Fuzzy optimizations of convex fuzzy mappings in the quotient space of fuzzy numbers

Abstract

In this paper, on the basis of an order in the quotient space of fuzzy numbers, the concepts of gradient and convexity of fuzzy mappings are presented. Then, the fuzzy optimizations of differentiable fuzzy mappings and convex fuzzy mappings are discussed and some examples are provided to illustrate the main results.

Keywords

Fuzzy optimizations fuzzy numbers convex fuzzy mappings quotient space

1 Introduction

The fuzzy set theory was introduced initially in 1965 by Zadeh [30] with a view to reconcile mathematical modeling and human knowledge in the engineering science. Since most of practical optimization problems are fuzzy and approximate, the concepts of convexity and differentiability of fuzzy mappings have been studied extensively to find the optimal solutions.

In 1992, Nanda and Kar [12] firstly introduced the concept of convexity for fuzzy mappings and proved that a fuzzy mapping is convex if and only if its epigraph is a convex set. Yan and Xu [29] described characteristics of the convex fuzzy mappings and quasi-convex fuzzy mappings by considering the concept of ordering due to Goetschel and Voxman [3]. In addition they discussed the properties of convex fuzzy optimizations. In [20, 21], Syau introduced the concepts of pseudo-convexity, invexity and pseudo-invexity for fuzzy mappings of one variable and investigated the relationships among them by using notion of differentiability and the results proposed by Goetschel and Voxman [3]. In [22], Syau defined a differentiable fuzzy mappings of several variables in ways that parallel the definition, proposed by Goetschel and Voxman [3], for a fuzzy mapping of one variable. In [23], Syau introduced and investigated a new kind of generalized convex fuzzy mappings known as a B-vex fuzzy mappings and proved that a strictly B-vex, or B-preinvex fuzzy mappings has at most one global minimum point. Wang and Wu [25] proposed the concepts of directional derivative, differential and subdifferential of fuzzy mappings from $ℝ^{n}$ into the set of fuzzy numbers and discussed the characterizations of directional derivative and differential. Panigrahi et al. [16] extended and generalized these concepts to fuzzy mappings of several variables using the approach due to Buckley and Feuring [2] for fuzzy differential and derived a K-K-T condition for the constrained fuzzy minimization problems. Wu and Xu [27] introduced the concepts of fuzzy pseudoconvex, fuzzy invex, fuzzy pseudoinvex and fuzzy preinvex mapping from $ℝ^{n}$ to the set of fuzzy numbers and relations were discussed between the fuzzy variational-like inequality and the fuzzy optimizations. Wu [26] discussed the optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued objective functions. Bao and Wu [1] introduced a new class of fuzzy mappings called semistrictly convex fuzzy mappings and proved that a local minimum of a semistrictly convex fuzzy mappings is also a global minimum.

In [17], Qiu et al. intuitively showed a method of finding the inverse operation in the quotient space of fuzzy numbers based on the Mareš equivalence relation [10, 11], which have the desired group properties for the addition operation [6 , 28]. Jamison [6 –8] proposed the midpoint functions of fuzzy numbers and pointed out that a fuzzy number $\tilde{x}$ is equivalent to a fuzzy number $\tilde{y}$ if and only if they have the same midpoint function. In this paper, on the basis of an order in the quotient space of fuzzy numbers, we will study the basic theory of fuzzy optimizations. In Section 2, we recall some related concepts and lemmas. In Section 3, we will deal with the convexity and gradient of a fuzzy mapping in the quotient space of fuzzy numbers. In Section 4, we will discuss the properties of fuzzy optimizations of differentiable fuzzy mappings and convex fuzzy mappings.

2 Preliminaries

This section is to recall some pertinent concepts and key lemmas from the function of bounded variation, fuzzy numbers and fuzzy equivalence classes which will be used later.

Definition 2.1. [9] Let $f : [a, b] \to ℝ$ be a function. f is said to be of bounded variation if there exists a C > 0 such that $\sum_{i = 1}^{n} | f (x_{i - 1}) - f (x_{i}) | \leq C$ for every partition a = x₀ < x₁ < x₂ < ⋯ < x_n = b on [a, b]. The set of all functions of bounded variation on [a, b] is denoted by BV [a, b].

Definition 2.2. [9] Let $f : [a, b] \to ℝ$ be a function of bounded variation. The total variation of f on [a, b], denoted by $V_{a}^{b} (f)$ , is defined by $V_{a}^{b} (f) = sup_{p} \sum_{i = 1}^{n} | f (x_{i - 1}) - f (x_{i}) |,$ where p represents all partitions of [a, b].

Lemma 2.3. [9] Let f, g ∈ BV [a, b], then we have (1) cf + dg ∈ BV [a, b] and $V_{a}^{b} (cf + dg) \leq | c | V_{a}^{b} (f) + | d | V_{a}^{b} (g)$ for any contents $c, d \in ℝ .$ (2) fg ∈ BV [a, b] and $V_{a}^{b} (f \cdot g) \leq V_{a}^{b} (f) sup_{x \in [a, b]} | g (x) | + V_{a}^{b} (g) sup_{x \in [a, b]} | f (x) | .$

A fuzzy set $\tilde{x}$ in $ℝ$ is characterized by a membership function $μ_{\tilde{x}} : ℝ \to [0, 1]$ . The α-level set of $\tilde{x}$ is denoted by $[\tilde{x}]_{α} = {x \in ℝ : μ_{\tilde{x}} (x) \geq α}$ for each α ∈ (0, 1]. The 0-level set $[\tilde{x}]_{0}$ is defined as the closure of the set ${x \in ℝ : μ_{\tilde{x}} (x) > 0}$ , i.e., $[\tilde{x}]_{0} = cl ({x \in ℝ : μ_{\tilde{x}} (x) > 0})$ . A fuzzy set $\tilde{x}$ is said to be a fuzzy number if it satisfies the following conditions:

$\tilde{x}$ is normal, i.e., there exists an $x_{0} \in ℝ$ such that $μ_{\tilde{x}} (x_{0}) = 1$ ;

$\tilde{x}$ is convex, i.e., $μ_{\tilde{x}} (λ x_{1} + (1 - λ) x_{2}) \geq min {μ_{\tilde{x}} (x_{1}), μ_{\tilde{x}} (x_{2})} for all x_{1}, x_{2} \in ℝ and$ λ ∈ (0, 1);

$\tilde{x}$ is upper semicontinuous, i.e., the α-level set $[\tilde{x}]_{α}$ is a closed subset of $ℝ$ for all α ∈ [0, 1];

the 0-level set $[\tilde{x}]_{0}$ is a compact subset of $ℝ$ .

