u ( x ) = 1 , x = u , 0 , otherwise .

In particular, the fuzzy number 0 ˜ is defined as 0 ˜ ( x ) = 1 if x = 0 , and 0 ˜ ( x ) = 0 otherwise.

Example 2.1

[17] Let u ∈ E 2 be defined by

(2.1) u ( x , y ) = 1 − x 2 − y 2 , x 2 + y 2 ≤ 1 , 0 , otherwise ,

then [ u ] r = { ( x , y ) : x 2 + y 2 ≤ 1 − r 2 } , r ∈ [ 0 , 1 ] .

A special kind of n-dimension fuzzy number is the fuzzy n-cell number proposed in [20]. If u ∈ E n , and [ u ] r is a cell, i.e., [ u ] r can be represented by [ u ] r = ∏ i = 1 n [ u i − ( r ) , u i + ( r ) ] = [ u 1 − ( r ) , u 1 + ( r ) ] × [ u 2 − ( r ) , u 2 + ( r ) ] × ⋯ × [ u n − ( r ) , u n + ( r ) ] for any r ∈ [ 0 , 1 ] , where u i − ( r ) , u i + ( r ) ∈ R with u i − ( r ) ≤ u i + ( r ) ,   i = 1 , 2 , … , n , then we call u a fuzzy n-cell number. And we denote the collection of all fuzzy n-cell numbers by L ( E n ) .

Example 2.2

F : [ 1 , ∞ ) → L ( E 2 ) is a fuzzy 2-cell number function, which is defined by

(2.2) F ( s ) ( x 1 , x 2 ) = e 2 s − x 1 2 e s , 0 ≤ x 1 ≤ e s ,  0 ≤ x 2 ≤ 3 , 0 , otherwise,

where the parameter s ∈ R . Then for all r ∈ [ 0 , 1 ] ,

F r ( s ) = ( x 1 , x 2 ) : 0 ≤ x 1 ≤ 1 − r 2 ⋅ e s ,  0 ≤ x 2 ≤ 3 .

Theorem 2.3

[3] If u ∈ E n , then

  1. [ u ] r is a nonempty compact convex subset of R n for any r ∈ ( 0 , 1 ] ,

  2. [ u ] r 1 ⊆ [ u ] r 2 , whenever 0 ≤ r 2 ≤ r 1 ≤ 1 ,

  3. if r n > 0 and r n converging to r ∈ [ 0 , 1 ] is nondecreasing, then ⋂ n = 1 ∞ [ u ] r n = [ u ] r .

Conversely, suppose for any r ∈ [ 0 , 1 ] , there exists an A r ⊆ R n which satisfies (i)-(iii), then there exists a unique u ∈ E n such that [ u ] r = A r , r ∈ ( 0 , 1 ] , [ u ] 0 = ⋃ r ∈ ( 0 , 1 ] [ u ] r ¯ ⊆ A 0 .

Note that when u ∈ E 1 , [ u ] r is a nonempty closed interval on [ 0 , 1 ] for any r ∈ [ 0 , 1 ] ; when u ∈ L ( E n ) , [ u ] r is a nonempty n-dimensional closed polyhedron for any r ∈ [ 0 , 1 ] .

Theorem 2.4

[20] If u ∈ L ( E n ) , then for i = 1 , 2 , … , n , u i − ( r ) , u i + ( r ) are real-valued functions on [ 0 , 1 ] and satisfy

  1. u i − ( r ) are non-decreasing and left continuous,

  2. u i + ( r ) are non-increasing and left continuous,

  3. u i − ( r ) ≤ u i + ( r ) (it is equivalent to u i − ( 1 ) ≤ u i + ( 1 ) ) and

  4. u i − ( r ) , u i + ( r ) are right continuous at r = 0 .

Conversely, if a i ( r ) , b i ( r ) ,   i = 1 , 2 , … , n , are real-valued functions on [ 0 , 1 ] which satisfy conditions (i)–(iv), then there exists a unique u ∈ L ( E n ) such that [ u ] r = ∏ i = 1 n [ a i ( r ) , b i ( r ) ] for any r ∈ [ 0 , 1 ] .

Let u , v ∈ E n , k ∈ R . For any x ∈ R n , the addition and scalar multiplication can be defined, respectively, as

(2.3) ( u + v ) ( x ) = sup s + t = x min { u ( s ) , v ( t ) } ,

(2.4) ( k u ) ( x ) = u x k , k ≠ 0 ,

(2.5) ( 0 u ) ( x ) = 1 , x = 0 , 0 ˜ , x ≠ 0 .

It is well known that for any u , v ∈ E n and k ∈ R ,

(2.6) [ u + v ] r = [ u ] r + [ v ] r = { x + y : x ∈ [ u ] r , y ∈ [ v ] r } ,

(2.7) [ k u ] r = k [ u ] r = { k x : x ∈ [ u ] r } .

If u , v ∈ L ( E n ) and k ∈ R , then for all r ∈ [ 0 , 1 ] , [20]

(2.8) [ u + v ] r = [ u ] r + [ v ] r = ∏ i = 1 n [ u i − ( r ) + v i − ( r ) , u i + ( r ) + v i + ( r ) ] ,

(2.9) [ k u ] r = k [ u ] r = w ⇔ ∏ i = 1 n [ k u i − ( r ) , k u i + ( r ) ] , k ≥ 0 , ∏ i = 1 n [ k u i + ( r ) , k u i − ( r ) ] ,    k < 0 .

Proposition 2.5

[19] If u , v ∈ E n , k , k 1 , k 2 ∈ R , then

  1. k ( u + v ) = k u + k v ,

  2. k 1 ( k 2 u ) = ( k 1 k 2 ) u ,

  3. ( k 1 + k 2 ) u = k 1 u + k 2 u when k 1 ≥ 0 and k 2 ≥ 0 .

Give two subsets A , B ⊆ R n and k ∈ R , the Minkowski difference is given by A − B = A + ( − 1 ) B = { a − b : a ∈ A , b ∈ B } . However, in general, A + ( − A ) ≠ 0 , i.e., the opposite of A is not the inverse of A in Minkowski addition (unless A = { a } is a singleton). The spaces K n and K C n are not linear spaces since they do not contain inverse elements and therefore subtraction is not defined. To partially overcome this situation, Hukuhara [21] introduced the following H-difference A ⊖ B = C ⇔ A = B + C and an important property of ⊖ is that A ⊖ A = { 0 } ,   ∀ A ∈ R n and ( A + B ) ⊖ B = A ,   ∀ A , B ∈ R n . The H-difference is unique, but a necessary condition for A ⊖ H B to exist is that A contains a translation { c } + B of B . In order to overcome this situation, Stefanini [22] defined the generalized Hukuhara difference of two sets A , B ∈ K n as follows:

A ⊖ g H B = C ⇔ ( 1 ) A = B + C , or ( 2 ) B = A + ( − 1 ) C .

For any A ⊆ R n , the support function associated with A is s A : R n → R defined by s A ( x ) = sup a ∈ A 〈 a , x 〉 , x ∈ R n . Let S n − 1 = { x ∈ R n : ∥ x ∥ = 1 } be the unit sphere of R n and 〈 ⋅ , ⋅ 〉 be the inner product in R n , i.e., 〈 x , y 〉 = ∑ i = 1 n x i y i , where x = ( x 1 , x 2 , … , x n ) T ∈ R n , y = ( y 1 , y 2 , … , y n ) T ∈ R n . If A ∈ K C n , then s A ( x ) = sup a ∈ A 〈 a , x 〉 for any x ∈ S n − 1 . Let s A ( x ) , s B ( x ) , s C ( x ) , s ( − 1 ) A ( x ) and s ( − 1) B ( x ) be the support functions of A , B , C , ( − 1 ) A and ( − 1) B , respectively. Then, for any x ∈ S n − 1 , we have [7]

s C ( x ) = ( 1 ) s A ( x ) − s B ( x ) , or ( 2 ) s ( − 1 ) B ( x ) − s ( − 1 ) A ( x ) .

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

u ⊖ g H v = w ⇔ ( 1 ) u = v + w , or ( 2 ) v = u + ( − 1 ) w .

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

(2.10) [ u ⊖ g v ] r = c l ( conv ⋃ β ≥ r [ u ] β ⊖ g H [ v ] β , ∀ r ∈ [ 0 , 1 ] ,

where the gH-difference ⊖ g H is with interval operands [ u ] β and [ v ] β .

Definition 2.7

[18] For u ∈ E n , r ∈ [ 0 , 1 ] and x ∈ S n − 1 , the support function of u is defined by

(2.11) u ∗ ( r , x ) = sup a ∈ [ u ] r 〈 a , x 〉 .

Obviously, if u , v ∈ L ( E n ) , for any x ∈ S n − 1 and r ∈ [ 0 , 1 ] , then

u ∗ ( r , x ) = sup a ∈ [ u ] r 〈 a , x 〉 = ∑ x i > 0 x i u i + ( r ) + ∑ x i < 0 x i u i − ( r ) .

Theorem 2.8

[23] Suppose u ∈ E n , r ∈ [ 0 , 1 ] , then

(2.12) [ u ] r = { y ∈ R n : 〈 y , x 〉 ≤ u ∗ ( r , x ) , x ∈ S n − 1 } .

Theorem 2.9

[9] Let u , v ∈ E n . If the g-difference u ⊖ g v of u and v exists, then for any r ∈ [ 0 , 1 ] and x ∈ S n − 1 , we have

(2.13) ( u ⊖ g v ) ∗ ( r , x ) = ( 1 ) sup β ≥ r ( u ∗ ( β , x ) − v ∗ ( β , x ) ) , or ( 2 ) sup β ≥ r ( ( − v ) ∗ ( β , x ) − ( − u ) ∗ ( β , x ) ) , = ( 1 ) sup β ≥ r ( u ∗ ( β , x ) − v ∗ ( β , x ) ) , or ( 2 ) sup β ≥ r ( v ∗ ( β , − x ) − u ∗ ( β , − x ) ) .

Remark 2.10

Let u , v ∈ L ( E n ) , we have

(2.14) [ u ⊖ g v ] r = ∏ i = 1 n [ inf β ≥ r min u i − ( β ) − v i − ( β ) , u i + ( β ) − v i + ( β ) , sup β ≥ r max u i − ( β ) − v i − ( β ) , u i + ( β ) − v i + ( β )

(2.15) = ∏ i = 1 n [ u i ⊖ g v i ] r .

Proposition 2.11

[9] Let u , v ∈ E n . Then

  1. if the g-difference exists, it is unique,

  2. u ⊖ g u = 0 ,

  3. ( u + v ) ⊖ g v = u , ( u + v ) ⊖ g u = v and

  4. u ⊖ g v = − ( v ⊖ g u ) .

Given u , v ∈ E n , the distance D : E n × E n → [ 0 , + ∞ ) between u and v is defined by the equation:

D ( u , v ) = sup r ∈ [ 0 , 1 ] d ( [ u ] r , [ v ] r ) ,

where d is the Hausdorff metric given by

d ( [ u ] r , [ v ] r ) = inf ε : [ u ] r ⊂ N ( [ v ] r , ε ) , [ v ] r ⊂ N ( [ u ] r , ε ) = max { sup a ∈ [ u ] r inf b ∈ [ v ] r ∥ a − b ∥ , sup b ∈ [ v ] r inf a ∈ [ u ] r ∥ a − b ∥ .

N ( [ u ] r , ε ) = x ∈ R n : d ( x , [ u ] r ) = inf y ∈ [ u ] r d ( x , y ) ≤ ε is the ε -neighborhood of [ u ] r . Then, ( E n , D ) is a complete metric space and satisfies D ( u + w , v + w ) = D ( u , v ) , D ( k u , k v ) = | k | D ( u , v ) for any u , v , w ∈ E n and k ∈ R .

Remark 2.12

If u , v ∈ L ( E n ) , i.e., [ u ] r = ∏ i = 1 n [ u i − ( r ) , u i + ( r ) ] , [ v ] r = ∏ i = 1 n [ v i − ( r ) , v i + ( r ) ] , r ∈ [ 0 , 1 ] ,

(2.16) D ( u , v ) = sup r ∈ [ 0 , 1 ] d ( [ u ] r , [ v ] r ) = sup r ∈ [ 0 , 1 ] max 1 ≤ i ≤ n | u i − ( r ) − v i − ( r ) | , | u i + ( r ) − v i + ( r ) | .

