Generalized convexity of n-dimensional fuzzy number-valued functions and its application in fuzzy optimization

Abstract

In this paper, we mainly introduced the invexity and generalized invexity of n-dimensional fuzzy number-valued functions based on the new ordering which defined by Gong and Hai in [9]. Simultaneously, we discussed the relationship between semicontinuous and preinvex fuzzy number-valued functions, and some properties among invexity and generalized invexity of n-dimensional fuzzy number-valued functions. Finally, we studied the necessary and sufficient conditions for weakly efficient point of fuzzy optimization.

Keywords

Invex fuzzy number-valued functions n-dimensional fuzzy number fuzzy optimization semicontinuity Gordan’s Theorem

1 Introduction

Since in 1965, the American scholar Zadeh proposed the concept of fuzzy sets in [12], the operation and application of fuzzy numbers and fuzzy sets have attracted widespread attention in the world. Because of the rapid development of computer science, control theory, and system science, computers are required to have the functions of fuzzy logic mind and image mind like the human brain. Gradually, fuzzy mathematics has become an independent discipline and has been widely studied by a large number of scholars.

Due to the development of the optimization theory, the convex analysis Plays an extremely important role and gets further development in recent years. In fuzzy mathematics, the concept of convex fuzzy mappings was first proposed by Nanada and Kar in [13] and established criteria for convex fuzzy mappings. Later, in the literature [14–16 , 18], many scholars have studied the convex fuzzy mappings more extensively and deeply and applied them to fuzzy optimization. In 1994, the concept of fuzzy preinvex functions as generalized convexities were proposed in [19], and some of their properties were studied. Subsequently, in [17 , 21], these concepts of pseudoconvex, invex, and pseudoinvex are proposed for the fuzzy mapping of one variable. Based on the differentiability of fuzzy mappings proposed by Wang and Wu in [22], Wu and Xu [10, 11] introduced convex, preinvex invex, pseudoinvex, prequasiinvex fuzzy mappings from Rⁿ to E, and applied these concepts to study the relationship between fuzzy variational-like inequality and fuzzy optimization. There is no doubt that the proposal of invex sets has expanded the scope of research and brought many new results to fuzzy optimization.

In 1986, Goetschel and Voxman proposed linear ordering from R to E in [1], this ordering is reflexive and transitive, and it turns the comparison of two fuzzy interval numbers into a comparison of two numbers. Based on this linear ordering, Yan and Xu in [2] proposed graphs, convex and quasi-convex of fuzzy mapping, describe characteristics of the convex, quasi-convex fuzzy mappings, and discussed the properties of convex fuzzy optimizations. In [3], Syau and Lee given concepts of convex, quasi-convex fuzzy mappings and continuous, and extended local-global minimum property of real-valued convex functions to convex fuzzy mappings. In [4], they given concepts of preinvex and prequasiinvex fuzzy mappings. In [5], Syau et al. discussed the properties of semi-continuous fuzzy mapping from Rⁿ to E. More research on the properties of convex and generalized fuzzy mappings based on the linear ordering in recent years, we can refer to [6, 7].

Since most of the previous studies of convex fuzzy functions are based on one-dimensional fuzzy numbers, few people have studied the properties of n-dimensional fuzzy numbers. Therefore, it is necessary to study the properties of convex and generalized convex fuzzy mappings under n-dimensional fuzzy numbers and related fuzzy optimization problems. Based on the study of the n-dimensional convex fuzzy number-valued functions in [9], in this paper, we give concepts of preinvex, (strictly) invex, (strictly) pseudoinvex and (strictly, strongly) prequasiinvex fuzzy number-valued functions from Rⁿ to Eⁿ and their properties are discussed. Since the function T_F is a real-valued vector function when mapping T acts on fuzzy mapping F, therefore, we discuss the relationship between the weakly efficient point and the vector critical point of the unconstrained n-dimensional generalized convex fuzzy number-valued functions. And we obtained some necessary and sufficient conditions for the vector critical point is a weakly efficient point.

The structure of this paper is as follows. In the section 2, some preliminary knowledge about fuzzy numbers and n-dimensional fuzzy numbers are introduced. In the section 3, the relationship between n-dimensional preinvex and semicontinuous fuzzy number-valued functions is discussed. The definitions of preinvex, (strictly) invex, (strictly) pseudoinvex and (strictly, strongly) prequasiinvex fuzzy number-valued functions are presented, and some corresponding examples are given. At the same time, we also discuss the relationship between n-dimensional generalized convex fuzzy number-valued functions; Finally, we give some results based on an application to generalized convex fuzzy programming in section 5.

2 Preliminaries

In this paper, we use Rⁿ to denotes the n-dimensional Euclidean. A fuzzy set μ on Rⁿ is a mapping μ : Rⁿ → [0, 1]. We call [μ] ^r ={ x ∈ Rⁿ : μ (x) ≥ r } r-cut for any r ∈ (0, 1], and suppμ ={ x ∈ Rⁿ : μ (x) >0 } is support of μ. We defined [μ] ⁰ is the closure of supp μ.

Definition 2.1. (See [11]) A compact and convex fuzzy set μ on Rⁿ is a fuzzy set with the following properties:

(1) μ is normal, i.e. there exists x₀ ∈ Rⁿ such that μ (x₀) =1;

(2) μ is an upper semi-continuous function;

(3) μ (λx + (1 - λ) y) ≥ min {μ (x) , μ (y)} , x, y ∈ Rⁿ, λ ∈ [0, 1];

(4) [μ] ⁰ is compact.

We denote the set of all fuzzy numbers by E.

A real number a is a special case of fuzzy number encoded as $\tilde{a} (t) = {\begin{matrix} 1, t = a \\ 0, t \neq a \end{matrix}$ In particular, the fuzzy number $\tilde{0}$ is defined as $\tilde{0} (t) = 1$ if t = 0, and $\tilde{0} (t) = 0$ if t ≠ 0.

In the following, Eⁿ denotes the n-dimensional fuzzy number space.

Theorem 2.1. (See [23, 24]) If μ ∈ Eⁿ, then (1) [μ] ^r is a nonempty compact convex subset of Rⁿ for any r ∈ (0, 1], (2) [μ] ^{r
₁} ⊆ [μ] ^{r
₂}, whenever 0 ≤ r₂ ≤ r₁ ≤ 1, (3) if r_n > 0 and r_n converging to r ∈ [0, 1] is nondecreasing, then $\cap_{n = 1}^{\infty} [μ]^{r_{n}} = [μ]^{r}$ . Conversely, suppose that for any r ∈ [0, 1], an A^r ⊆ Rⁿ exists that satisfies (1) - (3) above, then a unique μ ∈ Eⁿ exists such that [μ] ^r = A^r, r ∈ (0, 1], [μ] ⁰ = ∪ _r∈(0,1] [μ] ^r ⊆ A⁰. For any μ, ν ∈ Eⁿ, k ∈ R, the addition and scalar multiplication of n-dimensional fuzzy numbers are defined as following, $(μ + ν) (x) = \sup_{s + t = x} \min {μ (s), ν (t)}$ , $(k μ) (x) = μ (\frac{x}{k})$ , $(0 μ) (x) = {\begin{matrix} 0, x \neq 0, \\ 1, x = 0 . \end{matrix}$

Theorem 2.2. (See [25]) If μ, ν ∈ Eⁿ, k, k₁, k₂ ∈ R, then (1) k (μ + ν) = kμ + kν, (2) k₁ (k₂μ) = (k₁k₂) μ, (3) (k₁ + k₂) μ = k₁μ + k₂μ when k₁ ≥ 0 and k₂ ≥ 0. Given μ, ν ∈ Eⁿ, the distance D : Eⁿ × Eⁿ → [0, + ∞) between μ and ν is defined by the equation $D (μ, ν) = \sup_{r \in [0, 1]} d ([μ]^{r}, [ν]^{r}),$ where d is the Hausdorff metric given by $\begin{matrix} d ([μ]^{r}, [ν]^{r}) \\ = inf {ɛ : [μ]^{r} \subset N ([ν]^{r}, ɛ), [ν]^{r} \subset N ([μ]^{r}, ɛ)} \\ = \max {\sup_{a \in [μ]^{r}} \inf_{b \in [ν]^{r}} ∥ a - b ∥, \sup_{b \in [ν]^{r}} \inf_{a \in [μ]^{r}} ∥ a - b ∥} \end{matrix}$ Let μ, ν ∈ Eⁿ. If there exists ω ∈ Eⁿ, such that μ = ν + ω, then we say μ, ν having H-difference and ω is called the H-difference of μ and ν, denoted μ - ν.