Let ℱ be the set of all fuzzy numbers on $ℝ$ . Then for any $\tilde{x} \in ℱ$ , it is well known that the α-level set $[\tilde{x}]_{α} = [{\tilde{x}}_{L} (α), {\tilde{x}}_{R} (α)]$ is a non-empty bounded closed interval in $ℝ$ for all α ∈ [0, 1], where ${\tilde{x}}_{L} (α)$ denotes the left-hand end point of $[\tilde{x}]_{α}$ and the ${\tilde{x}}_{R} (α)$ denotes the right one.

For any $\tilde{x}, \tilde{y} \in ℱ$ and $λ \in ℝ$ , owing to Zadeh’s extension principle [31], the addition and scalar multiplication can be respectively defined for any $x \in ℝ$ by $μ_{\tilde{x} + \tilde{y}} (x) = sup_{x_{1}, x_{2} : x_{1} + x_{2} = x} min {μ_{\tilde{x}} (x_{1}), μ_{\tilde{y}} (x_{2})}$ and $μ_{λ \times \tilde{x}} (x) = μ_{λ \tilde{x}} (x) = {\begin{matrix} μ_{\tilde{x}} (\frac{x}{λ}), λ \neq 0, \\ 0, λ = 0 . \end{matrix}$

We say that a fuzzy number $\tilde{s} \in ℱ$ is symmetric, if $\tilde{s} = - \tilde{s}$ , i.e., $μ_{\tilde{s}} (x) = μ_{- \tilde{s}} (x) = μ_{\tilde{s}} (- x)$ for all $x \in ℝ$ . Denote the set of all symmetric fuzzy number by §.

Definition 2.4. [4] Let $\tilde{x}, \tilde{y} \in ℱ$ , we say that $\tilde{x}$ is equivalent to $\tilde{y}$ , if there exist two symmetric fuzzy numbers ${\tilde{s}}_{1}, {\tilde{s}}_{2} \in §$ such that $\tilde{x} + \tilde{s_{1}} = \tilde{y} + \tilde{s_{2}}$ and then we denote this by $\tilde{x} \sim \tilde{y} .$

It is easy to verify that the equivalence relation defined above is reflexive, symmetric and transitive [10]. Let $〈 \tilde{x} 〉$ denote the fuzzy equivalence class containing the element $\tilde{x}$ and ℱ/§ the set of all fuzzy equivalence classes.

Definition 2.5. [17] Let $\tilde{x} \in ℱ$ , we define the midpoint function ${\tilde{x}}_{M} : [0, 1] \to ℝ$ by assigning the midpoint of each α-level set to ${\tilde{x}}_{M} (α)$ for all α ∈ [0, 1], i.e., ${\tilde{x}}_{M} (α) = \frac{{\tilde{x}}_{L} (α) + {\tilde{x}}_{R} (α)}{2} .$ Then the function ${\tilde{x}}_{M} : [0, 1] \to ℝ$ will be called the midpoint function of the fuzzy number $\tilde{x}$ .

Lemma 2.6. [17] For any $\tilde{x} \in ℱ$ , the midpoint function ${\tilde{x}}_{M}$ is continuous from the right at 0 and continuous from the left on [0, 1]. Furthermore, it is a function of bounded variation on [0, 1].

Definition 2.7. [11] Let $\tilde{x} \in ℱ$ and let $\hat{x}$ be a fuzzy number such that $\tilde{x} = \hat{x} + \tilde{s}$ for some $\tilde{s} \in §$ , if $\hat{x} = \tilde{y} + {\tilde{s}}_{1}$ for some $\tilde{y} \in ℱ$ and ${\tilde{s}}_{1} \in §$ , then ${\tilde{s}}_{1} = \tilde{0}$ . Then the fuzzy number $\hat{x}$ will be called the Mareš core of the fuzzy number $\tilde{x}$ .

Definition 2.8. [19] Let $〈 \tilde{x} 〉 \in ℱ / §$ , we define the midpoint function $M_{〈 \tilde{x} 〉} : [0, 1] \to ℝ$ by $M_{〈 \tilde{x} 〉} (α) = {\hat{x}}_{M} (α)$ for all α ∈ [0, 1], where $\hat{x}$ is the Mareš core of $〈 \tilde{x} 〉$ .

Next, we will discuss some kinds of operations on the quotient space of fuzzy numbers ℱ/§.

Definition 2.9. [19] Let $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in ℱ / §$ , we define the sum of this two fuzzy equivalence classes as a fuzzy equivalence class $〈 \tilde{z} 〉 \in ℱ / §$ , which satisfies the condition $M_{〈 \tilde{x} 〉} (α) + M_{〈 \tilde{y} 〉} (α) = M_{〈 \tilde{z} 〉} (α)$ for all α ∈ [0, 1] and we denote this by $〈 \tilde{x} 〉 + 〈 \tilde{y} 〉 = 〈 \tilde{x} + \tilde{y} 〉 = 〈 \tilde{z} 〉 .$

Remark 2.10. [7, 8] The addition operation defined by Definition 2.9 is a group operation over the set of fuzzy equivalence classes ℱ/§ up to the equivalence relation in Definition 2.4.

Now, we define operation of multiplication and scalar-multiplication on the quotient space of fuzzy numbers.

Definition 2.11. [11] Let $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in ℱ / §$ , we say that $〈 \tilde{z} 〉 \in ℱ / §$ is the product of $〈 \tilde{x} 〉$ and $〈 \tilde{y} 〉$ if their midpoint functions satisfy $M_{〈 \tilde{x} 〉} (α) \cdot M_{〈 \tilde{y} 〉} (α) = M_{〈 \tilde{z} 〉} (α)$ for all α ∈ [0, 1] and denote this by $〈 \tilde{x} 〉 \cdot 〈 \tilde{y} 〉 = 〈 \tilde{z} 〉 .$

Definition 2.12. [18] For any $〈 \tilde{x} 〉 \in ℱ / §$ and $λ \in ℝ$ , we define $λ \cdot 〈 \tilde{x} 〉 = λ 〈 \tilde{x} 〉$ by $λ 〈 \tilde{x} 〉 = 〈 \tilde{x} 〉 λ = 〈 λ \tilde{x} 〉 .$