For u ∈ E n , we denote the centroid of [ u ] r , r ∈ [ 0 , 1 ] as

∫ ⋯ ∫ [ u ] r x 1 d x 1 d x 2 ⋯ d x n ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋯ d x n , ∫ ⋯ ∫ [ u ] r x 2 d x 1 d x 2 ⋯ d x n ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋯ d x n , … , ∫ ⋯ ∫ [ u ] r x n d x 1 d x 2 ⋯ d x n ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋯ d x n T ,

where ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋯ d x n is the solidity of [ u ] r , r ∈ [ 0 , 1 ] and ∫ ⋯ ∫ [ u ] r x i d x 1 d x 2 ⋯ d x n ( i = 1 , 2 , … , n ) is the multiple integral of x i on measurable sets [ u ] r , r ∈ [ 0 , 1 ] . Let τ : E n → R n be a real vector-valued function defined by

(2.17) τ ( u ) = 2 ∫ 0 1 r ∫ ⋯ ∫ [ u ] r x 1 d x 1 d x 2 ⋯ d x n ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋯ d x n d r , 2 ∫ 0 1 r ∫ ⋯ ∫ [ u ] r x 2 d x 1 d x 2 ⋯ d x n ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋯ d x n d r , … 2 ∫ 0 1 r ∫ ⋯ ∫ [ u ] r x n d x 1 d x 2 ⋯ d x n ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋯ d x n d r T ,

where ∫ 0 1 r ∫ ⋯ ∫ [ u ] r x i d x 1 d x 2 ⋅ ⋅ ⋅ d x n ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋅ ⋅ ⋅ d x n d r ( i = 1 , 2 , … , n ) is the Lebesgue integral of r ∫ ⋯ ∫ [ u ] r x i d x 1 d x 2 ⋅ ⋅ ⋅ d x n ∫ ⋯ ∫ [ u ] r 1 d x 1 d x 2 ⋅ ⋅ ⋅ d x n ( i = 1 , 2 , … , n ) on [ 0 , 1 ] . The vector-valued function τ is called a ranking value function defined on E n .

Definition 2.13

[16] Let u , v ∈ E n , C ⊆ R n be a closed convex cone with 0 ∈ C and C ≠ R n . We say that u ≼ c v (u precedes v) if

τ ( v ) ∈ τ ( u ) + C .

We say that u ≺ c v if u ≼ c v and τ ( u ) ≠ τ ( v ) . Sometimes we may write v ≽ c u (resp. v ≻ c u ) instead of u ≼ c v (resp. u ≺ c v ). If either u ≼ c v or v ≼ c u , we say that u and v are comparable; otherwise, they are non-comparable. In addition, ε ˜ ∈ E n is said to be an arbitrary positive fuzzy-number if ε ˜ ≻ c 0 ˜ and D ( ε ˜ , 0 ˜ ) < ε , where ε is an arbitrary positive real number.

If u , v ∈ E 1 , then τ ( u ) = ∫ 0 1 r ( u r − + u r + ) d r ,   τ ( v ) = ∫ 0 1 r ( v r − + v r + ) d r . Suppose C = R + = [ 0 , + ∞ ) ⊆ R , u ≼ c v if and only if τ ( u ) ≤ τ ( v ) , i.e., τ ( v ) ∈ τ ( u ) + [ 0 , + ∞ ) , which coincides with the definition of ordering of u , v proposed by Goetschel and Voxman [10].

Remark 2.14

  1. Let u , v ∈ L ( E n ) , then

    (2.18) τ ( u ) = ∫ 0 1 r ( u 1 − ( r ) + u 1 + ( r ) ) d r , ∫ 0 1 r ( u 2 − ( r ) + u 2 + ( r ) ) d r , ⋯ , ∫ 0 1 r ( u n − ( r ) + u n + ( r ) ) d r T .

  2. Let u , v ∈ L ( E n ) , for k 1 , k 2 ∈ R ,

    (2.19) τ ( k 1 u + k 2 v ) = k 1 τ ( u ) + k 2 τ ( v ) .

  3. Let u 1 , u 2 , v 1 , v 2 ∈ L ( E n ) . If u 1 ≼ c v 1 and u 2 ≼ c v 2 , then for k 1 , k 2 ∈ [ 0 , ∞ ) ,

(2.20) k 1 u 1 + k 2 u 2 ≼ c k 1 v 1 + k 2 v 2 .

Definition 2.15

A subset S of E n is said to be bounded above if there exists a fuzzy number u ∈ E n , called an upper bound of S, such that v ≼ c u for all v ∈ S . Furthermore, a fuzzy number u 0 ∈ E n is called the least upper bound for S if

  1. u 0 is an upper bound of S,

  2. u 0 ≼ c u for every upper bound u of S.

Similarly, we can define the lower bound and the greatest lower bound of a subset of E 1 .

We call F : M → E n a n-dimensional fuzzy-number-valued function. We denote its lower u-level set as L u = { z ∈ M : F ( z ) ≼ c u } , and the strict lower u-level set as L ˜ u = { z ∈ M : F ( z ) ≺ c u } , and the epigraph as epi ( F ) = { ( x , u ) : x ∈ M , u ∈ E n , F ( x ) ≼ c u } . For F : M → L ( E n ) , we denote [ F ( x ) ] r = ∏ i = 1 n F i − ( x ) ( r ) , F i + ( x ) ( r ) as [ F ( x ) ] r = ∏ i = 1 n F i − ( x , r ) , F i + ( x , r ) , r ∈ [ 0 , 1 ] ,

Definition 2.16

[16] Let F : M → E n . F is said to be lower semicontinuous at x 0 if for any ε ˜ , a neighborhood U of x 0 exists when x 0 ∈ U , and we have F ( x 0 ) ≺ c F ( x ) + ε ˜ ; F is said to be upper semicontinuous at x 0 if for any ε ˜ , a neighborhood U of x 0 exists when x 0 ∈ U , and we have F ( x ) ≺ c F ( x 0 ) + ε ˜ . F is continuous at x 0 ∈ M 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.

Let Φ : R m → R n be a set-valued function. The graph of Φ is the set g r Φ = { ( x , y ) ∈ R m × R n : y ∈ Φ ( x ) } . Φ is said to be closed at x ∈ R m if y ∈ Φ ( x ) whenever there exists a sequence { ( x k , y k ) } contained in g r Φ and converging to ( x , y ) .

Definition 2.17

[16] Let M ⊆ R m be a convex set and F : M → E n .

  1. F is said to be convex on M if for any x 1 , x 2 ∈ M and λ ∈ [ 0 , 1 ]

    F ( λ x 1 + ( 1 − λ ) x 2 ) ≼ c λ F ( x 1 ) + ( 1 − λ ) F ( x 2 ) .

  2. F is said to be quasi-convex on M if for any x 1 , x 2 ∈ M and λ ∈ [ 0 , 1 ]

F ( λ x 1 + ( 1 − λ ) x 2 ) ≼ c max { F ( x 1 ) , F ( x 2 ) } .

Proposition 2.18

[16] Let F i : M → E n ( i = 1 , 2 , … , k ) be convex. If a 1 , a 2 , … , a k > 0 , then F ( x ) = ∑ i = 1 k a i F i ( x ) is a convex fuzzy-number-valued function.

Proposition 2.19

(Interpolation property) Let F : M → L ( E n ) be an n-cell fuzzy-number-valued function. F is convex if and only if ∀ x 1 , x 2 ∈ M , λ ∈ [ 0 , 1 ] and ∀ u , v ∈ L ( E n ) with F ( x 1 ) ≺ c u ,   F ( x 2 ) ≺ c v ,

(2.21) F ( λ x 1 + ( 1 − λ ) x 2 ) ≺ c λ u + ( 1 − λ ) v .

Proof

Only if: if F is convex, then ∀ x 1 , x 2 ∈ M and λ ∈ [ 0 , 1 ] ,

F ( λ x 1 + ( 1 − λ ) x 2 ) ≼ c λ F ( x 1 ) + ( 1 − λ ) F ( x 2 ) .

For any u , v ∈ L ( E n ) , if F ( x 1 ) ≺ c u ,   F ( x 2 ) ≺ c v , then we have by (2.20)

λ F ( x 1 ) + ( 1 − λ ) F ( x 2 ) ≺ c λ u + ( 1 − λ ) v .

Therefore, F ( λ x 1 + ( 1 − λ ) x 2 ) ≺ c λ u + ( 1 − λ ) v .

If: ∀ x 1 , x 2 ∈ M , for an arbitrary positive fuzzy-number ε ˜ , setting u = F ( x 1 ) + ε ˜ , v = F ( x 2 ) + ε ˜ , then F ( x 1 ) ≺ c u ,   F ( x 2 ) ≺ c v . If F ( λ x 1 + ( 1 − λ ) x 2 ) ≺ c λ u + ( 1 − λ ) v , then τ ( λ u + ( 1 − λ ) v ) − τ ( F ( λ x 1 + ( 1 − λ ) x 2 ) ) ∈ C , where C ⊆ R n is a closed convex cone with 0 ∈ C and C ≠ R n , that is,

λ τ ( F ( x 1 + ε ˜ ) ) + ( 1 − λ ) τ ( F ( x 2 ) ) − τ ( F ( λ x 1 + ( 1 − λ ) x 2 ) ) + τ ( ε ˜ ) ∈ C .

Since ε ˜ ≻ c 0 , then we have τ ( ε ˜ ) ∈ C . Thus, we have

λ τ ( F ( x 1 + ε ˜ ) ) + ( 1 − λ ) τ ( F ( x 2 ) ) − τ ( F ( λ x 1 + ( 1 − λ ) x 2 ) ) ∈ C .

It follows that F ( λ x 1 + ( 1 − λ ) x 2 ) ≼ c λ F ( x 1 ) + ( 1 − λ ) F ( x 2 ) , therefore, F is convex.□

Theorem 2.20

[16] Let F : M → E n and u ∈ E n . Then F is quasi-convex on M if and only if its lower u-level set L u (or the strict lower u-level set L ˜ u ) is a convex subset of R m .

Definition 2.21

Let F : M → E 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 L F ( x ) and L ˜ F ( x ) at x, respectively, i.e.,

(2.22) N L F ( x ) ( x ) = x ⁎ ∈ R m : ( x ⁎ ) T ( z − x ) ≤ 0 ,   ∀ z ∈ L F ( x ) ,

(2.23) N ˜ L F ( x ) ( x ) = x ⁎ ∈ R m : ( x ⁎ ) T ( z − x ) ≤ 0 ,   ∀ z ∈ L ˜ F ( x ) .

Obviously, N L F ( x ) ( x ) and N ˜ L F ( x ) ( x ) are convex sets.

Theorem 2.22

Let F : M → E n be quasi-convex. If F is lower semicontinuous at x, then N ˜ L F ( x ) ( x ) is a closed convex set.

Proof

Let { ( x k , x k ⁎ ) } be a sequence contained in g r N ˜ and converge to ( x , x ⁎ ) .

Let z ∈ L ˜ F ( x ) , i.e., F ( z ) ≺ c F ( x ) . Since F is lower semicontinuous at x, then F ( z ) ≺ c F ( x k ) for k large enough. Therefore, x k ⁎ ∈ N ˜ L F ( x k ) ( x k ) implies that ( x k ⁎ ) T ( z − x k ) ≤ 0 . Let k → ∞ , we have ( x ⁎ ) T ( z − x ) ≤ 0 , thus, N ˜ L F ( x ) ( x ) is closed at x.□

Definition 2.23

Let F : R m → E n and S ∗ = { y ∈ R m × n : sup x ∈ R m y T x − τ ( F ( x ) ) < ∞ . The function F ∗ : S ∗ → E n , defined as

(2.24) F ∗ ( y ) = sup x ∈ R m { u ∈ E n | y T x − τ ( F ( x ) ) = τ ( u ) } ,

is called the conjugate of the fuzzy-number-valued function F.

Obviously, τ ( F ( x ) ) + τ ( F ∗ ( y ) ) = y T x , ∀ x ∈ R m ,   y ∈ S ∗ .

Theorem 2.24

Let F : R m → L ( E n ) . Then F ∗ is convex on S ∗ .

