Directional derivatives and subdifferential of convex fuzzy mapping on n-dimensional time scales and applications to fuzzy programming

Abstract

In this paper, we address the notions of directional derivative, differential and subdifferential of fuzzy mapping $f:{\mathrm{\Lambda }}^n\to {\mathbb{F}}^1$, where ${\mathrm{\Lambda }}^n$ denotes a n-dimensional time scale and ${\mathbb{F}}^1$ is the fuzzy number space. Through using the directional derivative and differential of two crisp functions that are determined by the fuzzy mapping on n-dimensional time scales, some characterizations of directional derivative and differential are discussed. Moreover, the existence of directional derivative for convex fuzzy mapping is considered on time scales and the relations among directional derivative, differential and subdifferential of fuzzy mapping are established. As applications, some examples of the convex fuzzy programming are given to show the feasibility of our obtained results.

Introduction

In 1972, Chang and Zadeh introduced the concept of fuzzy mappings (see [11]) and applied it to control theory. Since the wide applications of fuzzy theory, the calculus of fuzzy functions and metric spaces of fuzzy sets have been greatly developed (see [13], [19], [32], [34]) and some new generalized fuzzy arithmetic and differentiability were introduced to study fuzzy differential equations and dynamic fuzzy models (see [6], [16], [22], [23]).

With the development of theory of convex and uncertainty optimization (see [9], [12], [20]), the convexity and optimization of fuzzy mappings and fuzzy sets attract many researchers (see [5], [10], [17], [24], [25]). In [31], Wang and Wu studied the fuzzy n-cell numbers and established the basic properties of fuzzy n-cell number-valued mappings. Next, the authors introduced the concept of derivatives and subdifferential of convex fuzzy mappings and applied it to investigate the problems of convex fuzzy programming (see [30]). Since the significance of the derivative and differential of convex fuzzy functions on fuzzy nonlinear programming, some basic studies were conducted to establish the fundamental results (see [35], [36], [39]). In [14], [15], Gong and Hai et al. studied the convexity and generalized differentiability of n-dimensional fuzzy number-valued functions and used them to investigate fuzzy optimization. In 2019-2020, Xie and Gong proved some variational-like inequalities for high-dimensional fuzzy functions and applied them to solve the problems of fuzzy optimization (see [36], [38]), then a new class of generalized derivatives was put forward to consider the optimization problems of n-dimensional fuzzy number-valued functions (see [37]).

As is well known that, the problems of nonlinear programming on discrete domains are as important as those on continuous domains. In 2018, Adıvar and Fang established some basic results of the convex analysis and duality on discrete domains. Time scale theory was proposed by Hilger and used to unify the discrete and continuous analysis in mathematics (see [7], [8]). A time scale $\mathbb{T}$ is any subset of $\mathbb{R}$, the theory of which is a powerful tool to combine the mathematical analysis in discrete and continuous situations, one may obtain the results of the continuous case when $\mathbb{T}=\mathbb{R}$ and the results for the discrete case when $\mathbb{T}=\mathbb{Z}$, the more interesting results for the cases which are “in between” the continuous and discrete domains (or called the cases on hybrid domains) can be obtained. Time scales have been widely applied to study different mathematical subjects such as mathematical physics (see [3]), fuzzy dynamic equations (see [21], [26], [27], [29]), convex optimization and measure theory (see [28]), etc. By virtue of time scale theory, Adıvar and Fang (see [2]) studied the convex optimization on mixed domains which covers the optimization problems on continuous and discrete domains. In 2018, Al-Salih and Bohner proposed and solved the linear programming problems on time scales (see [4]) which contains the continuous and discrete linear programming. Indeed, the solving optimization problems on time scales provides a new and effective method to solve the optimization problems on hybrid domains. However, there is no notion of the directional derivatives and subdifferential of convex fuzzy mapping on time scales, it is impossible to study the convex fuzzy programming on hybrid domains and include the fuzzy programming problems in continuous and discrete cases through this method.

Motivated by the above, in this paper, we address the notions of directional derivative, differential and subdifferential of fuzzy mapping on n-dimensional time scales and some basic results are established, through which some characterizations of directional derivative and differential are discussed. Moreover, the existence of directional derivative for convex fuzzy mapping is considered on time scales and the relations among directional derivative, differential and subdifferential of fuzzy mapping are presented. As applications of our obtained results, some examples of the convex fuzzy programming are given.