Definition 2.13. Let $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in ℱ / §$ , we say that $〈 \tilde{x} 〉 ⪯ 〈 \tilde{y} 〉$ if $M_{〈 \tilde{x} 〉} (α) \leq M_{〈 \tilde{y} 〉} (α)$ for all α ∈ [0, 1]. If $〈 \tilde{x} 〉 ⪯ 〈 \tilde{y} 〉$ and $〈 \tilde{y} 〉 ⪯ 〈 \tilde{x} 〉$ , then $〈 \tilde{x} 〉 = 〈 \tilde{y} 〉$ . We say that $〈 \tilde{x} 〉 ≺ 〈 \tilde{y} 〉$ , if $〈 \tilde{x} 〉 ⪯ 〈 \tilde{y} 〉$ and there exists at least one α₀ ∈ [0, 1] such that $M_{〈 \tilde{x} 〉} (α_{0}) < M_{〈 \tilde{y} 〉} (α_{0}) .$

For any $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in ℱ / §$ , if either $〈 \tilde{x} 〉 ⪯ 〈 \tilde{y} 〉$ or $〈 \tilde{y} 〉 ⪯ 〈 \tilde{x} 〉$ , then we say that $〈 \tilde{x} 〉$ and $〈 \tilde{y} 〉$ are comparable, otherwise non-comparable. Note that ⪯ is a partial order relation on ℱ/§. In the field of fuzzy optimizations, ranking of fuzzy numbers is an important and prerequisite procedure. In general, we rank two fuzzy numbers by considering their left-hand functions and right-hand functions [5 , 16], which is called the fuzzy-max order. We will show that two fuzzy numbers are not comparable by using the general method but the corresponding equivalence classes are comparable in the sense of Definition 2.13.

Example 2.14. Define $〈 \tilde{x} 〉, 〈 \tilde{y} 〉$ respectively by $M_{〈 \tilde{x} 〉} (α) = 1 - \frac{α}{2} and M_{〈 \tilde{y} 〉} (α) = 1 - α, α \in [0, 1] .$

Then by Theorem 3.10 and 3.13 in [19], we can get that the level sets of the Mareš core of $〈 \tilde{x} 〉 and 〈 \tilde{y} 〉$ are ${[\hat{x}]}_{α} = [\frac{1}{2}, \frac{3}{2} - α] and {[\hat{y}]}_{α} = [0, 2 - 2 α], α \in [0, 1] .$

It is obvious that $〈 \tilde{x} 〉 ≻ 〈 \tilde{y} 〉$ in the sense of Definition 2.13. However, the fuzzy numbers $\hat{x}$ and $\hat{y}$ are not comparable with respect to the fuzzy-max order [5, 16].

Definition 2.15. [17] Let $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in ℱ / §$ , we define $d_{sup} : ℱ / § \times ℱ / § \to ℝ^{+} \cup {0}$ by $d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉) = sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) - M_{〈 \tilde{y} 〉} (α) | .$

It is easy to see that (ℱ/§ , d_sup) is a metric space [17].

3 Fuzzy mappings

The range of fuzzy mappings we discussed is the set of all fuzzy equivalence classes ℱ/§. Furthermore, since ℱ/§ is not totally ordered, we have to define the class of comparable fuzzy mappings.

Definition 3.1. Let F : T→ ℱ/§ be a fuzzy mapping, where $T = [a, b] \subseteq ℝ$ . F is said to be comparable if for each pair s, t ∈ T with s ≠ t, F (s) and F (t) are comparable. Otherwise, F is said to be a non-comparable fuzzy mapping. Let ℰ denote the set of all comparable fuzzy mappings.

Example 3.2. We give a fuzzy mapping $F : T \subseteq ℝ \to ℱ / §$ , defined by M_F(t) (α) = f (t) g (α) for all α ∈ [0, 1], where $f : T \to ℝ$ is a real-valued mapping and $g : [0, 1] \to ℝ$ is a non-negative real-valued mapping. For each pair s, t ∈ T with s ≠ t, we have M_F(s) (α) - M_F(t) (α) = g (α) (f (s) - f (t)) for all α ∈ [0, 1]. Then it is easy to get that F is comparable.

Definition 3.3. [18] Let F : T→ ℱ/§ be a fuzzy mapping, where $T = [a, b] \subseteq ℝ$ . Then F is said to be continuous at t ∈ T with respect to d_sup if for any h ≠ 0 with t + h ∈ T such that $lim_{h \to 0} d_{sup} (F (t + h), F (t)) = 0 .$ If t = a (orb), then we consider only h → 0⁺ or (h → 0^-).

Definition 3.4. [19] Let F : T→ ℱ/§ be a fuzzy mapping, where $T = [a, b] \subseteq ℝ$ . Then F is said to be differentiable at t ∈ T if there exists an F′ (t)∈ ℱ/§ such that $lim_{h \to 0} d_{sup} (\frac{F (t + h) - F (t)}{h}, F^{'} (t)) = 0 .$

If t = a (orb), then we consider only h → 0⁺ or (h → 0^-).

Lemma 3.5. [19] F : T→ ℱ/§ is differentiable on T if and only if

M_F(t) (α) is differentiable with respect to t ∈ T for all α ∈ [0, 1], i.e., $\frac{\partial}{\partial t} M_{F (t)} (α)$ exists and is of bounded variation with respect to α ∈ [0, 1] for all t ∈ T.

The mappings {M_F(t) (α) } _α∈[0,1] are uniformly differentiable with the derivatives $\frac{\partial}{\partial t} M_{F (t)} (α)$ , i.e., for each t ∈ T and ɛ > 0, there exists a δ > 0 such that $| \frac{M_{F (t + h)} (α) - M_{F (t)} (α)}{h} - \frac{\partial}{\partial t} M_{F (t)} (α) | < ɛ$ for all |h| ∈ (0, δ) and α ∈ [0, 1].

Lemma 3.6. [19] If F : T→ ℱ/§ is differentiable, then it is continuous with respect to d_sup.

Definition 3.7. Let $〈 \tilde{a} 〉 = {(〈 {\tilde{a}}_{1} 〉, 〈 {\tilde{a}}_{2} 〉, \dots, 〈 {\tilde{a}}_{n} 〉)}^{T} \in {(ℱ / §)}^{n}$ and $t = {(t_{1}, t_{2}, \dots, t_{n})}^{T} \in ℝ^{n}$ be an n-dimensional fuzzy equivalence class vector and n-dimensional real vector respectively. Define their product as $〈 \tilde{a} 〉^{T} t = \sum_{i = 1}^{n} 〈 {\tilde{a}}_{i} 〉 t_{i} = 〈 {\tilde{a}}_{1} 〉 t_{1} + 〈 {\tilde{a}}_{2} 〉 t_{2} + \dots + 〈 {\tilde{a}}_{n} 〉 t_{n},$ which is a fuzzy equivalence class.