Proof

S ∗ = y ∈ R m × n : sup x ∈ R m y T x − τ ( F ( x ) ) < ∞ is a convex set. In fact, ∀ a ∗ , b ∗ ∈ S ∗ ,

sup x ∈ R m ( a ∗ ) T x − τ ( F ( x ) ) < ∞ , sup x ∈ R m ( b ∗ ) T x − τ ( F ( x ) ) < ∞ ,

thus,

sup x ∈ R m { ( λ a ∗ + ( 1 − λ ) b ∗ ) T x − τ ( F ( x ) ) } = sup x ∈ R m ( λ a ∗ + ( 1 − λ ) b ∗ ) T x − λ τ ( F ( x ) ) − ( 1 − λ ) τ ( F ( x ) ) = sup x ∈ R m λ ( a ∗ ) T x − τ ( F ( x ) ) + ( 1 − λ ) ( b ∗ ) T x − τ ( F ( x ) ) ≤ λ sup x ∈ R m { ( a ∗ ) T x − τ ( F ( x ) ) + ( 1 − λ ) sup x ∈ R m ( b ∗ ) T x − τ ( F ( x ) ) < ∞ ,

therefore, λ a ∗ + ( 1 − λ ) b ∗ ∈ S ∗ , that is, S ∗ is a convex set.

For any y 1 , y 2 ∈ R m × n ,

F ∗ ( y 1 ) = sup x ∈ R m u 1 ∈ L ( E n ) | y 1 T x − τ ( F ( x ) ) = τ ( u 1 ) , F ∗ ( y 2 ) = sup x ∈ R m u 2 ∈ L ( E n ) | y 2 T x − τ ( F ( x ) ) = τ ( u 2 ) ,

then for any λ ∈ [ 0 , 1 ] , by (2.19), we have

F ∗ ( λ y 1 + ( 1 − λ ) y 2 ) = sup x ∈ R m u ∈ L ( E n ) | ( λ y 1 + ( 1 − λ ) y 2 ) T x − τ ( F ( x ) ) = τ ( u ) = sup x ∈ R m u ∈ L ( E n ) | λ y 1 T x − λ τ ( F ( x ) ) + ( 1 − λ ) y 2 T x − ( 1 − λ ) τ ( F ( x ) ) = τ ( u ) = sup x ∈ R m { u ∈ L ( E n ) | λ τ ( u 1 ) + ( 1 − λ ) τ ( u 2 ) = τ ( u ) } = sup x ∈ R m λ u 1 + ( 1 − λ ) u 2 ∈ L ( E n ) | y 1 T x − τ ( F ( x ) ) = τ ( u 1 ) ,   y 2 T x − τ ( F ( x ) ) = τ ( u 2 ) ,   u 1 , u 1 ∈ L ( E n ) ≼ c sup x ∈ R m u 1 ∈ L ( E n ) | y 1 T x − τ ( F ( x ) ) = τ ( u 1 ) } + ( 1 − λ ) sup x ∈ R m { u 2 ∈ L ( E n ) | y 2 T x − τ ( F ( x ) ) = τ ( u 2 ) = λ F ∗ ( y 1 ) + ( 1 − λ ) F ∗ ( y 2 ) ,

therefore, F ∗ is convex on S ∗ .□

Proposition 2.25

Let F , G : R m → L ( E n ) . We denote S f ∗ = { y ∈ R m × n : sup x ∈ R m { y T x − τ ( F ( x ) ) } < ∞ , S k f ∗ = { y ∈ R m × n : sup x ∈ R m y T x − τ ( k F ( x ) ) < ∞ and S f g ∗ = { y ∈ R m × n : sup x ∈ R m { y T x − τ ( F ( x ) ) } < ∞ , sup x ∈ R m { y T x − τ ( G ( x ) ) } < ∞ .

  1. If F ( x ) ≼ c G ( x ) , ∀ x ∈ R m , then F ∗ ( y ) ≽ c G ∗ ( y ) , y ∈ S f g ∗ .

  2. ( k F ) ∗ ( y ) = k F ∗ ( k − 1 y ) , y ∈ S f ∗ , k ∈ R .

Proof

  1. Since F ( x ) ≼ c G ( x ) , ∀ x ∈ R m , then τ ( G ( x ) ) − τ ( F ( x ) ) ∈ C , where C ⊆ R n is a closed convex cone with 0 ∈ C and C ≠ R n . It follows that

    τ ( G ( x ) ) − τ ( F ( x ) ) = ( y T x − τ ( F ( x ) ) ) − ( y T x − τ ( G ( x ) ) ) ∈ C ,   ∀ x ∈ R m ,   y ∈ S f g ∗ ,

    thus, let u , v ∈ L ( E n ) , for τ ( u ) = y T x − τ ( F ( x ) ) ,   τ ( v ) = y T x − τ ( G ( x ) ) , we have τ ( u ) ∈ τ ( v ) + C , that is, u ≽ c v . Therefore,

    F ∗ ( y ) = sup x ∈ R m { u ∈ L ( E n ) | y T x − τ ( F ( x ) ) = τ ( u ) } ≼ c sup x ∈ R m { v ∈ L ( E n ) | y T x − τ ( G ( x ) ) = τ ( v ) } = G ∗ ( y ) .

  2. Since sup x ∈ R m y T x − τ ( k F ( x ) ) = sup x ∈ R m y T x − k τ ( F ( x ) ) = k sup x ∈ R m k − 1 y T x − τ ( F ( x ) ) , then y ∈ S k f ∗ ⇔ k − 1 y ∈ S f ∗ . Therefore, for k ∈ R , we have

( k F ) ∗ ( y ) = sup x ∈ R m u ∈ L ( E n ) | y T x − τ ( k F ( x ) ) = τ ( u ) = sup x ∈ R m k k − 1 u ∈ L ( E n ) | k − 1 y T x − τ ( F ( x ) ) = τ ( k − 1 u ) = k sup x ∈ R m k − 1 u ∈ L ( E n ) | k − 1 y T x − τ ( F ( x ) ) = τ ( k − 1 u ) = k F ∗ ( k − 1 y ) . □

For any u i ∈ E n , i = 1 , 2 , … , n , we call the ordered n-dimensional fuzzy number class u 1 , u 2 , … , u n (i.e., the Cartesian product of n-dimensional fuzzy number u 1 , u 2 , … , u n ) a n-dimensional fuzzy vector, denoted it as ( u 1 , u 2 , … , u n ) , and call the collection of all n-dimensional fuzzy vectors (i.e., the Cartesian product E n × E n × ⋯ × E n ︸ n ) n-dimensional fuzzy vector space, and denote it as ( E n ) n .

For any n-dimensional fuzzy vectors u = ( u 1 , u 2 , … , u n ) T ∈ ( E n ) n and v = ( v 1 , v 2 , … , v n ) T ∈ ( E n ) n , let k = ( k 1 , k 2 , … , k n ) T ∈ R n , we define u + v = ( u 1 + v 1 , u 2 + v 2 , … , u n + v n ) , k T u = ∑ i = 1 n k i u i , where + is the fuzzy addition, ⋅ is the fuzzy multiplication, and [ u ] r = ( [ u 1 ] r , [ u 2 ] r , … , [ u n ] r ) , r ∈ [ 0 , 1 ] . We use the following convention for equalities and inequalities: (i) u ≤ v ⇔ u i ≼ c v i ,  i = 1 , 2 , … , n , (ii) u = v ⇔ u i = c v i ,   i = 1 , 2 , … , n and (iii) u < v ⇔ u i ≺ c v i ,   i = 1 , 2 , … , n .

3 Directional g-derivative and subdifferential for fuzzy-number-valued functions

In this section, we generalize the concepts of g-derivative proposed by Hai et al. [9] from [ a , b ] ⊆ R 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 M ⊆ R m , F : M → E 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.

Definition 3.1

Let F : M → E n be a fuzzy-number-valued function, x 0 = x 1 0 , x 2 0 , … , x m 0 ∈ M and h ∈ R with x 1 0 , … , x j 0 + h , … , x m 0 ∈ M . If g-difference F x 1 0 , … , x j 0 + h , … , x m 0 ⊖ g F x 1 0 , … , x j 0 , … , x m 0 exists and there exists u j ∈ E n   ( j = 1 , 2 , … , m ) such that

lim h → 0 F ( x 1 0 , … , x j 0 + h , … , x m 0 ) ⊖ g F ( x 1 0 , … , x j 0 , … , x m 0 ) h = u j ,

then we say that F has the jth partial generalized derivative (g-derivative for short) at x 0 , denoted by u j = ∂ F g ( x 0 ) ∂ x j 0 . 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 g ′ ( x 0 ) .

Here the limit is taken in the metric space ( E n , D ) . If F is g-differentiable at any point of M, then F is said to be g-differentiable on M. The fuzzy vector ( u 1 , u 2 , … , u m ) ∈ ( E n ) m is said to be the gradient of F at x 0 , denoted by ∇ g F ( x 0 ) , i.e., ∇ g F ( x 0 ) = ( u 1 , u 2 , … , u m ) T = ∂ F g ( x 0 ) ∂ x 1 0 , ∂ F g ( x 0 ) ∂ x 2 0 , … , ∂ F g ( x 0 ) ∂ x m 0 T .

Remark 3.2

We give an equivalent form of Definition 3.1. Let F : M → E n be a fuzzy-number-valued function, x 0 ∈ M and h ∈ R with x 0 + h ∈ M . If g-difference F ( x 0 + h ) ⊖ g F ( x 0 ) and there exists a fuzzy vector u = ( u 1 , u 2 , … , u m ) ∈ ( E n ) m such that

lim x → x 0 D ( F ( x ) ⊖ g F ( x 0 ) , ( x − x 0 ) T u ) d ( x , x 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 ∇ g F ( x 0 ) .

Definition 3.3

Let F : M → E 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

F g ′ ( x , y ) = lim λ → 0 + F ( x + λ y ) ⊖ g F ( x ) λ ,

if it exists. Note that

− F g ′ ( x , − y ) = lim λ → 0 − F ( x + λ y ) ⊖ g F ( x ) λ .

We say the one-sided directional g-derivative F g ′ ( x , y ) is two-sided if and only if F g ′ ( x , − y ) exists and

F g ′ ( x , y ) = − F g ′ ( x , − y ) .

We also say F is g-differentiable in the direction y at x. Here the limit is taken in the metric space ( E n , D ) .

Theorem 3.4

Let F : M → E n be a fuzzy-number-valued function. If F is g-differentiable at x, then the directional g-derivatives F g ′ ( x , y ) are two-sided, and

(3.1) F g ′ ( x , y ) = y T ∇ g F ( x ) , ∀ y ∈ R m .

Proof

Since F is g-differentiable at x, then for any y ∈ R m and y ≠ 0 ,

0 = lim λ → 0 + D ( F ( x + λ y ) ⊖ g F ( x ) , ( λ y ) T ∇ g F ( x ) ) | λ | | y | = 1 | y | lim λ → 0 + D F ( x + λ y ) ⊖ g F ( x ) λ , λ y T ∇ g F ( x ) λ = 1 | y | D ( F ′ g ( x , y ) , y T ∇ g F ( x ) ) .

Thus, F g ′ ( x , y ) exists and F g ′ ( x , y ) = y T ∇ g F ( x ) ,   ∀ y ∈ R m . Similarly, we can prove F g ′ ( x , y ) are two-sided.□

Theorem 3.5

Let F : M → L ( E n ) , x 0 = x 1 0 , x 2 0 , … , x m 0 ∈ M . Then F is g-differentiable at x 0 if and only if for any r ∈ [ 0 , 1 ] , the real-valued functions F i − ( x , r ) and F i + ( x , r ) , i = 1 , 2 , … , n , are differentiable at x 0 , and inf β ≥ r min ∂ F i − ( x 0 , β ) ∂ x j 0 , ∂ F i + ( x 0 , β ) ∂ x j 0 and sup β ≥ r max ∂ F i − ( x 0 , β ) ∂ x j 0 , ∂ F i + ( x 0 , β ) ∂ x j 0 satisfy conditions (1)–(4) of Theorem 2.4, where ∂ F i − ( x 0 , r ) ∂ x j 0 and ∂ F i + ( x 0 , r ) ∂ x j 0 , j = 1 , 2 , … , m , are the partial derivatives of F i − ( x , r ) and F i + ( x , r ) at x 0 with respect to the jth component, and

(3.2) ∂ F g ( x 0 ) ∂ x j 0 r = ∏ i = 1 n inf β ≥ r min ∂ F i − ( x 0 , β ) ∂ x j 0 , ∂ F i + ( x 0 , β ) ∂ x j 0 , sup β ≥ r max ∂ F i − ( x 0 , β ) ∂ x j 0 , ∂ F i + ( x 0 , β ) ∂ x j 0 .