Definition 2.2. (See [9]) Let τ : Eⁿ → Rⁿ be a vector-valued function defined by $\begin{matrix} τ (μ) = & (2 \int_{0}^{1} r \frac{\int \dots \int_{[μ]^{r}} x_{1} {dx}_{1} {dx}_{2} \dots {dx}_{n}}{\int \dots \int_{[μ]^{r}} 1 {dx}_{1} {dx}_{2} \dots {dx}_{n}} dr, \\ 2 \int_{0}^{1} r \frac{\int \dots \int_{[μ]^{r}} x_{2} {dx}_{1} {dx}_{2} \dots {dx}_{n}}{\int \dots \int_{[μ]^{r}} 1 {dx}_{1} {dx}_{2} \dots {dx}_{n}} dr, \dots, \\ 2 \int_{0}^{1} r \frac{\int \dots \int_{[μ]^{r}} x_{n} {dx}_{1} {dx}_{2} \dots {dx}_{n}}{\int \dots \int_{[μ]^{r}} 1 {dx}_{1} {dx}_{2} \dots {dx}_{n}} dr) \end{matrix}$ where $2 \int_{0}^{1} r \frac{\int \dots \int_{[μ]^{r}} x_{1} {dx}_{1} {dx}_{2} \dots {dx}_{n}}{\int \dots \int_{[μ]^{r}} 1 {dx}_{1} {dx}_{2} \dots {dx}_{n}} dr (i = 1, 2,$ ⋯, n) is the Lebesgue integral of $r \frac{\int \dots \int_{[μ]^{r}} x_{1} {dx}_{1} {dx}_{2} \dots {dx}_{n}}{\int \dots \int_{[μ]^{r}} 1 {dx}_{1} {dx}_{2} \dots {dx}_{n}} (i = 1, 2, \dots, n)$ on [0, 1]. The vector-valued function τ is called a ranking value function defined on Eⁿ. τ (μ) represents a centroid of the n-dimensional fuzzy number μ.

Definition 2.3. (See [9]) Let μ, ν ∈ Eⁿ, C ⊆ Rⁿ be a closed convex cone with 0 ∈ C and C ≠ Rⁿ. We say that μ ⪯ _cν(μ precedes ν) if $τ (ν) \in τ (μ) + C,$ where ⪯_c is a partially ordered relation on Eⁿ. For μ, ν ∈ Eⁿ, if either μ ⪯ _cν or ν ⪯ _cμ, then we say that μ and ν are comparable; otherwise, they are non-comparable. And it is obviously, when μ, ν ∈ E¹, C = [0, + ∞) ⊆ R, then Definition 2.3 coincides with the Definition 2.5 from [1]. Also, we say that μ ≺ _cν if μ ⪯ _cν and τ (μ) ≠ τ (ν). In some cases, we may write ν ⪰ _cμ(ν ≻ _cμ) instead of μ ⪯ _cν(μ ≺ _cν).

Definition 2.4. Let t₂ ∈ K (⊂ Rⁿ), we say K is invex with respect to η : K × K → Rⁿ. If for each t₁ ∈ K, λ ∈ [0, 1], t₂ + λη (t₁, t₂) ∈ K. K is said to be an invex set with respect to η if K is invex at each t₂ ∈ K. Considering fuzzy mappings F from K onto Eⁿ, we call this mapping a n-dimensional fuzzy number-valued function, and for any r ∈ [0, 1], denote F_r (t) = [F (t)] ^r = ([F₁ (t)] ^r, [F₂ (t)] ^r, ⋯ , [F_n (t)] ^r), t ∈ K. We say F (t) is differentiable (See [26]) at t ∈ K if and only if F₁ (t) , F₂ (t) , ⋯ , F_n (t) are all differentiable at t ∈ K. And when F (t) is differentiable, $F_{r}^{'} (t) = [F^{'} (t)]^{r} = ([F_{1}^{'} (t)]^{r}, [F_{2}^{'} (t)]^{r}, \dots, [F_{n}^{'} (t)]^{r}),$ for any r ∈ [0, 1].

Definition 2.5. Let F : K → Eⁿ. T_F : K → Rⁿ is defined by $\begin{matrix} T_{F} (t) & = T (F (t)) \\ = (2 \int_{0}^{1} r \frac{\int \dots \int_{[μ]^{r}} x_{1} {dx}_{1} {dx}_{2} \dots {dx}_{n}}{\int \dots \int_{[μ]^{r}} 1 {dx}_{1} {dx}_{2} \dots {dx}_{n}} dr, \\ 2 \int_{0}^{1} r \frac{\int \dots \int_{[μ]^{r}} x_{2} {dx}_{1} {dx}_{2} \dots {dx}_{n}}{\int \dots \int_{[μ]^{r}} 1 {dx}_{1} {dx}_{2} \dots {dx}_{n}} dr, \dots, \\ 2 \int_{0}^{1} r \frac{\int \dots \int_{[μ]^{r}} x_{n} {dx}_{1} {dx}_{2} \dots {dx}_{n}}{\int \dots \int_{[μ]^{r}} 1 {dx}_{1} {dx}_{2} \dots {dx}_{n}} dr) \end{matrix}$ Through the above Definition, the T_F (t) can be seen as a real-value vector. Thus for any T_F (t₁) , T_F (t₂) ∈ Rⁿ, we denote

(1)T_F (t₁) ≦ T_F (t₂) iff $T_{F} (t_{1}^{(i)}) ⩽ T_{F} (t_{2}^{(i)})$ , i = 1, ⋯ , n;

(2)T_F (t₁) ≤ T_F (t₂) iff $T_{F} (t_{1}^{(i)}) ⩽ T_{F} (t_{2}^{(i)})$ with T_F (t₁) ≠ T_F (t₂), i = 1, ⋯ , n;

(3)T_F (t₁) < T_F (t₂) iff $T_{F} (t_{1}^{(i)}) < T_{F} (t_{2}^{(i)})$ , i = 1, ⋯ , n;

(4)T_F (t₁) </T_F (t₂) is the negation of $T_{F} (t_{1}^{(i)}) < T_{F} (t_{2}^{(i)})$ , i = 1, ⋯ , n.

3 Semicontinuity and preinvexity of fuzzy number-valued functions

As we all known, semicontinuity plays a very important role in fuzzy function. In this section, we will discuss the relationship between semicontinuity and preinvexity of n-dimensional fuzzy number-valued functions. Since the function T_F is a real-valued vector function when mapping T acts on fuzzy mapping F, therefore, there are some difference in definitions and proofs and some properties may be limited.