Notice that in this paper later the vectors are all column vectors, whose components will be denoted by the transpose symbol T as t = (t₁, t₂, ⋯ , t_n)^T. The vector components will be denoted by t_i and the sequence of vectors will be denoted by {t^k}.

Definition 3.8. Let F : Ω→ ℱ/§ be a fuzzy mapping, where Ω is an open subset in $ℝ^{n}$ . We say that F has a partial derivative at t = (t₁, t₂, ⋯ , t_n)^T∈ Ω with respect to the i-th variable t_i if there exists an $\frac{\partial}{\partial t_{i}} F (t) \in ℱ / §$ such that $lim_{h \to 0} d_{sup} (\frac{F (t + h e^{i}) - F (t)}{h}, \frac{\partial}{\partial t_{i}} F (t)) = 0,$ where eⁱ stands for the unit vector that the i-th component is 1 and the others are 0.

Definition 3.9. Let F : Ω→ ℱ/§ be a fuzzy mapping, where Ω is an open subset in $ℝ^{n}$ . We say that F is differentiable at t = (t₁, t₂, ⋯ , t_n)^T∈ Ω if F has continuous partial derivatives $\frac{\partial}{\partial t_{i}} F (t)$ with respect to the i-th variable t_i (i = 1, 2, ⋯ , n) and satisfies $F (t + h) = F (t) + \tilde{\nabla} F {(t)}^{T} h + o (∥ h ∥),$ where $\tilde{\nabla} F (t) \in (ℱ / §)^{n}$ is an n-dimensional fuzzy equivalence class vector defined by $\tilde{\nabla} F (t) = {(\frac{\partial F (t)}{\partial t_{1}}, \frac{\partial F (t)}{\partial t_{2}}, \dots, \frac{\partial F (t)}{\partial t_{n}})}^{T},$ ∥h∥ is the usual Euclid norm of $h \in ℝ^{n}$ and o : [0, + ∞) → ℱ/§ is a fuzzy mapping that satisfies $lim_{t \to 0} d_{sup} (\frac{o (t)}{t}, 〈 \tilde{0} 〉) = 0 .$

Then we call $\tilde{\nabla} F (t)$ , the gradient of the fuzzy mappings F at t.

Theorem 3.10. Let F : Ω→ ℱ/§ be differentiable at t = (t₁, t₂, ⋯ , t_n)^T∈ Ω, where F ∈ ℰ and Ω is an open subset in $ℝ^{n}$ . If there is a vector $d \in ℝ^{n}$ such that $M_{\tilde{\nabla} F (t)} {(α_{0})}^{T} d < 0$ for some α₀ ∈ [0, 1], then there exists a δ > 0 such that F (t + λd) ≺ F (t) for each λ ∈ (0, δ) and we say that d is a descent direction of F at t.

Proof. Since F is differentiable at t, by Definition 3.10 we have $F (t + λ d) = F (t) + λ \tilde{\nabla} F {(t)}^{T} d + o (λ ∥ d ∥)$ for any $λ \neq 0, d \in ℝ^{n}$ , where $d_{sup} (\frac{o (λ ∥ d ∥)}{λ}, 〈 \tilde{0} 〉) \to 0$ as λ → 0. Since λ ≠ 0, we get

$\begin{matrix} \frac{M_{F (t + λ d)} (α) - M_{F (t)} (α)}{λ} \\ = & M_{\tilde{\nabla} F (t)} {(α)}^{T} d + \frac{M_{o (λ ∥ d ∥)} (α)}{λ} \end{matrix}$ for all α ∈ [0, 1]. Since $M_{\tilde{\nabla} F (t)} {(α_{0})}^{T} d < 0$ for some α₀ ∈ [0, 1] and $d_{sup} (\frac{o (λ ∥ d ∥)}{λ}, 〈 \tilde{0} 〉) = sup_{α \in [0, 1]} | \frac{M_{o (λ ∥ d ∥)} (α)}{λ} | \to 0$ as λ → 0, there exists a δ > 0 such that $M_{\tilde{\nabla} F (t)} {(α_{0})}^{T} d + \frac{M_{o (λ ∥ d ∥)} (α_{0})}{λ} < 0$ for α₀ and each λ ∈ (0, δ). Then we have $\frac{M_{F (t + λ d)} (α_{0}) - M_{F (t)} (α_{0})}{λ} < 0 .$ Sinceλ > 0, we get M_F(t+λd) (α₀) - M_F(t) (α₀) < 0, that is M_F(t+λd) (α₀) < M_F(t) (α₀) for α₀ and each λ ∈ (0, δ). Furthermore, since F ∈ ℰ, we have F (t + λd) ≺ F (t).

Definition 3.11. Let $〈 \tilde{A} 〉 = (〈 {\tilde{a}}_{ij} 〉)$ be an n × n fuzzy equivalence class matrix in which the entries $〈 {\tilde{a}}_{ij} 〉 (i, j = 1, 2, \dots, n)$ are all fuzzy equivalence classes. $〈 \tilde{A} 〉$ is said to be symmetric if $〈 {\tilde{a}}_{ij} 〉 = 〈 {\tilde{a}}_{ji} 〉$ for all i, j = 1, 2, ⋯ , n. $〈 \tilde{A} 〉$ is said to be positive semi-definite if the midpoint-value matrices $M_{〈 \tilde{A} 〉} (α) = (M_{〈 {\tilde{a}}_{ij} 〉} (α))$ are positive semi-definite for all α ∈ [0, 1]. $〈 \tilde{A} 〉$ is said to be positive definite if $〈 \tilde{A} 〉$ is positive semi-definite and the midpoint-value matrices $M_{〈 \tilde{A} 〉} (α) = (M_{〈 {\tilde{a}}_{ij} 〉} (α))$ are positive definite for at least one α ∈ [0, 1].