Proof

F is g-differentiable at x 0 if and only if there exists u j ∈ L ( E n ) , j = 1 , 2 , … , m , such that

lim h → 0 F ( x 1 0 , … , x j 0 + h , … , x m 0 ) ⊖ g F ( x 1 0 , … , x j 0 , … , x m 0 ) h = u j .

If and only if

lim h → 0 D F ( x 1 0 , … , x j 0 + h , … , x m 0 ) ⊖ g F ( x 1 0 , … , x j 0 , … , x m 0 ) h , u j = 0 .

If and only if by (2.16)

lim h → 0 sup r ∈ [ 0 , 1 ] max 1 ≤ i ≤ n F ( x 1 0 , … , x j 0 + h , … , x m 0 ) ⊖ g F ( x 1 0 , … , x j 0 , … , x m 0 ) h i − ( r ) − u j i − ( r ) , F ( x 1 0 , … , x j 0 + h , … , x m 0 ) ⊖ g F ( x 1 0 , … , x j 0 , … , x m 0 ) h i + ( r ) − u j i + ( r ) = 0 .

If and only if by (2.14)

lim h → 0 sup r ∈ [ 0 , 1 ] max 1 ≤ i ≤ n inf β ≥ r min { F i − ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i − ( x 1 0 , … , x j 0 , … , x m 0 , β ) h , F i + ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i + ( x 1 0 , … , x j 0 , … , x m 0 , β ) h } − u j i − ( r ) , sup β ≥ r max F i − ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i − ( x 1 0 , … , x j 0 , … , x m 0 , β ) h , F i + ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i + ( x 1 0 , … , x j 0 , … , x m 0 , β ) h } − u j i + ( r ) = 0

⇔

lim h → 0 sup r ∈ [ 0 , 1 ] inf β ≥ r min F i − ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i − ( x 1 0 , … , x j 0 , … , x m 0 , β ) h ,   F i + ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i + ( x 1 0 , … , x j 0 , … , x m 0 , β ) h − u j i − ( r ) = 0 , lim h → 0 sup r ∈ [ 0 , 1 ] sup β ≥ r max F i − ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i − ( x 1 0 , … , x j 0 , … , x m 0 , β ) h , F i + ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i + ( x 1 0 , … , x j 0 , … , x m 0 , β ) h − u j i + ( r ) = 0 , i = 1 , 2 , … , n

⇔

inf β ≥ r min lim h → 0 F i − ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i − ( x 1 0 , … , x j 0 , … , x m 0 , β ) h , lim h → 0 F i + ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i + ( x 1 0 , … , x j 0 , … , x m 0 , β ) h = u j i − ( r ) , sup β ≥ r max lim h → 0 F i − ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i − ( x 1 0 , … , x j 0 , … , x m 0 , β ) h , lim h → 0 F i + ( x 1 0 , … , x j 0 + h , … , x m 0 , β ) − F i + ( x 1 0 , … , x j 0 , … , x m 0 , β ) h = u j i + ( r ) , i = 1 , 2 , … , n ,   r ∈ [ 0 , 1 ]

⇔

For any r ∈ [ 0 , 1 ] , F i − ( x , r ) , F i + ( x , r ) , i = 1 , 2 , … , n , are differentiable at x 0 , inf β ≥ r min { ∂ F i − ( x 0 , β ) ∂ x j 0 , ∂ F i + ( x 0 , β ) ∂ x j 0 } = u j i − ( r ) and sup β ≥ r max { ∂ F i − ( x 0 , β ) ∂ x j 0 , ∂ F i + ( x 0 , β ) ∂ x j 0 } = u j i + ( r ) , j = 1 , 2 , … , m , satisfy conditions (1)–(4) of Theorem 2.4, and

∂ F g ( x 0 ) ∂ x j 0 r = ∏ i = 1 n inf β ≥ r min ∂ F i − ( x 0 , β ) ∂ x j 0 , ∂ F i + ( x 0 , β ) ∂ x j 0 , sup β ≥ r max ∂ F i − ( x 0 , β ) ∂ x j 0 , ∂ F i + ( x 0 , β ) ∂ x j 0 . □

Remark 3.6

For r ∈ [ 0 , 1 ] , we denote

∇ g F − ( x 0 , r ) = ∏ i = 1 n inf β ≥ r min ∂ F i − ( x 0 , β ) ∂ x 1 0 , ∂ F i + ( x 0 , β ) ∂ x 1 0 , ∏ i = 1 n inf β ≥ r min ∂ F i − ( x 0 , β ) ∂ x 2 0 , ∂ F i + ( x 0 , β ) ∂ x 2 0 … , ∏ i = 1 n inf β ≥ r min ∂ F i − ( x 0 , β ) ∂ x m 0 , ∂ F i + ( x 0 , β ) ∂ x m 0 T ,

∇ g F + ( x 0 , r ) = ∏ i = 1 n sup β ≥ r max ∂ F i − ( x 0 , β ) ∂ x 1 0 , ∂ F i + ( x 0 , β ) ∂ x 1 0 , ∏ i = 1 n sup β ≥ r max ∂ F i − ( x 0 , β ) ∂ x 2 0 , ∂ F i + ( x 0 , β ) ∂ x 2 0 , … , ∏ i = 1 n sup β ≥ r max ∂ F i − ( x 0 , β ) ∂ x m 0 , ∂ F i + ( x 0 , β ) ∂ x m 0 T .

Theorem 3.7

Let F : M → L ( E n ) , x ∈ M , y ∈ R m . Then F is g-differentiable in the direction y at x if and only if for any r ∈ [ 0 , 1 ] , the real-valued functions from M into R, F i − ( x , r ) and F i + ( x , r ) , i = 1 , 2 , … , n , are differentiable in the direction y at x, and inf β ≥ r min { F i − ′ ( x , y , β ) , F i + ′ ( x , y , β ) } and sup β ≥ r max { F i − ′ ( x , y , β ) , F i + ′ ( x , y , β ) } satisfy conditions (i)–(iv) of Theorem 2.4, and

(3.3) [ F g ′ ( x , y ) r = ∏ i = 1 n [ inf β ≥ r min F i − ′ ( x , y , β ) , F i + ′ ( x , y , β ) , sup β ≥ r max F i − ′ ( x , y , β ) , F i + ′ ( x , y , β ) .

Proof

For u ∈ L ( E n ) ,

lim λ → 0 D F ( x + λ y ) ⊖ g F ( x ) λ , u = 0

⇔

lim λ → 0 sup r ∈ [ 0 , 1 ] max 1 ≤ i ≤ n inf β ≥ r min { F i − ( x + λ y , β ) − F i − ( x + λ y , β ) λ , F i + ( x + λ y , β ) − F i + ( x + λ y , β ) λ } − u i − ( r ) , sup β ≥ r max F i − ( x + λ y , β ) − F i − ( x + λ y , β ) λ , F i + ( x + λ y , β ) − F i + ( x + λ y , β ) λ − u i + ( r ) = 0

⇔

lim λ → 0 sup r ∈ [ 0 , 1 ] inf β ≥ r min F i − ( x + λ y , β ) − F i − ( x + λ y , β ) λ , F i + ( x + λ y , β ) − F i + ( x + λ y , β ) λ − u i − ( r ) = 0 , lim λ → 0 sup r ∈ [ 0 , 1 ] sup β ≥ r max F i − ( x + λ y , β ) − F i − ( x + λ y , β ) λ , F i + ( x + λ y , β ) − F i + ( x + λ y , β ) λ − u i + ( r ) = 0 , i = 1 , 2 , … , n

⇔

inf β ≥ r min lim λ → 0 F i − ( x + λ y , β ) − F i − ( x + λ y , β ) λ , lim λ → 0 F i + ( x + λ y , β ) − F i + ( x + λ y , β ) λ = u i − ( r ) , sup β ≥ r max lim λ → 0 F i − ( x + λ y , β ) − F i − ( x + λ y , β ) λ , lim λ → 0 F i + ( x + λ y , β ) − F i + ( x + λ y , β ) λ = u i + ( r ) , i = 1 , 2 , … , n ,   r ∈ [ 0 , 1 ]

⇔

For any r ∈ [ 0 , 1 ] ,   F i − ( x , r ) and F i + ( x , r ) ,   i = 1 , 2 , … , n , are differentiable in the direction y at x, and inf β ≥ r min F i − ′ ( x , y , β ) , F i + ′ ( x , y , β ) = u i − ( r ) and sup β ≥ r max F i − ′ ( x , y , β ) , F i + ′ ( x , y , β ) = u i + ( r ) satisfy conditions (i)–(iv) of Theorem 2.4, and

[ F g ' ( x , y ) ] r = ∏ i = 1 n [ inf β ≥ r min F i − ′ ( x , y , β ) , F i + ′ ( x , y , β ) , sup β ≥ r max F i − ′ ( x , y , β ) , F i + ′ ( x , y , β ) . □

Definition 3.8

Let M ⊆ R m be a convex set and F : M → L ( E n ) be an n-cell fuzzy-number-valued function. If for any x 1 , x 2 ∈ M and λ ∈ [ 0 , 1 ] , we have

(3.4) F i − ( λ x 1 + ( 1 − λ ) x 2 , r ) ≤ λ F i − ( x 1 , r ) + ( 1 − λ ) F i − ( x 2 , r ) ,   i = 1 , 2 , … , n ,

and

(3.5) F i + ( λ x 1 + ( 1 − λ ) x 2 , r ) ≤ λ F i + ( x 1 , r ) + ( 1 − λ ) F i + ( x 2 , r ) ,   i = 1 , 2 , … , n ,

uniformly for r ∈ [ 0 , 1 ] , that is, for any fixed r ∈ [ 0 , 1 ] , F i − ( x , r ) and F i + ( x , r ) are all convex functions of x, then F is said to be endpoint-wise convex (e-convex for short) on M.

Proposition 3.9

Let F : M → L ( E n ) be e-convex, then F is convex.

Proof

Since F is e-convex, then for any x 1 , x 2 ∈ M , λ ∈ [ 0 , 1 ] and for i = 1 , 2 , … , n , we have

F i − ( λ x 1 + ( 1 − λ ) x 2 , r ) ≤ λ F i − ( x 1 , r ) + ( 1 − λ ) F i − ( x 2 , r ) ,   r ∈ [ 0 , 1 ] ,

F i + ( λ x 1 + ( 1 − λ ) x 2 , r ) ≤ λ F i + ( x 1 , r ) + ( 1 − λ ) F i + ( x 2 , r ) ,   r ∈ [ 0 , 1 ] .

It follows that for i = 1 , 2 , … , n ,

F i − ( λ x 1 + ( 1 − λ ) x 2 , r ) + F i + ( λ x 1 + ( 1 − λ ) x 2 , r ) ≤ λ F i − ( x 1 , r ) + ( 1 − λ ) F i − ( x 2 , r ) + λ F i + ( x 1 , r ) + ( 1 − λ ) F i + ( x 2 , r ) .

Since r ∈ [ 0 , 1 ] , then for i = 1 , 2 , … , n ,

∫ 0 1 r ( F i − ( λ x 1 + ( 1 − λ ) x 2 , r ) + F i + ( λ x 1 + ( 1 − λ ) x 2 , r ) ) d r ≤ ∫ 0 1 r ( λ F i − ( x 1 , r ) + ( 1 − λ ) F i − ( x 2 , r ) + λ F i + ( x 1 , r ) + ( 1 − λ ) F i + ( x 2 , r ) ) d r ,

thus, τ ( F ( λ x 1 + ( 1 − λ ) x 2 ) ) ≤ τ ( λ F ( x 1 ) + ( 1 − λ ) F ( x 2 ) ) . Let C = R n + = { ( x 1 , x 2 , … , x n ) T ∈ R n : x 1 ⩾ 0 ,   x 2 ⩾ 0 , … , x n ⩾ 0 } ⊆ R n , we obtain τ ( λ F ( x 1 ) + ( 1 − λ ) F ( x 2 ) ) ∈ τ ( F ( λ x 1 + ( 1 − λ ) x 2 ) ) + C , that is, F ( λ x 1 + ( 1 − λ ) x 2 ) ≼ c λ F ( x 1 ) + ( 1 − λ ) F ( x 2 ) , therefore, F is convex.□

Definition 3.10

Let F : M → L ( E n ) , then we say F is endpoint-wise differentiable (e-differentiable for short) at x 0 , that is, if there exists u i j − , u i j + ∈ R ,   i = 1 , 2 , … , n ,   j = 1 , 2 , … , m , such that

lim h → 0 F i − ( x 1 0 , … , x j 0 + h , … , x m 0 , r ) − F i − ( x 1 0 , … , x j 0 , … , x m 0 , r ) h = u i j −

and

lim h → 0 F i + ( x 1 0 , … , x j 0 + h , … , x m 0 , r ) − F i + ( x 1 0 , … , x j 0 , … , x m 0 , r ) h = u i j + ,

uniformly for r ∈ [ 0 , 1 ] , then we say F has jth partial e-differentiable at x 0 , denoted by ∂ F i − ( x 0 , r ) ∂ x j 0 = u i j − , ∂ F i + ( x 0 , r ) ∂ x j 0 = u i j + . If all the partial e-derivatives at x 0 exist, then we say F is e-differentiable at x 0 , denoted by F i − ′ ,   F i + ′ , i = 1 , 2 , … , n , respectively. The endpoint-wise gradients of F at x 0 , denoted by ∇ e F i − ( x 0 ) , ∇ e F i + ( x 0 ) , are

∇ e F i − ( x 0 , r ) = ( u i 1 − , u i 2 − , … , u i m − ) T = ∂ F i − ( x 0 , r ) ∂ x 1 0 , ∂ F i − ( x 0 , r ) ∂ x 2 0 , … , ∂ F i − ( x 0 , r ) ∂ x m 0 T , i = 1 , 2 , … , n ,

and

∇ e F i + ( x 0 , r ) = ( u i 1 + , u i 2 + , … , u i m + ) T = ∂ F i + ( x 0 , r ) ∂ x 1 0 , ∂ F i + ( x 0 , r ) ∂ x 2 0 , … , ∂ F i + ( x 0 , r ) ∂ x m 0 T , i = 1 , 2 , … , n .