Condition C. Let K ⊆ Rⁿ be an invex set with respect to η that satisfies the following conditions for ∀t₁, t₂ ∈ K, λ ∈ [0, 1]:

(1) η (t₂, t₂ + λη (t₁, t₂)) = - λη (t₁, t₂),

(2) η (t₁, t₂ + λη (t₁, t₂)) = (1 - λ) η (t₁, t₂).

Definition 3.1. Let F : K → Eⁿ be a fuzzy number-valued function. T_F : K → Rⁿ.

(1) F is said to be upper semicontinuous at t₀ ∈ K, if for any $\tilde{ɛ} ≻_{c} 0$ , a neighborhood U of t₀ exists when t ∈ U, and we have $F (t) ≺_{c} F (t_{0}) + \tilde{ɛ} .$

(2) F is said to be lower semicontinuous at t₀ ∈ K, if for any $\tilde{ɛ} ≻_{c} 0$ , a neighborhood U of t₀ exists when t ∈ U, and we have $F (t_{0}) ≺_{c} F (t) + \tilde{ɛ} .$

A fuzzy number-valued function F : K → Eⁿ is continuous at t₀ ∈ K if it is both upper and lower semicontinuous, and that it is continuous if and only if it is continuous at every point of K, and T_F is also semicontinuous or continuous.

Definition 3.2. Let F : K → Eⁿ be a fuzzy number-valued function. F is said to be fuzzy preinvex on K with respect to a function η : K × K → Rⁿ, if for all t₁, t₂ ∈ K, λ ∈ [0, 1], $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≦ λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) .$

Example 1. Consider the 2-dimensional fuzzy number-valued function F : [0, 2) → E² be defined as F_r (t) = [90 + 5t + 10r, 110 + 5t - 10r] × [94 + t + 2r, 98 + t - 2r], r ∈ [0, 1].

According Definition 2.5, we have T_F (t) = (100 + 5t, 96 + t), let $\begin{matrix} η (t_{1}, t_{2}) = {\begin{matrix} t_{2} - t_{1}, & if 0 \leq t_{2} \leq t_{1} < 2, \\ 2 (t_{1} - t_{2}), & if 0 \leq t_{1} \leq t_{2} < 2 . \end{matrix} \end{matrix}$

If we choose t₁ = 1, t₂ = 0, then, η (t₁, t₂) = -1, T_F (t₂ + λη (t₁, t₂)) = (100 - 5λ, 96 - λ), and λT_F (t₁) + (1 - λ) T_F (t₂) = (100 + 5λ, 96 + λ), we can verify that (100 - 5λ, 96 - λ) ≦ (100 + 5λ, 96 + λ) for all λ ∈ [0, 1]. That is T_F (t₂ + λη (t₁, t₂)) ≦ λT_F (t₁) + (1 - λ) T_F (t₂), from Definition 3.2, F is a preinvex fuzzy number-valued function.

Lemma 3.1. Let K ⊆ Rⁿ be an invex set with respect to η that satisfies Condition C for t₁, t₂ ∈ K, λ ∈ [0, 1]. Let F : K → Eⁿ be a fuzzy number-valued function that satisfies T_F (t₂ + η (t₁, t₂)) ≦ T_F (t₁) , ∀ t₁, t₂ ∈ K. If there exists a λ ∈ (0, 1) such that $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≦ λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}),$ ∀t₁, t₂ ∈ K . then the set A ={ λ ∈ [0, 1] |T_F (t₂ + λη (t₁, t₂)) ≦ λT_F (t₁) + (1 - λ) T_F (t₂) , ∀ t₁, t₂ ∈ K } is dense in [0, 1]. The proof is similar to the proof of Theorem 2 in [8] and is omitted.

Thereom 3.1. Let K ⊆ Rⁿ be an invex set with respect to η that satisfies Condition C for any t₁, t₂ ∈ K, λ ∈ [0, 1], and let F : K → Eⁿ be an upper semicontinuous fuzzy number-valued function and satisfies T_F (t₂ + η (t₁, t₂)) ≦ T_F (t₁) , ∀ t₁, t₂ ∈ K. If there exists a λ ∈ (0, 1) such that $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≦ λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}),$ ∀t₁, t₂ ∈ K . then F is a preinvex fuzzy number-valued function.

Proof. Suppose that the F is not a preinvex fuzzy number-valued function on K. Hence there exist t₁, t₂ ∈ K and λ ∈ (0, 1), such that $T_{F} (t_{2} + λ η (t_{1}, t_{2})) > λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) .$ (1) Let A = { λ ∈ [0, 1] |T_F (t₂ + λη (t₁, t₂)) ≦ λT_F (t₁) + (1 - λ) T_F (t₂) , ∀ t₁, t₂ ∈ K } . Define $t_{2}^{(n)} = t_{2} + \frac{λ - λ_{n}}{1 - λ_{n}} η (t_{1}, t_{2})$ , then, $∥ t_{2}^{(n)} - t_{2} ∥ \to 0$ when n→ ∞. According Condition C, we can get $t_{2}^{(n)} + λ_{n} η (t_{1}, t_{2}^{(n)}) \to t_{2} + λ η (t_{1}, t_{2}), n \to \infty$

By the upper semicontinuous of F on K and Definition 3.1, for any ɛ > 0, there exists a positive integer N such that $T_{F} (t_{2}^{(n)}) < T_{F} (t_{2}) + \frac{ɛ}{1 - λ}, \forall n > N .$

From the Lemma 3.1, A is dense. Thus there exists a sequence {λ_n} with λ_n ∈ A, and for any $t_{1}, t_{2}^{(n)} \in K$ such that $\begin{matrix} T_{F} (t_{2}^{(n)} + λ_{n} η (t_{1}, t_{2}^{(n)})) \\ ≦ λ_{n} T_{F} (t_{1}) + (1 - λ_{n}) T_{F} (t_{2}^{(n)}) \\ < λ_{n} T_{F} (t_{1}) + (1 - λ_{n}) T_{F} (t_{2}) + (1 - λ_{n}) \frac{ɛ}{1 - λ} . \end{matrix}$

That is, $\begin{matrix} T_{F} (t_{2}^{(n)} + λ_{n} η (t_{1}, t_{2}^{(n)})) \\ < λ_{n} T_{F} (t_{1}) + (1 - λ_{n}) T_{F} (t_{2}) + (1 - λ_{n}) \frac{ɛ}{1 - λ}, \end{matrix}$ (2)

Let λ_n → λ (n → ∞), we get $T_{F} (t_{2} + λ η (t_{1}, t_{2})) < λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) + ɛ .$ (3)

Since ɛ is an arbitrary small positive real number, we get $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≦ λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) .$ which is contradicts (1). Therefore, F is a preinvex fuzzy number-valued function on K.

Thereom 3.2. Let K ⊆ Rⁿ be an invex set with respect to η that satisfies Condition C for any t₁, t₂ ∈ K, λ ∈ [0, 1], and let F : K → Eⁿ be an lower semicontinuous fuzzy number-valued function and satisfies T_F (t₂ + η (t₁, t₂)) ≦ T_F (t₁) , ∀ t₁, t₂ ∈ K. If there exists a λ ∈ (0, 1) such that $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≦ λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}),$ ∀t₁, t₂ ∈ K . then F is a preinvex fuzzy number-valued function.