Definition 3.12. Let F : Ω→ ℱ/§ be a fuzzy mapping, where Ω is an open subset in $ℝ^{n}$ . We say that F is twice differentiable at t = (t₁, t₂, ⋯ , t_n)^T∈ Ω if F has continuous twice partial derivatives $\frac{\partial^{2}}{\partial t_{i} \partial t_{j}} F (t)$ with respect to i-th variable t_i (i = 1, 2, ⋯ , n) and j-th variable t_j (j = 1, 2, ⋯ , n) such that $F (t + h) = F (t) + \tilde{\nabla} F {(t)}^{T} h + \frac{1}{2} h^{T} {\tilde{\nabla}}^{2} F (t) h + o ({∥ h ∥}^{2}),$ where ${\tilde{\nabla}}^{2} F (t)$ is an n × n fuzzy equivalence class matrix defined by ${\tilde{\nabla}}^{2} F (t) = [\begin{matrix} \frac{\partial^{2} F (t)}{\partial^{2} t_{1}} & \dots & \frac{\partial^{2} F (t)}{\partial t_{1} \partial t_{n}} \\ ⋮ & ⋮ \\ \frac{\partial^{2} F (t)}{\partial t_{n} \partial t_{1}} & \dots & \frac{\partial^{2} F (t)}{\partial^{2} t_{n}} \end{matrix}],$ and o : [0, + ∞) → ℱ/§ is a fuzzy mapping that satisfies $lim_{t \to 0} d_{sup} (\frac{o (t)}{t}, 〈 \tilde{0} 〉) = 0 .$

Then, we call ${\tilde{\nabla}}^{2} F (t)$ , the Hessian matrix of F at t.

Since the Hessian matrix of a fuzzy mapping is a fuzzy equivalence class matrix, we can denote the midpoint-value functions matrices as $M_{{\tilde{\nabla}}^{2} F (t)} (α) = [\begin{matrix} \frac{\partial^{2} M_{F (t)} (α)}{\partial^{2} t_{1}} & \dots & \frac{\partial^{2} M_{F (t)} (α)}{\partial t_{1} \partial t_{n}} \\ ⋮ & ⋮ \\ \frac{\partial^{2} M_{F (t)} (α)}{\partial t_{n} \partial t_{1}} & \dots & \frac{\partial^{2} M_{F (t)} (α)}{\partial^{2} t_{n}} \end{matrix}]$ for all α ∈ [0, 1].

Definition 3.13. Let F : Ω→ ℱ/§ be a fuzzy mapping, where Ω is a non-empty convex subset in $ℝ^{n}$ and F ∈ ℰ. F is said to be convex on Ω if for any s, t ∈ Ω and λ ∈ (0, 1), we always have F (λs + (1 - λ) t) ⪯ λF (s) + (1 - λ) F (t) . F is said to be concave if -F is convex.

4 Applications to fuzzy optimizations of convex fuzzy mappings with differentiability

In this section, we will discuss the properties of fuzzy optimizations of differentiable fuzzy mappings and convex fuzzy mappings.

Definition 4.1. Let $F : Ω \subseteq ℝ^{n} \to ℱ / §$ be a fuzzy mapping. Consider the problem $\begin{matrix} Minimize & F (t), \\ subjectto & t \in Ω . \end{matrix}$ (1) when $Ω = ℝ^{n}$ , the problem (1) is called unconstrained optimization problem. A point $\bar{t} \in Ω$ is called a feasible solution. A point $\bar{t} \in Ω$ is called a local optimal solution if there exists an ɛ > 0 such that $F (t) \underline{≻} F (\bar{t})$ for all $t \in Ω \cap N_{ɛ} (\bar{t})$ , where $N_{ɛ} (\bar{t})$ is an ɛ-neighborhood around $\bar{t}$ . Similarly, a point $\bar{t} \in Ω$ is called a strict local optimal solution if there exists an ɛ > 0 such that $F (t) ≻ F (\bar{t})$ for all $t (\neq \bar{t}) \in Ω \cap N_{ɛ} (\bar{t})$ . A point $\bar{t} \in Ω$ is called a global optimal solution if $F (t) \underline{≻} F (\bar{t})$ for all t ∈ Ω. A point $\bar{t} \in Ω$ is called a strict global optimal solution if $F (t) ≻ F (\bar{t})$ for all $t (\neq \bar{t}) \in Ω$ .

First, we will show some analytical properties of fuzzy optimizations with differentiable fuzzy-valued objective functions.

Theorem 4.2. Let F : Ω→ ℱ/§ be a differentiable fuzzy mappings at $\bar{t}$ , where F ∈ ℰ and Ω is an open subset in $ℝ^{n}$ . If $\bar{t}$ is a local optimal solution, then $\tilde{\nabla} F (\bar{t}) = 〈 \tilde{0} 〉$ , where $〈 \tilde{0} 〉 = (〈 \tilde{0} 〉, 〈 \tilde{0} 〉, \dots, 〈 \tilde{0} 〉)^{T} \in (ℱ / §)^{n}$ .

Proof. Suppose $\tilde{\nabla} F (\bar{t}) \neq 〈 \tilde{0} 〉$ , then there exists at least one α₀ ∈ [0, 1] such that $M_{\tilde{\nabla} F (\bar{t})} (α_{0}) \neq 0$ . Let $d = - M_{\tilde{\nabla} F (\bar{t})} (α_{0}) \neq 0$ , then we get that $\begin{matrix} M_{\tilde{\nabla} F (\bar{t})} {(α_{0})}^{T} d & = & - M_{\tilde{\nabla} F (\bar{t})} {(α_{0})}^{T} \cdot M_{\tilde{\nabla} F (\bar{t})} (α_{0}) \\ = & - {∥ M_{\tilde{\nabla} F (\bar{t})} (α_{0}) ∥}^{2} \\ < & 0 . \end{matrix}$ Since F is differentiable and F ∈ ℰ, by Theorem 3.10 there exists a δ > 0 such that $F (\bar{t} + λ d) ≺ F (\bar{t})$ for λ ∈ (0, δ), which contradicts the assumption that $\bar{t}$ is a local optimal solution. Hence, $\tilde{\nabla} F (\bar{t}) = 〈 \tilde{0} 〉$ .

Theorem 4.3. Let $F : Ω \subseteq ℝ^{n} \to ℱ / §$ be a twice differentiable fuzzy mappings at $\bar{t}$ and F ∈ ℰ. If $\bar{t}$ is a local optimal solution, then ${\tilde{\nabla}}^{2} F (\bar{t})$ is a positive semi-definite fuzzy equivalence class matrix.