Theorem 3.11

Let M ⊆ R m be a convex set and F : M → E n be an e-differentiable fuzzy-number-valued function on M. F is e-convex on M if and only if for any x 1 , x 2 ∈ M ,

F i − ( x 1 , r ) ≥ F i − ( x 2 , r ) + ∇ e F i − ( x 2 ) T ( x 1 − x 2 )

and

F i + ( x 1 , r ) ≥ F i + ( x 2 , r ) + ∇ e F i + ( x 2 ) T ( x 1 − x 2 ) ,

uniformly for r ∈ [ 0 , 1 ] , i = 1 , 2 , … , n .

The proof is similar to the proof of Theorem 6.1.2 in the study of Mangasarian [24].

Definition 3.12

Let M ⊆ R m be a convex set and F : M → E n be convex. Then a fuzzy vector x ∈ ( E n ) m is said to be a subgradient of F at a point x ∈ M if

(3.6) F ( z ) ≽ c F ( x ) + ξ T ( z − x ) , ∀ z ∈ M .

The set of all subgradients of F at x is called the subdifferential of F at x and is denoted by ∂ F ( x ) . The multivalued mapping ∂ F : x → ∂ F ( x ) is called the subdifferential of F. If ∂ F ( x ) is not empty, F is said to be subdifferentiable at x.

Proposition 3.13

Let F : M → L ( E n ) be a convex fuzzy n-cell number, x ∈ M . If ξ ∈ ∂ F ( x ) , then

(3.7) F ′ g ( x , y ) ≽ c ξ T y , ∀ y ∈ R m .

Proof

Setting z = x + λ y , λ > 0 ,   ∀ y ∈ R m . Since ξ ∈ ∂ F ( x ) , then we have by definition

F ( x + λ y ) ≽ c F ( x ) + λ ξ T y , ∀ y ∈ R m ,   λ > 0 .

By Proposition 2.11, we have F ( x ) + F ( x + λ y ) ⊖ g F ( x ) ≽ c F ( x ) + λ ξ T y . According to (2.20),

F ( x ) λ + F ( x + λ y ) ⊖ g F ( x ) λ ≽ c F ( x ) λ + ξ T y ,

that is, τ F ( x ) λ + F ( x + λ y ) ⊖ g F ( x ) λ ∈ τ F ( x ) λ + ξ T y + C , where C ⊆ R n is a closed convex cone with 0 ∈ C and C ≠ R n . By (2.19), we have τ F ( x ) λ + τ F ( x + λ y ) ⊖ g F ( x ) λ ∈ τ F ( x ) λ + τ ( ξ T y ) + C . It follows that τ F ( x + λ y ) ⊖ g F ( x ) λ ∈ τ ( ξ T y ) + C , thus, F ( x + λ y ) ⊖ g F ( x ) λ ≽ c ξ T y ,   ∀ y ∈ R m ,   λ > 0 . Therefore,

lim λ → 0 + F ( x + λ y ) ⊖ g F ( x ) λ ≽ c ξ T y , ∀ y ∈ R m ,

and we obtain by definition F ′ g ( x , y ) ≽ c ξ T y ,   ∀ y ∈ R m . □

Theorem 3.14

Let F : M → L ( E n ) be convex and x be a vector in M. Then ∂ F ( x ) is a convex set. Furthermore, if F is lower semicontinuous at x, then ∂ F ( x ) is a closed set.

Proof

If ∂ F ( x ) is empty, the theorem is trivial. If ∂ F ( x ) is not empty, then suppose ξ 1 , ξ 2 ∈ ∂ F ( x ) , for all λ ∈ [ 0 , 1 ] , we have

λ F ( z ) ≽ c λ F ( x ) + λ ( z − x ) T ξ 1 , ∀ z ∈ M ,

( 1 − λ ) F ( z ) ≽ c ( 1 − λ ) F ( x ) + ( 1 − λ ) ( z − x ) T ξ 2 , ∀ z ∈ M .

By (2.20), we have F ( z ) ≽ c F ( x ) + ( z − x ) T ( λ ξ 1 + ( 1 − λ ) ξ 2 ) ,   ∀ z ∈ M . Therefore, λ ξ 1 + ( 1 − λ ) ξ 2 ∈ ∂ F ( x ) , that is, ∂ F ( x ) is convex.

On the other hand, for ξ k = ( ξ k 1 , ξ k 2 , … , x k m ) ∈ ( E n ) m , k ∈ N + , ξ = ( ξ 1 , ξ 2 , … , ξ m ) , suppose that ξ n ∈ ∂ F ( x ) , and ξ k → ξ   ( k → ∞ ) , that is, ξ k i → ξ i   ( k → ∞ ) , i = 1 , 2 , … , m . Since ξ k i ∈ E n ( i = 1 , 2 , … , m ) and ( E n , D ) is a complete metric space, then ξ i ∈ E n , i = 1 , 2 , … , m , thus, ξ = ( ξ 1 , ξ 2 , … , ξ m ) ∈ ( E n ) m . Let { ( x k , ξ k ) } be a sequence contained in g r ∂ F and converge to ( x , ξ ) . Since ξ k ∈ ∂ F , then F ( z ) ≽ c F ( x k ) + ξ k T ( z − x k ) ,   ∀ z ∈ M . Since F is lower semicontinuous at x, then we obtain F ( z ) ≽ c F ( x ) + ξ k T ( z − x k ) ,   ∀ z ∈ M . Let k → ∞ , then we have F ( z ) ≽ c F ( x ) + ξ T ( z − x ) ,   ∀ z ∈ M , thus, ξ ∈ ∂ F ( x ) . Therefore, ∂ F ( x ) is a closed convex set.□

Proposition 3.15

Let F , G : M → L ( E n ) be convex and x be a vector in M. Then for all λ ≥ 0 ,

(3.8) ∂ ( λ F ) ( x ) = λ ∂ F ( x ) , ∂ ( F + G ) ( x ) = ∂ F ( x ) + ∂ G ( x ) .

It is not difficult to prove by (2.20) and Proposition 2.18.

Let M ⊆ R m be a convex set and F : M → E n . Consider the following fuzzy optimization problem with no constraints (FOP)

(3.9) minimize F ( x ) subject to x ∈ M .

A point x ∈ M is called a feasible solution to (FOP). Let x ★ ∈ M , if F ( x ★ ) ≼ c F ( x ) for any x ∈ M , then x ★ is said to be an optimal solution to the problem.

Theorem 3.16

Let F : M → E n be convex. Then x ★ is an optimal solution to (FOP) if and only if 0 ∈ ∂ F ( x ★ ) .

Proof

By Definition 3.12, 0 ∈ ∂ F ( x ★ ) if and only if

F ( x ) ≽ c F ( x ★ ) + ( z − x ) T 0 = F ( x ★ ) ,   ∀ x ∈ M .

□

4 Duality optimality conditions and saddle point optimality criteria in fuzzy optimization problems with constraints

Under the ordering ≼ c , for n-cell fuzzy-number-valued functions, the linear properties (2.19) and (2.20) hold. In this section, we investigate the Lagrange duality and the optimality conditions, including the KKT conditions and the saddle point optimality criteria, in fuzzy optimization problems with constraints for n-cell fuzzy-number-valued functions.

Let M ⊆ R m , F : M → L ( E n ) be the objective function, and g k ( x ) , h s ( x ) : M → L ( E n ) , k = 1 , 2 , … , l ,   s = 1 , 2 , … , t , be the constraint conditions. The fuzzy optimization problem with constraints (FOP1) is defined as

(4.1) minimize F ( x ) subject to g k ( x ) ≼ c 0 ,   k = 1 , 2 , … , l , h s ( x ) = 0 ,   s = 1 , 2 , … , t ,

with variable x ∈ M .

D = { x ∈ M : g k ( x ) ≼ c 0 ,   h s ( x ) = 0 ,   k = 1 , 2 , … , l ,   s = 1 , 2 , … , t } is said to be the feasible set of (FOP1). Let x ★ ∈ D , if F ( x ☆ ) ≼ c F ( x ) 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 D is nonempty. In the following, we denote G ( x ) = ( g 1 ( x ) , g 2 ( x ) , … , g l ( x ) ) T , H ( x ) = ( h 1 ( x ) , h 2 ( x ) , … , h t ( x ) ) T .

Definition 4.1

The Lagrangian L : R m × R l + × R t → L ( E n ) associated with problem 4.1 is defined as

(4.2) L ( x , α , β ) = F ( x ) + α T G ( x ) + β T H ( x ) ,

with dom L = D × R l + × R t , where α = ( α 1 , α 2 , … , α l ) T and β = ( β 1 , β 2 , … , β t ) T . We refer to α k as the Lagrange multiplier associated with the kth inequality constraint g k ( x ) ≼ c 0 ; similarly, we refer to β s as the Lagrange multiplier associated with the sth equality constraint h s ( x ) = 0 . The vectors α and β are called the dual variables or Lagrange multiplier vectors associated with problem 4.1.

We define the Lagrange dual function φ : R l + × R t → E n as the minimum value of the Lagrangian over x, i.e., for α ∈ R l + , β ∈ R t ,

(4.3) φ ( α , β ) = inf x ∈ D L ( x , α , β ) = inf x ∈ D ( F ( x ) + α T G ( x ) + β T H ( x ) ) .

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

(4.4) φ ( α , β ) ≼ c p ★ .

Proof

Suppose x ˜ is a feasible solution to problem 4.1, i.e., g k ( x ˜ ) ≼ c 0 and h s ( x ˜ ) = 0 , and α ≥ 0 , then

α T G ( x ˜ ) + β T H ( x ˜ ) ≼ c 0 .

Since each term in the first sum is nonpositive, and each term in the second sum is zero, then by (2.20)

L ( x ˜ , α , β ) = F ( x ˜ ) + α T G ( x ˜ ) + β T H ( x ˜ ) ≼ c F ( x ˜ ) .

Hence, φ ( α , β ) = inf x ∈ D L ( x , α , β ) ≼ c L ( x ˜ , α , β ) ≼ c F ( x ˜ ) . Since φ ( α , β ) ≼ c F ( x ˜ ) 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

(4.5) maximize φ ( α , β ) subject to α ≥ 0 .

This problem is called the Lagrange dual problem associated (DFOP1) with the problem (FOP1). The original problem (FOP1) is also called the primal problem.

The set D ℒ = { ( α , β ) : α ≥ 0 , φ ( α , β ) > − ∞ } 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). Let ( α ★ , β ★ ) ∈ D ℒ , if φ ( α , β ) ≼ c φ ( α ★ , β ★ ) for each ( α , β ) ∈ D ℒ , 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., d ★ = φ ( α ★ , β ★ ) .

Theorem 4.3

(Weak Duality Theorem) If x and ( α , β ) are feasible solutions to the primal problem (FOP1) and the Lagrange dual problem (DFOP1), respectively, then weak duality holds:

(4.6) F ( x ) ≽ c φ ( α , β ) .