Let $\begin{matrix} A & = {λ \in [0, 1] | T_{F} (t_{2} + λ η (t_{1}, t_{2})) \\ ≦ λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}), \forall t_{1}, t_{2} \in K} . \end{matrix}$

From the Lemma 3.1, A is dense. Thus there exists a sequence {λ_n} with λ ∈ A such that $T_{F} (t_{2} + λ_{n} η (t_{1}, t_{2})) ≦ λ_{n} T_{F} (t_{1}) + (1 - λ_{n}) T_{F} (t_{2}),$ (5)

By the lower semicontinuous of F on K, then, for any ɛ > 0, there exists a positive integer N such that $T_{F} (t_{2} + λ η (t_{1}, t_{2})) < T_{F} (t_{2} + λ_{n} η (t_{1}, t_{2})) + ɛ .$ (6)

By (5),(6) and λ_n → λ (n → ∞), we get $T_{F} (t_{2} + λ η (t_{1}, t_{2})) < λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) + ɛ .$

Since ɛ is an arbitrary small positive real number, we get $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≦ λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) .$ which is contradicts (4). Hence F is a preinvex fuzzy number-valued function on K.

Thereom 3.3. Let K ⊆ Rⁿ be an invex set with respect to η and let F : K → Eⁿ be a fuzzy number-valued function satisfies Condition C and T_F (t₂ + η (t₁, t₂)) ≦ T_F (t₁) , ∀ t₁, t₂ ∈ K. Then, the fuzzy number-valued function F is preinvex on K if and only if the fuzzy function φ (λ) = T_F (t₂ + λη (t₁, t₂)) is convex.

Proof. Suppose that the fuzzy number-valued function F is preinvex on K with respect to η, now, we verify the fuzzy function φ (λ) is convex.

For any λ₁ > λ₂, λ₁, λ₂ ∈ [0, 1], k ∈ [0, 1], and for ∀t₁, t₂ ∈ K, from Condition C, we have $\begin{matrix} η (t_{2} + k η (t_{1}, t_{2}), t_{2}) \\ = η (t_{2} + k η (t_{1}, t_{2}), t_{2} + k η (t_{1}, t_{2}) - k η (t_{1}, t_{2})) \\ = η (t_{2} + k η (t_{1}, t_{2}), t_{2} + k η (t_{1}, t_{2}) + η (t_{2}, t_{2} + \\ k η (t_{1}, t_{2}))) \\ = - η (t_{2}, t_{2} + k η (t_{1}, t_{2})) \\ = k η (t_{1}, t_{2}) \end{matrix}$ and $\begin{matrix} η (t_{2} + λ_{1} η (t_{1}, t_{2}), t_{2} + λ_{2} η (t_{1}, t_{2})) \\ = η (t_{2} + λ_{2} η (t_{1}, t_{2}) + (λ_{1} - λ_{2}) η (t_{1}, t_{2}), t_{2} + \\ λ_{2} η (t_{1}, t_{2})) \\ = η (t_{2} + λ_{2} η (t_{1}, t_{2}) + (\frac{λ_{1} - λ_{2}}{1 - λ_{2}}) η (t_{1}, t_{2} + \\ λ_{2} η (t_{1}, t_{2})), t_{2} + λ_{2} η (t_{1}, t_{2})) \\ = \frac{λ_{1} - λ_{2}}{1 - λ_{2}} η (t_{1}, t_{2} + λ_{2} η (t_{1}, t_{2})) \\ = \frac{λ_{1} - λ_{2}}{1 - λ_{2}} \cdot (1 - λ_{2}) η (t_{1}, t_{2}) \\ = (λ_{1} - λ_{2}) η (t_{1}, t_{2}) \end{matrix}$

Then, by the preinvexity of F, we have $\begin{matrix} φ (k λ_{1} + (1 - k) λ_{2}) \\ = φ (λ_{2} + k (λ_{1} - λ_{2})) \end{matrix}$ $\begin{matrix} = T_{F} (t_{2} + (λ_{2} + k (λ_{1} - λ_{2})) η (t_{1}, t_{2})) \\ = T_{F} (t_{2} + λ_{2} η (t_{1}, t_{2}) + k (λ_{1} - λ_{2}) η (t_{1}, t_{2})) \\ = T_{F} (t_{2} + λ_{2} η (t_{1}, t_{2}) + k η (t_{2} + λ_{1} η (t_{1}, t_{2}), t_{2} + \\ λ_{2} η (t_{1}, t_{2}))) \\ ≦ {kT}_{F} (t_{2} + λ_{1} η (t_{1}, t_{2})) + (1 - k) T_{F} (t_{2} + λ_{2} η (t_{1}, t_{2})) \\ = k φ (λ_{1}) + (1 - k) φ (λ_{2}) \end{matrix}$

So, φ (λ) is convex.

Conversely, let the fuzzy functioin φ (λ) is convex. Then, we have $\begin{matrix} T_{F} (t_{2} + λ η (t_{1}, t_{2})) \\ = φ (λ) \\ = φ (λ \cdot 1 + (1 - λ) \cdot 0) \\ ≦ λ φ (1) + (1 - λ) φ (0) \\ = λ T_{F} (t_{2} + η (t_{1}, t_{2})) + (1 - λ) T_{F} (t_{2}) \\ ≦ λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) \end{matrix}$

Thus, the fuzzy number-valued function F is preinvex on K.

4 Generalized convexity of fuzzy number-valued functions

In this section, we will discuss the relationship in (strictly) invex, (strictly, strongly) pseudoinvex, and (strictly) prequasiinvex n-dimensional fuzzy number-valued functions under η-directional differentiable. In the first, we give these definitions about generalized convexity of n-dimensional fuzzy number-valued functions.

Definition 4.1. A differentiable fuzzy number-valued function F : K → Eⁿ is said to be fuzzy invex on K with respect to a function η : K × K → Rⁿ, if for all t₁, t₂ ∈ K, $▽ T_{F} (t_{2}) η (t_{1}, t_{2}) ≦ T_{F} (t_{1}) - T_{F} (t_{2}) .$

Definition 4.2. A differentiable fuzzy number-valued function F : K → Eⁿ is said to be fuzzy strictly invex on K with respect to a function η : K × K → Rⁿ, if for all t₁, t₂ ∈ K, t₁ ≠ t₂, $▽ T_{F} (t_{2}) η (t_{1}, t_{2}) < T_{F} (t_{1}) - T_{F} (t_{2}) .$

Example 2. Consider the 2-dimensional fuzzy number-valued function F : (0, ∞) → E² be defined as following $F (t) (x_{1}, x_{2}) = {\begin{matrix} \frac{1}{e^{t}} \sqrt{e^{2 t} - x_{1}^{2} - x_{2}^{2}}, & x_{1}^{2} + x_{2}^{2} \leq e^{2 t}, \\ x_{1} \geq 0, x_{2} \geq 0, \\ 0, & otherwise, \end{matrix}$

Then $F_{r} (t) = {(x_{1}, x_{2}) : x_{1}^{2} + x_{2}^{2} \leq e^{2 t} (1 - r^{2}), x_{1} \geq 0, x_{2} \geq 0}$ , from Definition 2.5, we have $T_{F} (t) = (\frac{8 e^{t}}{9 π}, \frac{8 e^{t}}{9 π})$ , and $▽ T_{F} (t) = (\frac{8 e^{t}}{9 π}, \frac{8 e^{t}}{9 π})$ . Let t₁ = 3, t₂ = 1, and $η (t_{1}, t_{2}) = {\begin{matrix} 1 - e^{t_{1} - t_{2}}, & t_{1} \geq t_{2} > 0, \\ (e^{t_{1} - t_{2}} - 1), & t_{2} > t_{1} > 0 . \end{matrix}$ , we have $▽ T_{F} (1) η (3, 1) = (\frac{8 e}{9 π} (1 - e^{2}), \frac{8 e}{9 π} (1 - e^{2}))$ , $T_{F} (3) - T_{F} (1) = (\frac{8}{9 π} (e^{3} - e), \frac{8}{9 π} (e^{3} - e))$ , thus, ▽T_F (1) η (3, 1) < T_F (3) - T_F (1). If we let t₁ = 1, t₂ = 3 and with the same η, we get ▽T_F (3) η (1, 3) = T_F (1) - T_F (3). Therefore, for any t₁, t₂ ∈ (0, ∞), there exists a η, such that ▽T_F (t₂) η (t₁, t₂) ≦ T_F (t₁) - T_F (t₂), from Definition 4.1, F is a invex fuzzy number-valued function. But, from Definition 4.2, F is not a strictly invex fuzzy number-valued function.