Proof. Since F is twice differentiable at $\bar{t}$ , from Definition 3.12, $F (\bar{t} + λ d)$ is equal to $F (\bar{t}) + λ \tilde{\nabla} F {(\bar{t})}^{T} d + \frac{1}{2} λ^{2} d^{T} {\tilde{\nabla}}^{2} F (\bar{t}) d + o (λ^{2} {∥ d ∥}^{2})$ for any $λ \neq 0, d \in ℝ^{n}$ , where $d_{sup} (\frac{o (λ^{2} {∥ d ∥}^{2})}{λ^{2}}, 〈 \tilde{0} 〉) \to 0$ as λ → 0 . Since $\bar{t}$ is a local optimal solution, by Theorem 4.2 we have $\tilde{\nabla} F (\bar{t}) = 〈 \tilde{0} 〉$ . Divided by λ²> 0, we get $\frac{F (\bar{t} + λ d) - F (\bar{t})}{λ^{2}} = \frac{1}{2} d^{T} {\tilde{\nabla}}^{2} F (\bar{t}) d + \frac{o (λ^{2} {∥ d ∥}^{2})}{λ^{2}} .$

Since $\bar{t}$ is a local optimal solution, we have $F (\bar{t} + λ d) \underline{≻} F (\bar{t})$ for λ sufficiently small. That is $\frac{1}{2} t^{T} M_{{\tilde{\nabla}}^{2} F (\bar{t})} (α) d + \frac{M_{o (λ^{2} {∥ d ∥}^{2})} (α)}{λ^{2}} \geq 0$ for all α ∈ [0, 1] and λ sufficiently small. Since $sup_{α \in [0, 1]} | \frac{M_{o (λ^{2} {∥ d ∥}^{2})} (α)}{λ^{2}} | \to 0$ as λ → 0, by taking the limit as λ → 0, it follows that $d^{T} M_{{\tilde{\nabla}}^{2} F (\bar{t})} (α) d \geq 0$ for all α ∈ [0, 1]. Hence, by the arbitrariness of d, $M_{{\tilde{\nabla}}^{2} F (\bar{t})} (α)$ is positive semi-definite for all α ∈ [0, 1], so is ${\tilde{\nabla}}^{2} F (\bar{t})$ .

Theorem 4.4. Let $F : Ω \subseteq ℝ^{n} \to ℱ / §$ be a twice differentiable fuzzy mappings at $\bar{t}$ and F ∈ ℰ. If $\tilde{\nabla} F (\bar{t}) = 〈 \tilde{0} 〉$ and ${\tilde{\nabla}}^{2} F (\bar{t})$ is a positive definite fuzzy equivalence class matrix, then $\bar{t}$ is a strict local optimal solution.

Proof. Since F is twice differentiable at $\bar{t}$ , by Definition 3.12 we have

$\begin{matrix} F (t) = F (\bar{t}) + \tilde{\nabla} F {(\bar{t})}^{T} (t - \bar{t}) \\ + \frac{1}{2} {(t - \bar{t})}^{T} {\tilde{\nabla}}^{2} F (\bar{t}) (t - \bar{t}) + o ({∥ t - \bar{t} ∥}^{2}) \end{matrix}$ (2) for any $t \in Ω \cap N_{ɛ} (\bar{t})$ , where $d_{sup} (\frac{o ({∥ t - \bar{t} ∥}^{2})}{{∥ t - \bar{t} ∥}^{2}}, 〈 \tilde{0} 〉) \to 0$ as $t \to \bar{t}$ . Suppose that $\bar{t}$ is not a strict local optimal solution, then there exists a sequence ${t^{k}} \subseteq ℝ^{n}$ converging to $\bar{t}$ such that $F (t^{k}) ⪯ F (\bar{t}), t^{k} \neq \bar{t},$ for each $k \in ℕ^{+}$ . Denote $d^{k} = \frac{t^{k} - \bar{t}}{∥ t^{k} - \bar{t} ∥} .$ Since $\tilde{\nabla} F (\bar{t}) = 〈 \tilde{0} 〉$ , the equality (1) implies that $\begin{matrix} \frac{F (t^{k}) - F (\bar{t})}{{∥ (t^{k} - \bar{t}) ∥}^{2}} & = & \frac{1}{2} {d^{k}}^{T} {\tilde{\nabla}}^{2} F (\bar{t}) d^{k} + \frac{o ({∥ t^{k} - \bar{t} ∥}^{2})}{{∥ (t^{k} - \bar{t}) ∥}^{2}} \\ ⪯ & 〈 \tilde{0} 〉, \end{matrix}$ for each $k \in ℕ^{+}$ . That is $\frac{1}{2} {d^{k}}^{T} M_{{\tilde{\nabla}}^{2} F (\bar{t})} (α) d^{k} + \frac{M_{o ({∥ t - \bar{t} ∥}^{2})} (α)}{{∥ t - \bar{t} ∥}^{2}} \leq 0$ for all α ∈ [0, 1] and each $k \in ℕ^{+}$ . Since ∥d^k∥ = 1 and $sup_{α \in [0, 1]} | \frac{M_{o ({∥ t^{k} - \bar{t} ∥}^{2})} (α)}{{∥ t^{k} - \bar{t} ∥}^{2}} | \to 0$ as $t^{k} \to \bar{t}$ , when $t^{k} \to \bar{t} as k \to \infty$ , there exists a $d \in ℝ^{n}$ such that d^k→ d as k→ ∞. Considering this two sequences {t^k} and {d^k} and taking the limit as k→ ∞, we can get that $d^{T} M_{{\tilde{\nabla}}^{2} F (\bar{t})} (α) d \leq 0$ for all α ∈ [0, 1], which contradicts the assumption that ${\tilde{\nabla}}^{2} F (\bar{t})$ is positive definite. Therefore, $\bar{t}$ is indeed a strict local optimal solution.

Now, we will reveal the important necessary and sufficient conditions for convex fuzzy optimizations.

Theorem 4.5. Let $F : Ω \subseteq ℝ^{n} \to ℱ / §$ be a convex fuzzy mappings, where Ω is a non-empty open convex set and F ∈ ℰ. Then the any local optimal solution in the problem (1) is a global optimal solution.

Proof. Let $\bar{t} \in Ω$ be a local optimal solution in the problem(1). Suppose that $\bar{t}$ is not a global optimal solution, then there exists at least one t′ ∈ Ω such that $F (t^{'}) ≺ F (\bar{t}) .$ Since Ω is a convex set, we have $λ \bar{t} + (1 - λ) t^{'} \in Ω$ for any λ ∈ (0, 1). Considering that F is convex, we have $\begin{matrix} M_{F (λ t^{'} + (1 - λ) \bar{t})} (α) & \leq & M_{λ F (t^{'}) + (1 - λ) F (\bar{t})} (α) \\ < & M_{F (\bar{t})} (α) \end{matrix}$ for some α ∈ [0, 1]. Since λ > 0 can be sufficiently small, there exists an ɛ-neighborhood around $\bar{t}$ such that $M_{F (t)} (α) < M_{F (\bar{t})} (α)$ for all $t \in Ω \cap N_{ɛ} (\bar{t}),$ where ɛ > 0, which contradicts the condition that $\bar{t} \in Ω$ is a local optimal solution. Hence, $\bar{t}$ is also a global optimal solution.