Proof

By definition, we have

φ ( α , β ) = inf x ∈ D L ( x , α , β ) ≼ c F ( x ) + α T G ( x ) + β T H ( x ) ≼ c F ( x ) . □

Corollary 4.4

If x ★ and ( α ★ , β ★ ) are the optimal solutions to the primal problem (FOP1) and the Lagrange dual problem (DFOP1), respectively, then

(4.7) F ( x ★ ) ≽ c φ ( α ★ , β ★ ) ( i . e ., p ★ ≽ c d ★ ) .

We say that strong duality holds if

(4.8) p ★ = d ★ .

Remark 4.5

Let x ★ be a primal optimal solution, ( α ★ , β ★ ) a dual optimal solution with strong duality. Then the complementary slackness condition holds:

(4.9) α k ★ g k ( x ★ ) = 0 ,

namely, α k ★ > 0 ⇒ g k ( x ★ ) = 0 or g k ( x ★ ) ≺ c 0 ⇒ α k ★ = 0 .

Proof

By definition, we have

F ( x ★ ) = φ ( α ★ , β ★ ) = inf x ∈ D F ( x ) + ∑ k = 1 l α i ★ g k ( x ) + ∑ s = 1 t β i ★ h s ( x ) ≼ c F ( x ★ ) + ∑ k = 1 l α i ★ g k ( x ★ ) + ∑ s = 1 t β i ★ h s ( x ★ ) ≼ c F ( x ★ ) .

It follows that

∑ k = 1 l α i ★ g k ( x ★ ) + ∑ s = 1 t β i ★ h s ( x ★ ) = 0 .

According to h s ( x ★ ) = 0 ,   s = 1 , 2 , … , t , we have ∑ k = 1 l α i ★ g k ( x ★ ) = 0 , and since α k ★ ≥ 0 , g k ( x ★ ) ≼ c 0 , k = 1 , 2 , … , l , we obtain α k ★ g k ( x ★ ) = 0 .

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.

Setting C = R n + = { ( x 1 , x 2 , … , x n ) T ∈ R n : x 1 ⩾ 0 ,   x 2 ⩾ 0 , … ,   x n ⩾ 0 } ⊆ R n , the constraint conditions are equivalent to

G ( x ) ≤ 0 ⇔ τ ( g k ( x ) ) ≤ 0 ( 0 ∈ R n ) , k = 1 , 2 , … , l ,

where τ ( g k ( x ) ) = ∫ 0 1 r ( g k 1 − ( x ) ( r ) + g k 1 + ( x ) ( r ) ) d r , ∫ 0 1 r ( g k 2 − ( x ) ( r ) + g k 2 + ( x ) ( r ) ) d r , … , ∫ 0 1 r ( g k n − ( x ) ( r ) + g k n + ( x ) ( r ) d r T , k = 1 , 2 , … l , r ∈ [ 0 , 1 ] , thus, we obtain

G ( x ) ≤ 0 ⇔ ∫ 0 1 r ( g k j − ( x ) ( r ) + g k j + ( x ) ( r ) ) d r ≤ 0 ( 0 ∈ R ) , k = 1 , 2 , … , l , j = 1 , 2 , … , n .

Similarly, we have

H ( x ) = 0 ⇔ ∫ 0 1 r ( h s j − ( x ) ( r ) + h s j + ( x ) ( r ) ) d r ≤ 0 ( 0 ∈ R ) , s = 1 , 2 , … , t , j = 1 , 2 , … , n .

We denote G k ′ ( x ) = ∫ 0 1 r ( g k j − ( x ) ( r ) + g k j + ( x ) ( r ) ) d r , H s ′ ( x ) = ∫ 0 1 r ( h s j − ( x ) ( r ) + h s j + ( x ) ( r ) ) d r , k ′ = 1 , 2 , … , l × n ,   s ′ = 1 , 2 , … , t × n , and denote l × n = p ,   t × n = q , then the fuzzy optimization problem (FOP1) can be transformed into the following fuzzy optimization problem (FOP1′)

(4.10) minimize F ( x ) subject to G k ′ ( x ) ≤ 0 , H s ′ ( x ) = 0 ,

where x ∈ M ,   F : M → L ( E n ) ,   G k ′ , H s ′ : M → R . Obviously, the feasible set of (FOP1′) is equivalent to the feasible set of (FOP1).

Let

φ ′ ( α , β ) = inf x ∈ D L ( x , α , β ) = inf x ∈ D ( F ( x ) + α ( r ) T G ′ ( x ) + β ( r ) T H ′ ( x ) ) , r ∈ [ 0 , 1 ] ,

where G ′ ( x ) = ( G 1 ( x ) , G 2 ( x ) , … , G p ( x ) ) T , H ′ ( x ) = ( H 1 ( x ) , H 2 ( x ) , … , H q ( x ) ) T . The Lagrange dual problem (DFOP1′) associated with the problem (FOP1′) is

(4.11) maximize φ ′ ( α , β ) subject to α ≥ 0 ,

We refer to α ( r ) = ( α 1 ( r ) , α 2 ( r ) , … , α p ( r ) ) T ∈ R p + and β ( r ) = ( β 1 ( r ) , β 2 ( r ) , … , β q ( r ) ) T ∈ R q as the Lagrange multiplier vectors containing parameter.

We now assume that the feasible set of (FOP1′) D ′ = { x ∈ M : G k ′ ( x ) ≤ 0 ,   H s ′ ( x ) = 0 ,   k ′ = 1 , 2 , … , p , s ′ = 1 , 2 , … , q } ⊆ K C n , the real-valued functions G k ′ ( x ) are convex and differentiable on M, and H s ′ ( x ) are affine functions.□

Theorem 4.6

(KKT conditions) Let F be e-convex and e-differentiable on M. If x ★ = ( x 1 ★ , x 2 ★ , … , x m ★ ) is an optimal solution to (FOP1′) and ( α ★ ( r ) , β ★ ( r ) )   ( r ∈ [ 0 , 1 ] ) is an optimal solution to (DFOP1′) with strong duality, then x ★ , α ★ ( r ) , β ★ ( r ) satisfy the following conditions:

(4.12) ∂ F i − ( x ★ , r ) ∂ x j ★ + ∂ F i + ( x ★ , r ) ∂ x j ★ + ∑ k ′ = 1 p α k ′ ★ ( r ) ∂ G k ′ ( x ★ ) ∂ x j ★ + ∑ s ′ = 1 q β s ′ ★ ( r ) ∂ H s ′ ( x ★ ) ∂ x j ★ = 0 , i = 1 , 2 , … , n , j = 1 , 2 , … , m ,

(4.13) α k ′ ★ ( r ) G k ′ ( x ★ ) = 0 , k ′ = 1 , 2 , … , p ,

(4.14) G k ′ ( x ★ ) ≤ 0 ,   H s ′ ( x ★ ) = 0 , k ′ = 1 , 2 , … , p ,   s ′ = 1 , 2 , … , q ,

(4.15) α k ′ ★ ( r ) ≥ 0 , k ′ = 1 , 2 , … , p .

Conversely, if x ★ , α ★ , β ★ are any points that satisfy the KKT conditions (4.12–4.15), then x ★ and ( α ★ , β ★ ) are primal solution and dual optimal solution, and strong duality holds.

Proof

∀ r ∈ [ 0 , 1 ] , we denote F ¯ i ( x , r ) = F i − ( x , r ) + F i + ( x , r ) , i = 1 , 2 , … , n . Since F is e-convex and e-differentiable on M, then the real-valued function F i − ( x , r ) and F i + ( x , r ) ,   i = 1 , 2 , … , n , are convex and differentiable on M. Thus, ∀ r ∈ [ 0 , 1 ] , F ¯ i ( x , r ) is convex on M and differentiable at x ★ , furthermore, we have

∂ F ¯ i ( x ★ , r ) ∂ x j ★ = ∂ F i − ( x ★ , r ) ∂ x j ★ + ∂ F i + ( x ★ , r ) ∂ x j ★ , i = 1 , 2 , … , n , j = 1 , 2 , … , m .

Therefore, ∀ r ∈ [ 0 , 1 ] , (4.12) of the KKT conditions is equivalent to

(4.16) ∂ F ¯ i ( x ★ , r ) ∂ x j ★ + ∑ k ′ = 1 p α k ′ ★ ( r ) ∂ G k ′ ( x ★ ) ∂ x j ★ + ∑ s ′ = 1 q β s ′ ★ ( r ) ∂ H s ′ ( x ★ ) ∂ x j ★ = 0 , i = 1 , 2 , … , n , j = 1 , 2 , … , m ,

thus, (FOP1′) is equivalent to the problem of which the objective function is the real-valued function containing parameter F ¯ i ( x , r ) under the constraint conditions (4.13–4.16), and its Lagrangian is

L ( x , α ★ , β ★ ) ( r ) = inf x ∈ D ( F ¯ i ( x , r ) + α ( r ) T G ′ ( x ) + β ( r ) T H ′ ( x ) ) , r ∈ [ 0 , 1 ] .

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 ¯ i ( x , r ) , thus, x ★ is an optimal solution to the optimization problem and ( α ★ , β ★ ) is an optimal solution to its Lagrange dual problem, that is, ∀ x ∈ int M and ∀ ( α , β ) ∈ ( R l + , R t ) ,

(4.17) F ¯ i ( x ★ , r ) ≤ F ¯ i ( x , r ) , i = 1 , 2 , … , n ,

(4.18) φ ′ ( α ★ , β ★ ) ≥ φ ′ ( α , β ) .

By reductio ad absurdum, suppose that x ★ is not an optimal solution of (FOP1′), then there exists x ′ ∈ i n t M such that F ( x ′ ) < F ( x ★ ) . Let C = R n + ⊆ R n , according to Definition 2.13, we have

∫ 0 1 r ( F i − ( x ′ , r ) + F i + ( x ′ , r ) ) d r < ∫ 0 1 r ( F i − ( x ★ , r ) + F i + ( x ★ , r ) ) d r , i = 1 , 2 , … , n ,

that is, ∫ 0 1 r F ¯ i ( x ′ , r ) d r < ∫ 0 1 r F ¯ i ( x ★ , r ) d r , i = 1 , 2 , … , n , which is in contradiction to (4.17). Therefore, x ★ is an optimal solution to (FOP1′). Equation (4.18) can be proved similarly. This completes the proof.□

Theorem 4.7

(Strong Duality Theorem) If Slater’s condition holds in e-convex problem (FOP1), i.e., there exists x ˜ ∈ relint D with g k ( x ˜ ) ≺ c 0 , then strong duality holds.

Proof

Consider the real-valued convex optimization problem (OP1)

(4.19) minimize F ¯ i ( x , r ) subject to G k ′ ( x ) ≤ 0 , H s ′ ( x ) = 0 ,

where x ∈ M ,   r ∈ [ 0 , 1 ]   i = 1 , 2 , … , n . For x ˜ ∈ relint D, since g k ( x ˜ ) ≺ c 0 ⇔ G k ′ ( x ˜ ) < 0 , 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.□

Definition 4.8

Let f : R m × R l × R t → E n , X ⊆ R m ,  W ⊆ R l ,  Z ⊆ R t . A vector triplet ( x ¯ , α ¯ , β ¯ ) with x ¯ ∈ X ,  w ¯ ∈ W ,  z ¯ ∈ Z is said to be a saddle point of f if

(4.20) f ( x ¯ , w , z ) ≼ c f ( x ¯ , w ¯ , z ¯ ) ≼ c f ( x , w ¯ , z ¯ )

holds for all x ∈ X , w ∈ W , z ∈ Z .

In other words, x ¯ minimizes f ( x , w ¯ , z ¯ ) over x ∈ X and ( w ¯ , z ¯ ) maximizes f ( x ¯ , w , z ) over ( w , z ) ∈ ( W , Z ) , i.e.,

f ( x ¯ , w ¯ , z ¯ ) = inf x ∈ X f ( x , w ¯ , z ¯ ) , f ( x ¯ , w ¯ , z ¯ ) = sup w ∈ W , z ∈ Z f ( x ¯ , w , z ) .

Theorem 4.9

(Saddle Point Theorem) If x ★ and ( α ★ , β ★ ) are primal and dual optimal solutions to (FOP1) in which strong duality obtains, respectively, then ( 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.

Proof

For any x ∈ R m ,

(4.21) sup α ≥ 0 L ( x , α , β ) = sup α ≥ 0 F ( x ) + ∑ k = 1 l α k g k ( x ) + ∑ s = 1 t β s h s ( x ) = F ( x ) , x ∈ D , ∞ , otherwise .