Definition 4.3. A differentiable fuzzy number-valued function F : K → Eⁿ is called fuzzy pseudoinvex on K with respect to a function η : K × K → Rⁿ, if for all t₁, t₂ ∈ K, $▽ T_{F} (t_{2}) η (t_{1}, t_{2}) ≧ 0 \Rightarrow T_{F} (t_{1}) ≧ T_{F} (t_{2}), or$ $T_{F} (t_{1}) < T_{F} (t_{2}) \Rightarrow ▽ T_{F} (t_{2}) η (t_{1}, t_{2}) < 0 .$

Definition 4.4. A differentiable fuzzy number-valued function F : K → Eⁿ is called fuzzy strictly pseudoinvex on K with respect to a function η : K × K → Rⁿ, if for all t₁, t₂ ∈ K, t₁ ≠ t₂, $▽ T_{F} (t_{2}) η (t_{1}, t_{2}) ≧ 0 \Rightarrow T_{F} (t_{1}) > T_{F} (t_{2}), or$ $T_{F} (t_{1}) ≦ T_{F} (t_{2}) \Rightarrow ▽ T_{F} (t_{2}) η (t_{1}, t_{2}) < 0 .$

Example 3. Consider the Example 2, for any t₁ > 0, t₂ > 0, if T_F (t₁) < T_F (t₂), then there exist a η (t₁, t₂) = t₁ - t₂ such that ▽T_F (t₂) η (t₁, t₂) <0, according Definition 4.3, F is a pseudoinvex fuzzy number-valued function. If t₁ ≠ t₂, we have ▽T_F (t₂) · η (t₁, t₂) <0, from Definition 4.4, F is also a strictly pseudoinvex fuzzy number-valued function.

Definition 4.5. A differentiable fuzzy number-valued function F : K → Eⁿ is called fuzzy prequasiinvex on K with respect to a function η : K × K → Rⁿ, if for any t₁, t₂ ∈ K, λ ∈ [0, 1], $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≦ max {T_{F} (t_{1}), T_{F} (t_{2})} .$

Where T_F (t₁) and T_F (t₂) are comparable.

Example 4. Let the fuzzy number-valued function F : (1, ∞) × (1, ∞) → E² be defined by $\begin{matrix} F (ξ_{1}, ξ_{2}) (x_{1}, x_{2}) \\ = {\begin{matrix} \frac{\sqrt{{ln}^{2} (ξ_{1} + ξ_{2}) - x_{1}^{2}}}{ln (ξ_{1} + ξ_{2})}, & if 0 \leq x_{1} \leq ln (ξ_{1} + ξ_{2}), \\ 0 \leq x_{2} \leq 3, \\ 0 & if otherwise, \end{matrix} \end{matrix}$

Then, $F_{r} (ξ_{1}, ξ_{2}) = {(x_{1}, x_{2}) : 0 \leq x_{1} \leq ln (ξ_{1} + ξ_{2}) \cdot \sqrt{1 - r^{2}}, 0 \leq x_{2} \leq 3}$ , r ∈ [0, 1], and T_F (ξ₁, $ξ_{2}) = (\frac{1}{3} ln (ξ_{1} + ξ_{2}), \frac{3}{2})$ . Writing t = (ξ₁, ξ₂) ∈ (1, ∞) × (1, ∞), it follows that $t_{1} = (ξ_{1}^{(1)}, ξ_{2}^{(1)})$ , $t_{2} = (ξ_{1}^{(2)}, ξ_{2}^{(2)})$ , apparently, T_F (t₁) and T_F (t₂) are comparable. Let $η (t_{1}, t_{2}) = {\begin{matrix} \frac{1}{2} (t_{1} - t_{2}), & if 1 < t_{2} ≦ t_{1}, \\ 0, & if 1 < t_{1} < t_{2}, \end{matrix}$ . There is no loss of generality, suppose T_F (t₂) ≦ T_F (t₁), it implies that t₂ ≦ t₁, thus, let t₁ = (2, 3), t₂ = (2, 2), then, $η (t_{1}, t_{2}) = (0, \frac{1}{2})$ , $T_{F} (t_{2} + λ η (t_{1}, t_{2})) = (\frac{1}{3} ln (4 + \frac{1}{2} λ), \frac{3}{2})$ , $T_{F} (t_{1}) = (\frac{1}{3} ln 5, \frac{3}{2})$ . Obviously, T_F (t₂ + λη (t₁, t₂)) ≦ T_F (t₁), for all λ ∈ [0, 1], therefore, from Definition 4.4, F is a prequasiinvex fuzzy number-valued function.

Definition 4.6. A differentiable fuzzy number-valued function F : K → Eⁿ is called strictly prequasiinvex on K with respect to a function η : K × K → Rⁿ, if for any t₁, t₂ ∈ K with T_F (t₁) ≠ T_F (t₂), λ ∈ (0, 1), $T_{F} (t_{2} + λ η (t_{1}, t_{2})) < max {T_{F} (t_{1}), T_{F} (t_{2})} .$

Where T_F (t₁) and T_F (t₂) are comparable.

Example 5. Let the 2-dimensional fuzzy number-valued function F : (0, 1] → E² be defined by $F_{r} (t) = [(90 + 10 r) \sqrt{t}, (110 - 10 r) \sqrt{t}] \times [(0.9 + 0.05 r) \sqrt{t}, (1 - 0.05 r) \sqrt{t}]$ , r ∈ [0, 1], t ∈ (0, 1].

According Definition 2.5, $T_{F} (t) = (100 \sqrt{t}, 0.95 \cdot \sqrt{t})$ , let $η (t_{1}, t_{2}) = \frac{1}{2} (t_{1} - t_{2})$ , t₁, t₂ ∈ (0, 1]. Without loss of generality, if T_F (t₁) < T_F (t₂), let $t_{1} = \frac{1}{2}$ , t₂ = 1, then, $η (t_{1}, t_{2}) = - \frac{1}{4}$ , $T_{F} (t_{2} + λ η (t_{1}, t_{2})) = (100 \sqrt{1 - \frac{1}{4} λ}, 0.95 \sqrt{1 - \frac{1}{4} λ})$ , and T_F (t₂) = (100, 0.95). Obviously, T_F (t₂ + λη (t₁, t₂)) < T_F (t₂), for all λ ∈ (0, 1), therefore, from Definition 4.5, F is a strictly prequasiinvex fuzzy number-valued function.

Definition 4.7. A differentiable fuzzy number-valued function F : K → Eⁿ is called stongly prequasiinvex on K with respect to a function η : K × K → Rⁿ, if for any t₁, t₂ ∈ K with t₁ ≠ t₂, λ ∈ (0, 1), $T_{F} (t_{2} + λ η (t_{1}, t_{2})) < max {T_{F} (t_{1}), T_{F} (t_{2})} .$

Where T_F (t₁) and T_F (t₂) are comparable.

Example 6. Consider the Example 5, we can see, for any t₁ ≠ t₂, T_F (t₂+ λη (t₁, t₂)) < max { T_F (t₁) , T_F (t₂) }, therefore, F is also a stongly prequasiinvex fuzzy number-valued function respect to the same η.