Theorem 4.6. Let F : Ω→ ℱ/§ be a differentiable fuzzy mappings on Ω, where Ω is a non-empty open convex subset in $ℝ^{n}$ and F ∈ ℰ. Then F is convex if and only if for any $\bar{t} \in Ω$ we have $F (t) \underline{≻} F (\bar{t}) + \tilde{\nabla} F {(\bar{t})}^{T} (t - \bar{t})$ for each t ∈ Ω.

Proof. We suppose that F is convex, then for any $\bar{t} \in Ω$ we have $\begin{matrix} M_{F (λ t + (1 - λ) \bar{t})} (α) & \leq & λ M_{F (t)} (α) + (1 - λ) M_{F (\bar{t})} (α) \end{matrix}$ for each t ∈ Ω and all α ∈ [0, 1], which implies that $\frac{M_{F (\bar{t} + λ (t - \bar{t}))} (α) - M_{F (\bar{t})} (α)}{λ} \leq M_{F (t)} (α) - M_{F (\bar{t})} (α)$ for each t ∈ Ω and all α ∈ [0, 1]. Furthermore, by the Definition 3 and the proof of Theorem 3.10, it is easy to get that $\begin{matrix} \frac{M_{F (\bar{t} + λ (t - \bar{t}))} (α) - M_{F (\bar{t})} (α)}{λ} \\ = & M_{\tilde{\nabla} F (\bar{t})} {(α)}^{T} (t - \bar{t}) + \frac{M_{o (λ ∥ t - \bar{t} ∥)} (α)}{λ} \end{matrix}$ for each t ∈ Ω and all α ∈ [0, 1], where $d_{sup} (\frac{o (λ ∥ t - \bar{t} ∥)}{λ}, 〈 \tilde{0} 〉) \to 0$ as λ → 0. So, if we take the limit λ → 0, it follows that $M_{\tilde{\nabla} F (\bar{t})} {(α)}^{T} (t - \bar{t}) \leq M_{F (t)} (α) - M_{F (\bar{t})} (α)$ for each t ∈ Ω and all α ∈ [0, 1], that is, $F (t) \underline{≻} F (\bar{t}) + \tilde{\nabla} F {(\bar{t})}^{T} (t - \bar{t})$ for each t ∈ Ω.

Conversely, for any t¹, t²∈ Ω and λ ∈ (0, 1), let t = λt¹+ (1 - λ) t², then we have $F (t^{1}) \underline{≻} F (t) + \tilde{\nabla} F {(t)}^{T} (t^{1} - t),$ (3) and $F (t^{2}) \underline{≻} F (t) + \tilde{\nabla} F {(t)}^{T} (t^{2} - t) .$ (4) Now we get $λ F (t^{1}) + (1 - λ) F (t^{2}) \underline{≻} F (λ t^{1} + (1 - λ) t^{2}) .$ Hence, F is convex.

Theorem 4.7. Let F : Ω→ ℱ/§ be a twice differentiable fuzzy mapping, where Ω is a non-empty open convex subset in $ℝ^{n}$ and F ∈ ℰ. F is convex on Ω if and only if for each t ∈ Ω, ${\tilde{\nabla}}^{2} F (t)$ is a positive semi-definite fuzzy equivalence class matrix.

Proof. Suppose that F is convex. For any $\bar{t} \in Ω$ , by Theorem 4.6 we have $F (t) \underline{≻} F (\bar{t}) + \tilde{\nabla} F {(\bar{t})}^{T} (t - \bar{t})$ (5) for each t ∈ Ω. Since F is twice differentiable at $\bar{t}$ , by Definition 3.12 we have

Conversely, suppose that ${\tilde{\nabla}}^{2} F (t)$ is positive semi-definite for all t ∈ Ω, then $M_{{\tilde{\nabla}}^{2} F (t)} (α)$ is positive semi-definite for all t ∈ Ω and α ∈ [0, 1]. Since F is a differentiable fuzzy mappings on Ω and F is comparable, we have that M_F(t) (α) is differentiable with respect to t ∈ Ω for all α ∈ [0, 1]. Now, let any t¹, t²∈ Ω, by the mean value theorem in [14], there exists at least one λ ∈ (0, 1) such that

$\begin{matrix} F (t^{1}) = F (t^{2}) + \tilde{\nabla} F {(t^{2})}^{T} (t^{1} - t^{2}) \\ + \frac{1}{2} {(t^{1} - t^{2})}^{T} {\tilde{\nabla}}^{2} F (t) (t^{1} - t^{2}) \end{matrix}$ (7) for all α ∈ [0, 1], where t = λt¹+ (1 - λ) t². Notice that t ∈ Ω because Ω is convex and hence, by assumption, $M_{{\tilde{\nabla}}^{2} F (t)} (α)$ is positive semi-definite for all t ∈ Ω and α ∈ [0, 1]. Therefore, from (7) one can get $F (t^{1}) \underline{≻} F (t^{2}) + \tilde{\nabla} F {(t^{2})}^{T} (t^{1} - t^{2})$ for any t¹, t²∈ Ω. Hence,by Theorem 4.6 we have that F is convex on Ω.

Theorem 4.8. Let F : Ω→ ℱ/§ be a differentiable fuzzy mappings on Ω, where Ω is a non-empty open convex subset in $ℝ^{n}$ and F ∈ ℰ. Then F is convex if and only if for each t¹, t²∈ Ω we have ${(\tilde{\nabla} F (t^{2}) - \tilde{\nabla} F (t^{1}))}^{T} (t^{2} - t^{1}) \underline{≻} 〈 \tilde{0} 〉 .$

Proof. Suppose that F is convex. For any t¹, t²∈ Ω, by Theorem 4.6 we have $F (t^{1}) - F (t^{2}) \underline{≻} \tilde{\nabla} F {(t^{2})}^{T} (t^{1} - t^{2})$ and $F (t^{2}) - F (t^{1}) \underline{≻} \tilde{\nabla} F {(t^{1})}^{T} (t^{2} - t^{1}) .$ Adding the two inequalities above, we get ${(\tilde{\nabla} F (t^{2}) - \tilde{\nabla} F (t^{1}))}^{T} (t^{2} - t^{1}) \underline{≻} 〈 \tilde{0} 〉 .$