Indeed, if x is a feasible solution, i.e., g k ( x ) ≼ c 0 , h s ( x ) = 0 , k = 1 , 2 , … , l , s = 1 , 2 , … , t , then the optimal choice α is α = 0 for any β , thus sup α ≥ 0 L ( x , α , β ) = F ( x ) . If x is not a feasible solution, then there exists g k ( x ) ≻ c 0 for some k, let α m = 0 ,   m ≠ k and α k → ∞ , we obtain sup α ≥ 0 L ( x , α , β ) = ∞ .

Similarly, for any α ∈ R l ,  β ∈ R t ,

(4.22) inf x ∈ D L ( x , α , β ) = inf x ∈ D F ( x ) + ∑ k = 1 l α k g k ( x ) + ∑ s = 1 t β s h s ( x ) = inf x ∈ D F ( x ) + ∑ k = 1 l α k g k ( x ) + ∑ s = 1 t β s h s ( x ) , α k ≥ 0 , k = 1 , 2 , … , l , − ∞ , otherwise .

Therefore, we can express the optimal value of the primal problem as

(4.23) p ★ = inf x ∈ D sup α ≥ 0 L ( x , α , β ) .

If x ★ is a primal optimal solution, then we obtain p ★ = sup α ≥ 0 L ( x ★ , α , β ) . If the pair ( α ★ , β ★ ) is a dual optimal solution, then we have d ★ = φ ( α ★ , β ★ ) = sup α ≥ 0 φ ( α , β ) = sup α ≥ 0 inf x ∈ D L ( x , α , β ) = inf x ∈ D L ( x , α ★ , β ★ ) .

Since strong duality holds, then p ★ = d ★ , thus, we have

inf x ∈ D L ( x , α ★ , β ★ ) = L ( x ★ , α ★ , β ★ ) = sup α ≥ 0 L ( x ★ , α , β ) , ∀ x ∈ D , α ∈ R l + , β ∈ R t ,

that is, L ( x ★ , α , β ) ≼ c L ( x ★ , α ★ , β ★ ) ≼ c L ( x , α ★ , β ★ ) . Therefore, ( x ★ , α ★ , β ★ ) is a saddle point for the Lagrangian.

Conversely, since ( x ¯ , α ¯ , β ¯ ) is a saddle point of the Lagrangian, then

inf x ∈ D L ( x , α ¯ , β ¯ ) = sup α ≥ 0 L ( x ¯ , α , β ) , ∀ x ∈ D , α ∈ R l + , β ∈ R t .

It follows from (4.21) and (4.22) that x ¯ ∈ S ,   α ¯ ∈ R l + , inf x ∈ D L ( x , α ¯ , β ¯ ) = F ( x ¯ ) . For any x ∈ D , we have α ¯ k g k ( x ) ≼ c 0 , k = 1 , 2 , … , l , and consequently L ( x , α ¯ , β ¯ ) ≼ c F ( x ) , where α ∈ R l + . Therefore,

inf x ∈ D L ( x , α ¯ , β ¯ ) ≼ c inf x ∈ D F ( x ) = p ★ ≼ c F ( x ¯ ) .

It follows that inf L ( x , α ¯ , β ¯ ) = p ★ = F ( x ¯ ) . 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 1 , x 2 ) . However, the quantity is only an estimated quantity, then using a two-dimensional fuzzy number-valued function u ( x 1 , x 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 number-valued function u ( x 1 , x 2 ) , which is defined as follows:

u ( x 1 , x 2 ) = 20 x 2 − 18 , 0.9 ≤ x 2 ≤ 0.95 , 200 x 2 − 90 ≤ x 1 ≤ 290 − 200 x 2 , − 0.1 x 1 + 11 , 100 ≤ x 1 ≤ 110 , 1.45 − 0.005 x 1 ≤ x 2 ≤ 0.45 + 0.005 x 1 , − 20 x 2 + 20 , 0.95 ≤ x 2 ≤ 1 , 290 − 200 x 2 ≤ x 1 ≤ 200 x 2 − 90 , 0.1 x 1 − 9 , 90 ≤ x 1 ≤ 100 , 0.45 + 0.005 x 1 ≤ x 2 ≤ 1.45 − 0.005 x 1 , 0 , otherwise .

Let C = R 2 + ⊆ R 2 , then u r = [ 90 + 10 r , 100 − 10 r ] × [ 0.9 + 0.05 r , 1 − 0.05 r ] , r ∈ [ 0 , 1 ] . By (2.18), we have τ ( u ) = ( 95 , 0.95 ) T . Suppose that F ( t ) ( x 1 , x 2 ) = f ( t ) ⋅ u ( x 1 , x 2 ) , where f ( t ) = e t − t 2 , t ∈ [ 1 , ∞ ) . Consider the following fuzzy optimization problem

minimize F ( t ) subject to G 1 ( t ) = t 2 − 36 ≤ 0 , G 2 ( t ) = − t + 1 ≤ 0 .

Obviously, G 1 ( t ) , G 2 ( t ) are convex, and differentiable and continuous at t = 1 , F is e-convex, and e-differentiable and continuous at t = 1 . There exist α 1 ★ = 0 and α 2 ★ = e − 2 , satisfy the condition of Theorem 4.6, therefore, t = 1 is an optimal solution to the above fuzzy optimization problem.

Example 4.11

Let u 1 , u 2 : R 1 → E 1 be triangular fuzzy numbers, and u 1 = ( x 1 − 1 , x 1 , x 1 + 1 ) , u 2 = ( x 2 − 1 , x 2 , x 2 + 1 ) and 4 ˜ = ( 3 , 4 , 5 ) is a triangular fuzzy number. Consider the following fuzzy optimization problem

(4.24) minimize F ( x 1 , x 2 ) = u 1 ⋅ u 1 + u 2 ⋅ u 2 subject to G ( x 1 , x 2 ) = u 1 + u 2 + ( − 4 ˜ ) ± c 0 ,

where x 1 , x 2 ≥ 1 .

Since for u 1 , u 2 ∈ E 1 , and r ∈ [ 0 , 1 ] , [ u ⋅ v ] r = [ min { u − ( r ) ⋅ v − ( r ) , u − ( r ) ⋅ v + ( r ) , u + ( r ) ⋅ v − ( r ) , u + ( r ) ⋅ v + ( r ) } , max { u − ( r ) ⋅ v − ( r ) , u − ( r ) ⋅ v + ( r ) , u + ( r ) ⋅ v − ( r ) , u + ( r ) ⋅ v + ( r ) } ], then for any r ∈ [ 0 , 1 ] , we have

  [ u 1 ( x 1 ) ] r = [ x 1 − ( 1 − r ) ,   x 1 + ( 1 − r ) ] ,   [ u 2 ( x 2 ) ] r = [ x 2 − ( 1 − r ) ,   x 2 + ( 1 − r ) ] ,   [ 4 ˜ ] r = [ 4 − ( 1 − r ) ,  4 + ( 1 − r ) ] ,   [ F ( x 1 , x 2 ) ] r = [ ( x 1 − ( 1 − r ) ) 2 + ( x 2 − ( 1 − r ) ) 2 ,   ( x 1 + ( 1 − r ) ) 2 + ( x 2 + ( 1 − r ) ) 2 ] ,   [ G ( x 1 , x 2 ) ] r = [ x 1 − ( 1 − r ) + ( x 2 − ( 1 − r ) − 3 − r ,   x 1 + ( 1 − r ) + x 2 + ( 1 − r ) − 5 + r ] = [ x 1 + x 2 + r − 5 ,   x 1 + x 2 − r − 3 ] ,

thus, we obtain

  [ ∇ F ( x 1 , x 2 ) ] r = ( [ 2 ( x 1 − 1 + r ) , 2 ( x 1 + 1 − r ) ] ,   [ 2 ( x 2 − 1 + r ) , 2 ( x 2 + 1 − r ) ] ) T ,   [ ∇ G ( x 1 , x 2 ) ] r = ( [ 1 , 1 ] ,   [ 1 , 1 ] ) T ,

therefore, according to Theorem 4.6, the KKT conditions are as follows:

2 x 1 − α = 0 , 2 x 2 − α = 0 , α ( x 1 + x 2 − 4 ) = 0 , α ≥ 0 .

If α = 0 , then x 1 = 0 , x 2 = 0 ; if α = 4 , then x 1 = 2 , x 2 = 2 . Since x 1 , x 2 ≥ 1 , then x = ( 2 , 2 ) T is an optimal solution to (4.24).

5 Discussion

Based on another fuzzy ordering ≼ s proposed in [9], since the linear properties hold for n-dimensional fuzzy-number-valued functions, we can similarly investigate the Lagrange duality and the optimality conditions.

Proposition 5.1

[9,18,19,23] Suppose u ∈ E n , then

  1. u ∗ ( r , x + y ) ≤ u ∗ ( r , x ) + u ∗ ( r , y ) ,

  2. u ∗ ( r , x ) ≤ sup a ∈ [ u ] r ∥ a ∥ , i.e., u ∗ ( r , x ) is bounded on S n − 1 for each fixed r ∈ [0,1],

  3. u ∗ ( r , x ) is nonincreasing and left continuous in r ∈ [0,   1], right continuous at r = 0, for each fixed x ∈ S n − 1 ,

  4. u ∗ ( r , x ) is Lipschitz continuous in x , i.e., | u ∗ ( r , x ) − u ˜ ∗ ( r , y ) | ≤ ( sup a ∈ [ u ] r ∥ a ∥ ) ∥ x − y ∥ ,

  5. if u , v ∈ E n , r ∈ [ 0 , 1 ] , then d ( [ u ] r , [ v ] r ) = sup x ∈ S n − 1 | u ∗ ( r , x ) − u ∗ ( r , x ) | ,

  6. ( u + v ) ∗ ( r , x ) = u ∗ ( r , x ) + u ∗ ( r , x ) ,

  7. ( k u ) ∗ ( r , x ) = k u ∗ ( r , x ) , for any k ≥ 0,

  8. − u ∗ ( r , − x ) ≤ u ∗ ( r , x ) ,

  9. ( − u ) ∗ ( r , x ) = u ∗ ( r , − x ) .

Definition 5.2

Let F : M → E n be a fuzzy-number-valued function, x 0 = x 1 0 , x 2 0 , … , x m 0 ∈ M and h ∈ R with ( x 1 0 , … , x j 0 + h , … , x m 0 ) ∈ M . If there exists u j ∈ E n   ( j = 1 , 2 , … , m ) such that

lim h → 0 F ( x 1 0 , … , x j 0 + h , … , x m 0 ) ∗ ( r , x ) − F ( x 1 0 , … , x j 0 , … , x m 0 ) ∗ ( r , x ) h = u j ∗ ( r , x )

uniformly for r ∈ [ 0 , 1 ] and x ∈ S n − 1 , then we say that F has the jth partial support-function-wise derivative (s-derivative for short) at x 0 , denoted by ∂ F s ( x 0 ) ∂ x j 0 = u j . If all the partial s-derivatives at x 0 exist, then we say F is support-function-wise differentiable (s-differentiable for short) at x 0 , denoted by F ′ s ( x 0 ) .

If F is s-differentiable at any point of M, then F is said to be s-differentiable on M. The fuzzy vector ( u 1 , u 2 , … , u m ) ∈ ( E n ) m is said to be the support-function-wise gradient of F at x 0 , denoted by ∇ s F ( x 0 ) ,

∇ s F ( x 0 ) = ( u 1 , u 2 , … , u m ) T = ∂ F s ( x 0 ) ∂ x 1 0 , ∂ F s ( x 0 ) ∂ x 2 0 , … , ∂ F s ( x 0 ) ∂ x m 0 T .

Theorem 5.3

Let F : M → E n be s-differentiable at x 0 ∈ M . If there exists δ > 0 such that x 0 + h ∈ M and g-difference F ( x 0 + h ) ⊖ g F ( x 0 ) exists for any | h | < δ , then F is g-differentiable at x 0 and we have

(5.1) F g ′ ( x 0 ) ∗ ( r , x ) = ( 1 ) sup β ≥ r F g ′ ( x 0 ) ∗ ( β , x ) , or ( 2 ) sup β ≥ r ( − 1 ) ( − F g ′ ( x 0 ) ) ∗ ( β , x ) .

The proof is similar to the proof of Theorem 4.2 in the study of Hai et al. [9].

