# On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (1)
- u is normal, that is, there exists ${x}_{0}\in \mathbb{R}$ such that $u\left({x}_{0}\right)=1$;
- (2)
- u is upper semicontinuous;
- (3)
- u is convex, that is,$$u(\lambda x+(1-\lambda \left)y\right)\ge min\left\{u\right(x),u(y\left)\right\}$$
- (4)
- ${\left[u\right]}^{0}=\overline{\{x\in \mathbb{R}|u\left(x\right)>0\}}$ is compact, where $\overline{A}$ denotes the closure of A.

**Definition**

**2.**

- (1)
- $u\u2aafv$ if and only if ${\left[u\right]}^{\alpha}=[{u}_{L}\left(\alpha \right),$ ${u}_{R}{\left(\alpha \right)]\le \left[v\right]}^{\alpha}=[{u}_{L}\left(\alpha \right),{u}_{R}\left(\alpha \right)]$ for each $\alpha \in [0,1],$ where ${\left[u\right]}^{\alpha}\le {\left[v\right]}^{\alpha}$ if and only if ${u}_{L}\left(\alpha \right)\le {v}_{L}\left(\alpha \right)$ and ${u}_{R}\left(\alpha \right)\le {v}_{R}\left(\alpha \right).$
- (2)
- $u\prec v$ if and only if $u\u2aafv$ and there exists ${\alpha}_{0}\in [0,1]$ such that ${u}_{L}\left({\alpha}_{0}\right)<{v}_{L}\left({\alpha}_{0}\right)$ or ${u}_{R}\left({\alpha}_{0}\right)<{v}_{R}\left({\alpha}_{0}\right).$
- (3)
- if either $u\u2aafv$ or $v\u2aafu$, then u and v are comparable. Otherwise, u and v are non-comparable.

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 3. Directional Derivative of the Fuzzy Function

**Definition**

**8.**

**Remark**

**1.**

**Definition**

**9.**

- (1)
- F is convex on D if$$F(\lambda {x}_{1}+(1-\lambda ){x}_{2})\u2aaf\lambda F\left({x}_{1}\right)+(1-\lambda )F\left({x}_{2}\right)\phantom{\rule{2.em}{0ex}}$$
- (2)
- F is strictly convex on D if$$F(\lambda {x}_{1}+(1-\lambda ){x}_{2})\prec \lambda F\left({x}_{1}\right)+(1-\lambda )F\left({x}_{2}\right)\phantom{\rule{2.em}{0ex}}$$

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Definition**

**10.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Definition**

**11.**

**Theorem**

**9.**

**Proof.**

## 4. Subdifferential of Fuzzy Function

**Definition**

**12.**

- (1)
- A fuzzy function $l\left(d\right):{\mathbb{R}}^{m}\to {E}^{1}$ with$$l\left(d\right)\u2aaf{F}^{\prime}(\overline{x},d)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}for\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}all\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}d\in {\mathbb{R}}^{m}.$$
- (2)
- Define the set$$\partial F\left(\overline{x}\right)=\left\{l\left(d\right)\right|l\left(d\right)\u2aaf{F}^{\prime}(\overline{x},d)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}for\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}all\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}d\in {\mathbb{R}}^{m}\}$$

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Zhang, W.; Xing, Y.; Qiu, D.
On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions. *Symmetry* **2017**, *9*, 177.
https://doi.org/10.3390/sym9090177

**AMA Style**

Zhang W, Xing Y, Qiu D.
On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions. *Symmetry*. 2017; 9(9):177.
https://doi.org/10.3390/sym9090177

**Chicago/Turabian Style**

Zhang, Wei, Yumei Xing, and Dong Qiu.
2017. "On Characterizations of Directional Derivatives and Subdifferentials of Fuzzy Functions" *Symmetry* 9, no. 9: 177.
https://doi.org/10.3390/sym9090177