Definition 4.8. Let F : K → Eⁿ be a fuzzy number-valued function defined on an invex set K (≠ ∅), with respect to a function η : K × K → Rⁿ, if for any t₁, t₂ ∈ K, there exist δ > 0, such that T_F (t₂ + hη (t₁, t₂)) - T_F (t₂) exists for any real number h ∈ (0, δ), and $u_{η}^{(i)} \in R^{n}$ , i = 1, 2, ⋯ , n, such that $▽ T_{F_{η}} (t_{2}) η (t_{1}, t_{2}) = lim_{h \to 0^{+}} \frac{T_{F} (t_{2} + h η (t_{1}, t_{2})) - T_{F} (t_{2})}{h},$ then T_F is called η-extended directionally differentiable at t₂. The operator $u_{η}^{(i)} \in R^{n}$ , i = 1, 2, ⋯ , n assigns to η_i (t₁, t₂), i = 1, 2, ⋯ , n. ▽T_{F
_η} (t₂) η (t₁, t₂) is called the η-extended directional derivative at t₂ in the direction η (t₁, t₂) (denoted $▽ T_{F_{η}} (t_{2}) = (u_{η}^{(1)}, u_{η}^{(2)}, \dots, u_{η}^{(n)})$ ).

Definition 4.9. Let F : K → Eⁿ, T_F : K → Rⁿ. A fuzzy function is said to be nondecreasing(resp. nonincreasing) if $T_{F} (t_{1}) ≦ T_{F} (t_{2}) (resp . T_{F} (t_{1}) ≧ T_{F} (t_{2})),$ for any t₁, t₂ ∈ K, and t₁ ≤ t₂.

Thereom 4.1. Let K ⊆ Rⁿ be an invex set with respect to η and let F : K → Eⁿ be η-extended directionally differentiable fuzzy number-valued function satisfies Condition C. Then, the fuzzy mapping F is preinvex on K if and only if F is invex.

Proof. By the invexity and η-extended directionally differentiable of F. We have $T_{F} (t_{2} + λ η (t_{1}, t_{2})) - T_{F} (t_{2}) ≦ λ [T_{F} (t_{1}) - T_{F} (t_{2})] .$ and $\frac{T_{F} (t_{2} + λ η (t_{1}, t_{2})) - T_{F} (t_{2})}{λ} ≦ T_{F} (t_{1}) - T_{F} (t_{2}) .$

Let λ → 0⁺, we get $▽ T_{F_{η}} (t_{2}) η (t_{1}, t_{2}) ≦ T_{F} (t_{1}) - T_{F} (t_{2}) .$

Thus, F is invex fuzzy number-valued function.

Conversely, because F is invex fuzzy number-valued function, so, for any t₁, t₂, t₃ ∈ K, we have $T_{F} (t_{1}) - T_{F} (t_{3}) ≧ ▽ T_{F_{η}} (t_{3}) η (t_{1}, t_{3}),$ (7) $T_{F} (t_{2}) - T_{F} (t_{3}) ≧ ▽ T_{F_{η}} (t_{3}) η (t_{2}, t_{3}) .$ (8)

Let (7) multiply by λ, and (8) multiply by (1 - λ), we get $\begin{matrix} λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) - T_{F} (t_{3}) \\ ≧ λ ▽ T_{F} (t_{3}) η (t_{1}, t_{3}) + (1 - λ) ▽ T_{F} (t_{3}) η (t_{2}, t_{3}) . \end{matrix}$

Let t₃ = t₂ + λη (t₁, t₂), and by Condition C, we have $η (t_{1}, t_{3}) = η (t_{1}, t_{2} + λ η (t_{1}, t_{2})) = (1 - λ) η (t_{1}, t_{2}),$ $η (t_{2}, t_{3}) = η (t_{2}, t_{2} + λ η (t_{1}, t_{2})) = - λ η (t_{1}, t_{2}) .$

Thus, λT_F (t₁) + (1 - λ) T_F (t₂) - T_F (t₃) ≧0 . That is, $λ T_{F} (t_{1}) + (1 - λ) T_{F} (t_{2}) ≧ T_{F} (t_{2} + λ η (t_{1}, t_{2})) .$

Therefore, F is a preinvex fuzzy number-valued function.

Thereom 4.2. Let K ⊆ Rⁿ be an invex set with respect to η and let F : K → Eⁿ be η-extended directionally differentiable fuzzy number-valued function. Then, the fuzzy number-valued function F is prequasiinvex on K if and only if F is pseudoinvex.

Proof. Suppose F is not pseudoinvex, that is for any t₁, t₂ ∈ K and η, we have $T_{F} (t_{1}) ≦ T_{F} (t_{2}) \Rightarrow ▽ T_{F_{η}} (t_{2}) η (t_{1}, t_{2}) > 0 .$ (9)

By F is prequasiinvex and η-extended differentiable fuzzy number-valued function, and t₁, t₂ ∈ K, such that T_F (t₁) ≦ T_F (t₂), we have $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≦ max {T_{F} (t_{1}), T_{F} (t_{2})} = T_{F} (t_{2}),$ and $\begin{matrix} ▽ T_{F_{η}} (t_{2}) η (t_{1}, t_{2}) \\ = lim_{h \to 0^{+}} \frac{T_{F} (t_{2} + λ η (t_{1}, t_{2})) - T_{F} (t_{2})}{λ} ≦ 0, \end{matrix}$

So, we get ▽T_{F
_η} (t₂) η (t₁, t₂) ≦0 . Which contradicts (9), therefore, F is pseudoinvex fuzzy number-valued function.

Conversely, suppose F is not prequasiinvex fuzzy number-valued function, then for any t₁, t₂ ∈ K and η, such that T_F (t₁) ≦ T_F (t₂), we have $T_{F} (t_{2} + λ η (t_{1}, t_{2})) > max {T_{F} (t_{1}), T_{F} (t_{2})} = T_{F} (t_{2}) .$ (10) Because F is pseudoinvex, that is for any t₁, t₂ ∈ K and η, we have $T_{F} (t_{1}) ≦ T_{F} (t_{2}) \Rightarrow ▽ T_{F_{η}} (t_{2}) η (t_{1}, t_{2}) ≦ 0 .$

From F is η-extended differentiable fuzzy number-valued function and K is an invex set, thus, for t₁, t₂ ∈ K, t₂ + λη (t₁, t₂) ∈ K, we have $\begin{matrix} ▽ T_{F_{η}} (t_{2}) η (t_{1}, t_{2}) \\ = lim_{h \to 0^{+}} \frac{T_{F} (t_{2} + λ η (t_{1}, t_{2})) - T_{F} (t_{2})}{λ} ≦ 0, \end{matrix}$

It implies, T_F (t₂ + λη (t₁, t₂)) ≦ T_F (t₂) , Which is a contradiction with (10). Hence, F is prequasiinvex fuzzy number-valued function.

Thereom 4.3. Let K ⊆ Rⁿ be an invex set with respect to η and let F : K → Eⁿ be η-extended directionally differentiable, nondecreasing fuzzy number-valued function and satisfied Condition C. If the fuzzy number-valued function F is pseudoinvex on K, then, F is strictly prequasiinvex.