Definition 5.4

Let M ⊆ R m be a convex set and F : M → E n be a fuzzy-number-valued function. If for any x 1 , x 2 ∈ M and λ ∈ [ 0 , 1 ] ,

F ( λ x 1 + ( 1 − λ ) x 2 ) ∗ ( r , x ) ≤ λ F ( x 1 ) ∗ ( r , x ) + ( 1 − λ ) F ( x 2 ) ∗ ( r , x )

uniformly for r ∈ [ 0 , 1 ] and x ∈ S n − 1 , then F is said to be support-function-wise convex (s-convex) on M.

Theorem 5.5

Let M ⊆ R m be a convex set and F : M → E n be an s-differentiable fuzzy-number-valued function on M. F is s-convex on M if and only if for any x 1 , x 2 ∈ M ,

F ( x 1 ) ∗ ( r , x ) − F ( x 2 ) ∗ ( r , x ) ≥ ∇ s F ( x 2 ) ∗ ( r , x ) T ( x 1 − x 2 ) ,

uniformly for r ∈ [ 0 , 1 ] and x ∈ S n − 1 .

Note that for u = ( u 1 , u 2 , … , u n ) T ∈ ( E n ) n , u ∗ ( r , x ) T = ( u 1 ∗ ( r , x ) , u 2 ∗ ( r , x ) , … , u n ∗ ( r , x ) ) T . The proof is similar to the proof of Theorem 6.1.2 in the study of Mangasarian [24].

Definition 5.6

[9] For u , v ∈ E n , we say that u ≼ s v if for any r ∈ [ 0 , 1 ] and x ∈ S n − 1 , u ∗ ( r , x ) ≤ v ∗ ( r , x ) . We say that u ≺ s v if u ≼ s v and there exists r 0 ∈ [ 0 , 1 ] with u ∗ ( r 0 , x ) < v ∗ ( r 0 , x ) , or there exists x 0 ∈ S n − 1 with u ∗ ( r , x 0 ) < v ∗ ( r , x 0 ) .

Let M ⊆ R m be a convex set, F ( x ) , g k ( x ) , h p ( x ) ,   k = 1 , 2 , … , l ,   p = 1 , 2 , … , q , be s-convex and s-differentiable fuzzy-number-valued functions on M. We consider the following optimization problem (FOP2):

(5.2) minimize F ( x ) subject to g k ( x ) ≼ s 0 ,   k = 1 , 2 , … , l , h p ( x ) = 0 ,   p = 1 , 2 , … , q .

S = { x ∈ M : g k ( x ) ≼ s 0,  h p ( x ) = 0 ,   k = 1 , 2 , … , l ,   p = 1 , 2 , … , q } is said to be the feasible set of (FOP2). Let x ★ ∈ S , if F ( x ★ ) ≼ s F ( x ) for each x ∈ S , then x ★ is said to be an optimal solution to (FOP2).

We define

φ ( α , β ) = inf x ∈ D L ( x , α , β ) = inf x ∈ D ( F ( x ) + α T G ( x ) + β T H ( x ) ) ,

where α = ( α 1 , α 2 , … , α l ) T ∈ R l + ,   β = ( β 1 , β 2 , … , β t ) T ∈ R q , G ( x ) = ( g 1 ( x ) , g 2 ( x ) , … , g l ( x ) ) T , H ( x ) = ( h 1 ( x ) , h 2 ( x ) , … , h q ( x ) ) T . The Lagrange dual problem (DFOP2) associated with (FOP2) is

(5.3) maximize φ ( α , β ) subject to α ≥ 0 .

Theorem 5.7

(Weak Duality Theorem) If x and ( α , β ) are feasible solutions to the primal problem (FOP2) and the Lagrange dual problem (DFOP2), respectively, then weak duality holds: F ( x ) ≽ s φ ( α , β ) .

Theorem 5.8

(KKT Conditions) Let F be s-convex and s-differentiable on M. If x ˜ = ( x ˜ 1 , x ˜ 2 , … , x ˜ m ) is an optimal solution to (FOP2) and ( α ˜ , β ˜ ) is an optimal solution to (DFOP2) with strong duality, then x ˜ , α ˜ , β ˜ satisfy the following conditions:

(5.4) ∂ F s ( x ˜ ) ∂ x ˜ j + ∑ k = 1 l α ˜ k ∂ ( g k ) s ( x ˜ ) ∂ x ˜ j + ∑ p = 1 q β ˜ p ∂ ( h p ) s ( x ˜ ) ∂ x ˜ j = 0 , j = 1 , 2 , … , m ,

(5.5) α ˜ k g k ( x ˜ ) = 0 , k = 1 , 2 , … , l ,

(5.6) g k ( x ˜ ) ≼ s 0 ,   h p ( x ˜ ) = 0 , k = 1 , 2 , … , l ,   p = 1 , 2 , … , q ,

(5.7) α ˜ k ≥ 0 , k = 1 , 2 , … , l .

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

(5.8) g k ( x ) ≼ s 0 , h p ( x ) = 0 ,   k = 1 , 2 , … , l ,   p = 1 , 2 , … , q .

If x ˜ , α ˜ , β ˜ are any points that satisfy the KKT conditions (5.4)–(5.7), then by Theorem 5.5, (5.4)–(5.7) and (5.8),

F ( x ) ∗ ( r , t ) − F ( x ˜ ) ∗ ( r , t ) ≥ ∂ F s ( x ˜ ) ∂ x ˜ 1 ∗ ( r , t ) , ∂ F s ( x ˜ ) ∂ x ˜ 2 ∗ ( r , t ) , … , ( ∂ F s ( x ˜ ) ∂ x ˜ m ) ∗ ( r , t ) T ( x − x ˜ ) = − ∑ k = 1 l α ˜ k ∂ ( g k ) s ( x ˜ ) ∂ x ˜ 1 ∗ ( r , t ) + ∑ p = 1 q β ˜ p ∂ ( h p ) s ( x ˜ ) ∂ x ˜ 1 ) ∗ ( r , t ) , − ∑ k = 1 l α ˜ k ∂ ( g k ) s ( x ˜ ) ∂ x ˜ 2 ∗ ( r , t ) + ∑ p = 1 q β ˜ p ∂ ( h p ) s ( x ˜ ) ∂ x ˜ 2 ∗ ( r , t ) , … , − ∑ k = 1 l α ˜ k ∂ ( g k ) s ( x ˜ ) ∂ x ˜ m ∗ ( r , t ) + ∑ p = 1 q β ˜ p ∂ ( h p ) s ( x ˜ ) ∂ x ˜ m ∗ ( r , t ) T ( x − x ˜ ) = ( − α ˜ 1 ) ∂ ( g 1 ) s ( x ˜ ) ∂ x ˜ 1 , ∂ ( g 1 ) s ( x ˜ ) ∂ x ˜ 2 , … , ∂ ( g 1 ) s ( x ˜ ) ∂ x ˜ m ) ∗ ( r , t ) T ( x − x ˜ ) + ( − α ˜ 2 ) ∂ ( g 2 ) s ( x ˜ ) ∂ x ˜ 1 , ∂ ( g 2 ) s ( x ˜ ) ∂ x ˜ 2 , … , ∂ ( g 2 ) s ( x ˜ ) ∂ x ˜ m ) ∗ ( r , t ) T ( x − x ˜ ) + ⋯ + ( − α ˜ l ) ∂ ( g l ) s ( x ˜ ) ∂ x ˜ 1 , ∂ ( g l ) s ( x ˜ ) ∂ x ˜ 2 , … , ∂ ( g l ) s ( x ˜ ) ∂ x ˜ m ) ∗ ( r , t ) T ( x − x ˜ ) + ( − β ˜ 1 ) ∂ ( h 1 ) s ( x ˜ ) ∂ x ˜ 1 , ∂ ( h 1 ) s ( x ˜ ) ∂ x ˜ 2 , … , ∂ ( h 1 ) s ( x ˜ ) ∂ x ˜ m ) ∗ ( r , t ) T ( x − x ˜ ) + ( − β ˜ 2 ) ∂ ( h 2 ) s ( x ˜ ) ∂ x ˜ 1 , ∂ ( h 2 ) s ( x ˜ ) ∂ x ˜ 2 , … , ∂ ( h 2 ) s ( x ˜ ) ∂ x ˜ m ) ∗ ( r , t ) T ( x − x ˜ ) + ⋯ + ( − β ˜ q ) ∂ ( h q ) s ( x ˜ ) ∂ x ˜ 1 , ∂ ( h q ) s ( x ˜ ) ∂ x ˜ 2 , … , ∂ ( h q ) s ( x ˜ ) ∂ x ˜ m ) ∗ ( r , t ) T ( x − x ˜ ) = ( − α ˜ 1 ∇ s g 1 ( x ˜ ) T ( x − x ˜ ) ) + ( − α ˜ 2 ∇ s g 2 ( x ˜ ) T ( x − x ˜ ) ) + ⋯ + ( − α ˜ l ∇ s g l ( x ˜ ) T ( x − x ˜ ) ) + ( − β ˜ 1 ∇ s h 1 ( x ˜ ) T ( x − x ˜ ) ) + ( − β ˜ 2 ∇ s h 2 ( x ˜ ) T ( x − x ˜ ) ) + ⋯ + ( − β ˜ q ∇ s h q ( x ˜ ) T ( x − x ˜ ) ) ≥ ( α ˜ 1 g 1 ( x ˜ ) ∗ ( r , t ) − α ˜ 1 g 1 ( x ) ∗ ( r , t ) ) + ( α ˜ 2 g 2 ( x ˜ ) ∗ ( r , t ) − α ˜ 2 g 2 ( x ) ∗ ( r , t ) ) + ⋯ + ( α ˜ l g l ( x ˜ ) ∗ ( r , t ) − α ˜ l g l ( x ) ∗ ( r , t ) ) + ( β ˜ 1 h 1 ( x ˜ ) ∗ ( r , t ) − β ˜ 1 h 1 ( x ) ∗ ( r , t ) ) + ( β ˜ 2 h 2 ( x ˜ ) ∗ ( r , t ) − β ˜ 2 h 2 ( x ) ∗ ( r , t ) ) + ⋯ + ( β ˜ q h q ( x ˜ ) ∗ ( r , t ) − β ˜ q h q ( x ) ∗ ( r , t ) ) = ( − α ˜ 1 g 1 ( x ) ∗ ( r , t ) ) + ( − α ˜ 2 g 2 ( x ) ∗ ( r , t ) ) + ⋯ + ( − α ˜ l g l ( x ) ∗ ( r , t ) ) ≥ 0 ,

that is, ∀ x ∈ S , F ( x ˜ ) ≼ s F ( x ) , therefore, x ˜ is an optimal solution to (FOP2). Similarly, we can prove ( α ˜ , β ˜ ) is a dual optimal solution, and strong duality holds.□

Theorem 5.9

(Strong Duality Theorem) For s-convex problem (FOP2), if Slater’s condition holds, i.e., there exists x ˜ ∈ r e l i n t D with g k ( x ˜ ) ≺ c 0 , then strong duality holds.

Proof

For any r ∈ [0 , 1] , t ∈ S n − 1 , consider the real-valued convex optimization problem (OP2)

(5.9) minimize F ( x ) ∗ ( r , t ) subject to g k ( x ) ∗ ( r , t ) ≤ 0 , h p ( x ) ∗ ( r , t ) = 0 ,

where x ∈ M , the real-valued functions F ( x ) ∗ ( r , t ) , g k ( x ) ∗ ( r , t ) are convex and differentiable on M, and H p ( x ) ∗ ( r , t ) are affine functions. For x ˜ ∈ r e l i n t D, since g k ( x ˜ ) ≺ s 0 ⇔ g k ( x ) ∗ ( r , t ) ( x ˜ ) < 0 , the convex optimization problem (OP2) satisfies Slater’s condition, therefore, for (OP2) and its Lagrange dual problem, the 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.□

Theorem 5.10

(Saddle Point Theorem) If x ★ and ( α ★ , β ★ ) are primal and dual optimal solutions to (FOP2) in which strong duality obtains, respectively, then ( 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.

6 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 fuzzy-number-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.

Acknowledgments

This research was supported by the National Natural Science Foundation of China under grant number 61763044.

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Received: 2019-09-27
Revised: 2020-07-04
Accepted: 2020-09-10
Published Online: 2020-12-07

© 2020 Ting Xie et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0081
Keywords for this article
fuzzy n-cell number; n-dimensional fuzzy-number-valued function; directional generalized derivative; subdifferential; fuzzy optimization
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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