Conversely, let t¹, t²∈ Ω, Since F is a differentiable fuzzy mappings on Ω and F is comparable, we have that M_F(t) (α) is differentiable with respect to t ∈ Ω for all α ∈ [0, 1]. By the mean value theorem in [14], there exists at least one λ ∈ (0, 1) such that $M_{F (t^{2})} (α) - M_{F (t^{1})} (α) = M_{\tilde{\nabla} F (t)} {(α)}^{T} (t^{2} - t^{1})$ (8) for all α ∈ [0, 1], where t = λt¹+ (1 - λ) t². And by assumption, we have ${(\tilde{\nabla} F (t) - \tilde{\nabla} F (t^{1}))}^{T} (t - t^{1}) \underline{≻} 〈 \tilde{0} 〉,$ which implies that $M_{\tilde{\nabla} F (t)} (α) - M_{\tilde{\nabla} F (t^{1})} {(α)}^{T} (t - t^{1}) \geq 0$ for all α ∈ [0, 1]. That is ${M_{\tilde{\nabla} F (t)} (α)}^{T} (t^{2} - t^{1}) \geq {M_{\tilde{\nabla} F (t^{1})} (α)}^{T} (t^{2} - t^{1})$ for all α ∈ [0, 1]. Then, by (8) we get $M_{F (t^{2})} (α) \geq {M_{\tilde{\nabla} F (t^{1})} (α)}^{T} (t^{2} - t^{1}) + M_{F (t^{1})} (α)$ for all α ∈ [0, 1], which implies that for any t¹, t²∈ Ω, we have $F (t^{2}) \underline{≻} F (t^{1}) + {\tilde{\nabla} F (t^{1})}^{T} (t^{2} - t^{1}) .$ Hence, by Theorem 4.6 we have that F is convex.

Finally, we give the following example to illustrate the main results. It is also shown that we have improved some existing results in [12 , 27] since the order for the equivalence classes of fuzzy numbers is more general than the fuzzy-max order.

Example 4.9. Define a fuzzy mapping F : Ω→ ℱ/§ by the level sets ${[\hat{F (t_{1}, t_{2})}]}_{α} = [e^{\frac{1 + α}{2} ({t_{1}}^{2} + {t_{1}}^{2})} - \frac{1}{2} e^{{t_{1}}^{2} + {t_{1}}^{2}}, \frac{1}{2} e^{{t_{1}}^{2} + {t_{1}}^{2}}]$ for all α ∈ [0, 1], where $Ω = (- 1, 1) \times (- 1, 1) \subseteq ℝ^{2}$ and $\hat{F (t)}$ is the Mareš core of F (t), for all t = (t₁, t₂)^T∈ Ω. Thus, we have $M_{F (t_{1}, t_{2})} (α) = \frac{1}{2} e^{\frac{1 + α}{2} ({t_{1}}^{2} + {t_{1}}^{2})}$ for all α ∈ [0, 1] and t = (t₁, t₂)^T∈ Ω. It is obvious that M_F(t) (α) is continuous from the right at 0 and continuous form the left on [0, 1] with respect to α. Since M_F(t) (α) is increasing with respect to α, we get $V_{0}^{1} (M_{F (t_{1}, t_{2})}) < \frac{1}{2} e^{{t_{1}}^{2} + {t_{1}}^{2}} \leq \frac{1}{2} e^{2} .$ Thus, we find that M_F(t) (α) is of bounded variation with respect to α for all t = (t₁, t₂)^T∈ Ω. It is easy to verify that F (t) is twice differentiable at any t = (t₁, t₂)^T∈ Ω and F ∈ ℰ. In fact, $M_{\tilde{\nabla} F (t_{1}, t_{2})} (α)$ is equal to $(\frac{1 + α}{2} t_{1} e^{\frac{1 + α}{2} ({t_{1}}^{2} + {t_{1}}^{2})}, \frac{1 + α}{2} t_{2} e^{\frac{1 + α}{2} ({t_{1}}^{2} + {t_{1}}^{2})})$ for all α ∈ [0, 1] and t = (t₁, t₂)^T∈ Ω. Let $\tilde{\nabla} F (t) = 〈 \tilde{0} 〉$ , we have t = (0, 0)^T. Thus, at t = (0, 0)^T, $\tilde{\nabla} F (t_{1}, t_{2}) = 〈 \tilde{0} 〉 .$ Furthermore, the Hessian matrix of F at t = (0, 0)^T exists and we can get that the midpoint-value matrices $M_{{\tilde{\nabla}}^{2} F (0, 0)} (α) = [\begin{matrix} \frac{1 + α}{2} & 0 \\ 0 & \frac{1 + α}{2} \end{matrix}]$ are positive definite for all α ∈ [0, 1]. Hence, ${\tilde{\nabla}}^{2} F (0, 0)$ is a positive definite fuzzy equivalence matrix. Then by Theorem 4.4, we can get that F has a strict local optimal solution.

It is obvious that $\frac{\partial^{2} M_{F (t_{1}, t_{2})} (α)}{\partial^{2} t_{1}} = \frac{1 + α}{2} e^{\frac{1 + α}{2} ({t_{1}}^{2} + {t_{1}}^{2})} (1 + {t_{1}}^{2}) > 0$ and $| M_{{\tilde{\nabla}}^{2} F (t_{1}, t_{2})} (α) | > 0$ for all α ∈ [0, 1] and t = (t₁, t₂)^T∈ Ω, which implies that ${\tilde{\nabla}}^{2} F (t)$ is a positive definite fuzzy equivalence matrix for all t = (t₁, t₂)^T∈ Ω. By Theorem 4.7 we have that F is convex on Ω. By Theorem 4.5, the strict local optimal solution t = (0, 0)^T is indeed a strict global optimal solution of F.

5 Conclusions

In this present investigation, the concepts of gradient and convexity of the fuzzy mappings are presented in the quotient space of fuzzy numbers. The fuzzy optimizations of differentiable fuzzy mappings and convex fuzzy mappings are discussed. The research results on the quotient space of fuzzy numbers can be traced back to the works of Mareš [10, 11]. Hong and Do [4] improved this results and proposed a more refined equivalence relation. This equivalence relation can be used to partition of the set of fuzzy numbers into equivalence class having the desired group properties for the addition operation. Since the quotient space of fuzzy numbers is characterized by the midpoint-valued functions, there are more continuous fuzzy mappings and more differentiable fuzzy mappings. The development of fuzzy optimization theory has brought the convexity of fuzzy mappings into many theoretical and application problems recently. Thus the results of this paper may provide a background to ongoing work in related fields.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61472056), The Graduate Teaching Reform Research Program of Chongqing Municipal Education Commission (No. YJG143010) and The Natural Science Foundation Project of CQ CSTC (cstc2015jcyjA00034).

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