Proof. Suppose F is not strictly prequasiinvex, then, there exists t₁, t₂ ∈ K, such that T_F (t₁) ≠ T_F (t₂), and $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≧ max {T_{F} (t_{1}), T_{F} (t_{2})} .$

Without loss of generality, let T_F (t₁) < T_F (t₂), and t₃ = t₂ + λη (t₁, t₂),we have $T_{F} (t_{1}) < T_{F} (t_{2}) ≦ T_{F} (t_{3}),$ (11)

By F is pseudoinvex, then $▽ T_{F_{η}} (t_{3}) η (t_{1}, t_{3}) < 0 .$ (12)

From the Condition C, we have η (t₁, t₃) = (1 - λ) η (t₁, t₂) and η (t₂, t₃) = - λη (t₁, t₂),

Thus, we can get $η (t_{1}, t_{3}) = - \frac{1 - λ}{λ} η (t_{2}, t_{3}),$ (13)

From (12) and (13), we get $▽ T_{F_{η}} (t_{3}) η (t_{2}, t_{3}) > 0 .$ (14) Once again by condition C, we have $\begin{matrix} t_{3} + \bar{λ} η (t_{2}, t_{3}) \\ = t_{2} + λ η (t_{1}, t_{2}) + \bar{λ} η (t_{2}, t_{2} + λ η (t_{1}, t_{2})) \\ = t_{2} + (1 - \bar{λ}) λ η (t_{1}, t_{2}) \\ \leq t_{2} + λ η (t_{1}, t_{2}) = t_{3} \end{matrix}$

Since fuzzy function F is nondecreasing and η-extended directionally differentiable, we get $T_{F} (t_{3} + \bar{λ} η (t_{2}, t_{3})) ≦ T_{F} (t_{3}),$ and

$\begin{matrix} ▽ T_{F_{η}} (t_{3}) η (t_{2}, t_{3}) \\ = lim_{\bar{λ} \to 0^{+}} \frac{T_{F} (t_{3} + \bar{λ} η (t_{2}, t_{3})) - T_{F} (t_{3})}{\bar{λ}}, \end{matrix}$ therefore, $▽ T_{F_{η}} (t_{3}) η (t_{2}, t_{3}) ≦ 0 .$ (15)

Obviously, (15) is a contradiction to (14). Thus, F is strictly prequasiinvex fuzzy number-valued function.

Thereom 4.4. Let K ⊆ Rⁿ be an invex set with respect to η and let F : K → Eⁿ be η-extended directionally differentiable fuzzy number-valued function and satisfied Condition C. If the fuzzy number-valued function F is strictly pseudoinvex on K, then, F is strongly prequasiinvex.

Proof. By contradiction, suppose that F is not a strongly prequasiinvex, then there exist t₁, t₂ ∈ K, t₁ ≠ t₂, λ ∈ (0, 1), such that $T_{F} (t_{2} + λ η (t_{1}, t_{2})) ≧ max {T_{F} (t_{1}), T_{F} (t_{2})} .$

Let t₃ = t₂ + λη (t₁, t₂), since T_F (t₁) ≦ T_F (t₃), T_F (t₂) ≦ T_F (t₃), by F is strictly pseudoinvex fuzzy number-valued function, we have $▽ T_{F_{η}} (t_{3}) η (t_{1}, t_{3}) < 0,$ (16)

From Condition C, $η (t_{1}, t_{3}) = η (t_{1}, t_{2} + λ η (t_{1}, t_{2})) = (1 - λ) η (t_{1}, t_{2}),$ (17)

From (16) and (17), we can get $▽ T_{F_{η}} (t_{3}) η (t_{1}, t_{2}) < 0 .$ (18)

Similarly, since T_F (t₂) ≦ T_F (t₃), we have $▽ T_{F_{η}} (t_{3}) η (t_{1}, t_{2}) > 0 .$ (19)

The inequalities (18) and (19) are not compatible. Therefore, F is strongly prequasiinvex.

5 Main results of optimization under generalized convexity of fuzzy number-valued functions

The generalized convex function is an extremely important condition in the optimization theory. Many scholars have studied the fuzzy programming problem and put forward many meaningful programming models and solving methods. For the application of fuzzy numbers, we can refer to the literature [30 –32]. In the following, we mainly consider the optimization theory of the unconstrained n-dimensional fuzzy function under the condition of the generalized convex functions: $(FP) min_{t \in K} F (t),$ where K is a subset of Rⁿ, F : K (⊆ Rⁿ) → Eⁿ is a n-dimensional fuzzy number-valued function. According to Definition 2.5, T_F : K → Rⁿ is a vector-valued function, and ▽T_F (t) is a function with a m × n matrix as its Jacobian. Since getting the efficient points is quite costly, therefore, the usual way to solve vector programming problems or multi-objective programming problems is to combine weakly effective points with vector critical points. In the following, we give these definitions of (weakly) efficient point and vector critical point for the problem (FP).

Definition 5.1. Let F : K (⊆ Rⁿ) → Eⁿ be a fuzzy number-valued function, and T_F : K → Rⁿ be a vector-valued function.

(1) A point $\bar{t} \in K$ is said to be efficient point, if there exist no t ∈ K such that $T_{F} (t) \leq T_{F} (\bar{t})$ .

(2) A point $\bar{t} \in K$ is said to be weakly efficient point, if there exist no t ∈ K such that $T_{F} (t) < T_{F} (\bar{t})$ .

Definition 5.2. A feasible solution $\bar{t} \in K$ is said to be a vector critical point for the problem (FP), if there exists a vector u ∈ Rⁿ, u ≥ 0, such that $u^{T} ▽ T_{F} (\bar{t}) = 0$ .

We should noticed that scalar stationary points are those whose vector gradients are zero. But for vector problems, the vector critical points are those such that there exists a non-negative linear combination of the gradient vectors of each component of objective function, valued at that point, equal to zero.

Theorem 5.1. Let K be an invex set of Rⁿ, $\bar{t} \in K$ , F : K (⊆ Rⁿ) → Eⁿ be a differentiable and (strictly) invex fuzzy number-valued function with respect to η. If $\bar{t}$ is a vector critical point of (FP), then $\bar{t}$ is a weakly efficient point of (FP).

Proof. Suppose $\bar{t}$ is not a weakly efficient point of (FP), then there exists a t ∈ K, such that $T_{F} (t) < T_{F} (\bar{t})$ . Since F is a differentiable and invex fuzzy number-valued function, that is, for all $t, \bar{t} \in K$ , we have $▽ T_{F} (\bar{t}) η (t, \bar{t}) ≦ T_{F} (t) - T_{F} (\bar{t}) < 0,$

By Gordan’s Theorem, the system $▽ T_{F} (\bar{t}) η (t, \bar{t}) < 0$ has a solution, that is, the system $u^{T} T_{F} (\bar{t}) = 0,$ has no solution at u ∈ Rⁿ. It implies that $\bar{t}$ is not a vector critical point of (FP), this contradicts the fact $\bar{t}$ is a vector critical point of (FP).

Theorem 5.2. Let K be an invex set of Rⁿ, $\bar{t} \in K$ , F : K (⊆ Rⁿ) → Eⁿ be a differentiable and pseudoinvex fuzzy number-valued function with respect to η. If $\bar{t}$ is a vector critical point of (FP), then $\bar{t}$ is a weakly efficient point of (FP).

Proof. Suppose $\bar{t}$ is not a weakly efficient point of (FP), then there exists a t ∈ K, such that $T_{F} (t) < T_{F} (\bar{t})$ . Since F is a differentiable and pseudoinvex fuzzy number-valued function, that is, for all $t, \bar{t} \in K$ , we have $T_{F} (t) < T_{F} (\bar{t}) \Rightarrow ▽ T_{F} (\bar{t}) η (t, \bar{t}) < 0,$

By Gordan’s Theorem, the system $▽ T_{F} (\bar{t}) η (t, \bar{t}) < 0$ has a solution, that is, the system $u^{T} T_{F} (\bar{t}) = 0$ , has no solution at u ∈ Rⁿ. It implies that $\bar{t}$ is not a vector critical point of (FP), this contradicts the fact $\bar{t}$ is a vector critical point of (FP).

Theorem 5.3. Let K be an invex set of Rⁿ, $\bar{t} \in K$ , F : K (⊆ Rⁿ) → Eⁿ be a η-extended directionally differentiable and strictly prequasiinvex fuzzy number-valued function with respect to η. $\bar{t}$ is a vector critical point of (FP) if and only if $\bar{t}$ is a weakly efficient point of (FP).

Proof. Suppose $\bar{t}$ is not a weakly efficient point of (FP), then, ∃t ∈ K, s.t. $T_{F} (t) < T_{F} (\bar{t})$ . Since F is a strictly prequasiinvex fuzzy number-valued function, we have $T_{F} (\bar{t} + λ η (t, \bar{t})) < T_{F} (\bar{t}),$

By F is η-extended directionally differentiable, thus, from the above inequality, we get $\frac{T_{F} (\bar{t} + λ η (t, \bar{t})) - T_{F} (\bar{t})}{λ} < 0,$ let λ → 0⁺, the system $▽ T_{F_{η}} (\bar{t}) η (t, \bar{t}) < 0,$ has a solution at η ∈ Rⁿ. From Gordan’s Theorem, the system $u^{T} T_{F} (\bar{t}) = 0$ , has no solution at u ∈ Rⁿ. this contradicts the fact $\bar{t}$ is a vector critical point of (FP).

Conversely, suppose $\bar{t}$ is not a vector critical point of (FP), then, the system $u^{T} T_{F} (\bar{t}) = 0$ , has no solution at u ∈ Rⁿ. By Gordan’s Theorem, there exists a vector ν ∈ Rⁿ such that, the system $▽ T_{F} (\bar{t}) ν < 0$ , has a solution. That is, there exists a vector function $η (t, \bar{t})$ , such that $▽ T_{F_{η}} (\bar{t}) η (t, \bar{t}) < 0 .$ Since F is η-extended directionally differentiable, thus, we have $lim_{λ \to 0^{+}} \frac{T_{F} (\bar{t} + λ η (t, \bar{t})) - T_{F} (\bar{t})}{λ} = ▽ T_{F} (\bar{t}) η (t, \bar{t}) < 0,$ that is $T_{F} (\bar{t} + λ η (t, \bar{t})) < T_{F} (\bar{t}) .$ Let $\hat{t} = \bar{t} + λ η (t, \bar{t}) \in K$ , then, $\exists \hat{t} \in K$ , s.t. $T_{F} (\hat{t}) < T_{F} (\bar{t})$ . This contradicts the fact that $\bar{t}$ is a weakly efficient point of (FP).

Theorem 5.4. Let K be an invex set of Rⁿ, $\bar{t} \in K$ , F : K (⊆ Rⁿ) → Eⁿ be a η-extended directionally differentiable and preinvex fuzzy number-valued function with respect to η. $\bar{t}$ is a vector critical point of (FP) if and only if $\bar{t}$ is an weakly efficient point of (FP).

Proof. Suppose $\bar{t}$ is not a weakly efficient point of (FP), then, ∃t ∈ K, s.t. $T_{F} (t) < T_{F} (\bar{t})$ . Since F is a preinvex fuzzy number-valued function and η-extended directionally differentiable, thus, we have $\frac{T_{F} (\bar{t} + λ η (t, \bar{t})) - T_{F} (\bar{t})}{λ} ≦ T_{F} (t) - T_{F} (\bar{t}) < 0,$ and, let λ → 0⁺, the system $▽ T_{F_{η}} (\bar{t}) η (t, \bar{t}) < 0$ , has a solution at η ∈ Rⁿ. By Gordan’s Theorem, there doesn’t exists a vector u ∈ Rⁿ such that, the system $u^{T} T_{F} (\bar{t}) = 0$ , has solution. This is contradicts the fact $\bar{t}$ is a vector critical point of (FP).

Conversely, suppose $\bar{t}$ is not a vector critical point of (FP), then, the system $u^{T} T_{F} (\bar{t}) = 0,$ has no solution at u ∈ Rⁿ. By Gordan’s Theorem, there exists a vector η ∈ Rⁿ such that, the system $▽ T_{F_{η}} (\bar{t}) η (t, \bar{t}) < 0,$ has a solution. Since F is η-extended directionally differentiable, thus, we have $lim_{λ \to 0^{+}} \frac{T_{F} (\bar{t} + λ η (t, \bar{t})) - T_{F} (\bar{t})}{λ} = ▽ T_{F} (\bar{t}) η (t, \bar{t}) < 0,$ that is $T_{F} (\bar{t} + λ η (t, \bar{t})) < T_{F} (\bar{t}) .$ Let $\hat{t} = \bar{t} + λ η (t, \bar{t}) \in K$ , then, $\exists \hat{t} \in K$ , s.t. $T_{F} (\hat{t}) < T_{F} (\bar{t})$ . This contradicts the fact that $\bar{t}$ is a weakly efficient point of (FP).

Example 7. Consider the 2-dimensional fuzzy optimization problem $\begin{matrix} Min F (t) (r) \end{matrix}$

Where, F (t) (r) = [-5t + 10r, 20 - 5t - 10r] × [t + 2r, 4 + t - 2r] , r ∈ [0, 1] , t ∈ [0, 3].

According to Definition 2.5, we have T_F (t) = (10 - 5t, 2 + t). Let η (t₁, t₂) = t₁ - t₂ for all t₁, t₂ ∈ [0, 3], then, from Definition 3.2, F is a preinvex fuzzy number-valued function. We can find μ = (1, 1) and $\bar{t} = 3$ is a critical point of F, and it’s also a weakly efficient point of F.

From this example, the optimization problem of multidimensional fuzzy numerical function based on a new ordering is essentially a classic multi-objective programming, which greatly reduces the computational complexity.

6 Conclusion

Convexity and generalized convexity play an important role in mathematical economics, engineering, management science, and optimization theory. Research on generalized convexity is one of the most important direction in mathematical programming. With the appearance of fuzzy optimization problems, fuzzy convex analysis has also gradually attracted more and more attention. As a fuzzy generalized convexity, fuzzy invexity has been studied extensively and deeply.

In this study, firstly, we mainly discussed the relationship between n-dimensional preinvex and semicontinuous fuzzy number-valued functions based on a new ordering, which proposed in [9]. Then, discussed some properties about preinvex, (strictly) invex, (strictly) pseudoinvex and (strictly, strongly) prequasiinvex fuzzy number-valued functions based on this new ordering. From the previous discussion, we can easily see that the minimization problem of n-dimensional fuzzy numerical functions can be transformed into a classic multi-objective minimization problem under this new ordering. Therefore, at the end of the article, we discussed the relationship between weakly efficient points and critical points under some conditions and using this new ordering, and give an example to illustrate. These results are useful to fuzzy optimization.

In addition, the fuzzy number-valued functions based on a new ordering are more likely to find the middle point [27] on each fuzzy interval. Research on the middle points and quotient spaces can be found in the literature [28, 29]. Therefore, in the next work, we can continue to discuss the relationship between the results n-dimensional fuzzy number-valued functions based on a new ordering and the results in the quotient space of the fuzzy number. Of course, there are still many issues to study, such as properties of weakly differentials of generalized convex fuzzy number-valued functions, and constrained fuzzy optimization. These studies may bring to us more novel results.

Footnotes

Acknowledgment

The research was supported by the National Natural Science Foundation of China (Grant No. 71672013).

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