Optimality conditions for fuzzy optimization problems under granular convexity concept
Introduction
Fuzzy optimization models have been usually used to tackle uncertainty and inaccuracy in the coefficients of mathematical programming. As a class of useful tool, there are a lot of works on this topic has been widely studied by many authors [1], [2], [3], [12]. It is well known that the stationary point concept plays a crucial role as a necessary optimality condition for the fuzzy optimization problems defined by differentiable functions, since it can identify the potential candidates as the optimal solutions. The differentiability and convexity play an important role in deriving sufficient optimality conditions and duality results for fuzzy optimization problems. In recent years, considerable progress has been made to weaken the differentiability and convexity conditions in fuzzy optimization problems. The Karush-Kuhn-Tucker optimality conditions were studied for fuzzy optimization problems under the assumptions of different differentiable conditions.
Under the assumptions of Hukuhara differentiability, Wu [1], [2], [3] has studied the fuzzy convex functions and proposed the Karush-Kuhn-Tucker optimality conditions for the optimization problem with a fuzzy-valued objective function. Sun, Xu and Wang [4] defined the saddle point of Lagrangian function and presented saddle point optimality conditions for the interval-valued programming. Zhang et al. [5] extended the concepts of preinvexity and invexity to the interval-valued functions, and studied the KKT optimality conditions for LU-preinvex and invex optimization problems with an interval-valued objective function under the conditions of weakly continuous differentiability and Hukuhara differentiability. Under smooth and non-smooth LU-convexity assumptions, Zhang et al. [6] also identified the weakly efficient points of the interval-valued vector optimization problems and the solutions of the weak vector variational inequalities. Based on generalized convexity and generalized differentiability of interval valued functions, the KKT optimality conditions for aforesaid problems were obtained by Ahmad et al. [7] and Jayswal et al. [8].
In 2013, Bede and Stefanini [9] proposed the concept of generalized Hukuhara differentiability (or gH-differentiability) of fuzzy-valued functions. Singh et al. [10] proposed KKT optimality theorems with gH-differentiability. Based on the gH-differentiability, Arana-Jiménez et al. [12] introduced the concepts of fuzzy pseudoinvex-I and fuzzy pseudo-invex-II, and proved that pseudoinvexity is the necessary and sufficient condition for a fuzzy vector optimization problem with a stationary point as a solution. In [13], Chalco-Cano et al. introduced a class of gH-differentiable pseudoinvex fuzzy-valued functions and proposed the Karush-Kuhn-Tucker optimality conditions for fuzzy optimization problems. Based on the GH-differentiability, Osuna-Gómez et al. [14] introduced the concepts of weak pseudoinvex fuzzy functions and strict pseudoinvex fuzzy functions, they also studied the necessary and sufficient conditions for fuzzy optimality problems. For the relationships between the above generalized differentiabilities of fuzzy functions, refer to [12], [13], [14]. Li, Liu and Zhang [15] have extended the univexity to the fuzzy functions and studied the optimality conditions for fuzzy optimization problems. In a recent study, Osuna-Gomez and Hernandez-Jimenez et al. [16] studied necessary efficiency conditions for the fuzzy multiobjective optimization problems based on gH-differentiability concept.
Although there have been many meaningful results in the research of the H-differentiable fuzzy optimization problem and the gH-differentiable fuzzy optimization problem, the H-differentiability or gH-differentiability of the fuzzy optimization problems still have some drawbacks as Son, Long and Dong discussed in the paper [17], such as: the increasing support closure length of fuzzy functions; the existence of unnatural behavior in the modeling phenomenon and the existence of multi-solutions with different geometric presentations.
In [30], Mazandarani and Li discussed at least 6 drawbacks in dealing with fuzzy differential equations (FDEs) based on the concepts of H-derivative, SGH-derivative, gH-derivative, and g-derivative or any other concepts which are equivalent to those mentioned concepts. These 6 drawbacks are as follows, (1) In almost all cases, FDEs are solved based on the characterization theorem which helps find the solutions related to the first and second forms of the differentiability. These solutions correspond to the cases where the diameter of fuzzy function is non-decreasing and non-increasing, respectively. However, in a general setting, for the analysis or prediction of behavior of a phenomenon or a dynamical system, a unique solution is required depending on which decision is made. It should be understood that by the “unique solution”, a single fuzzy solution whose diameter is not necessarily monotonic. (2) Determining the first form solution of a first order FDE equates to solving a system of two differential equations. This is also the case with the second form solution. This fact raises a challenge for higher order FDEs. Due to this fact, based on the earlier explanations about the unique solution, even the analysis of the simplest dynamical systems on the basis of the family of Hukuhara derivatives is challenging. (3) In order to find solutions of FDEs under a concept of fuzzy derivative by the use of the characterization theorem, we need to determine the lower and upper α-level cuts of functions in questions involved in the FDEs. Whereas characterizing such α-level cuts in simple cases such as linear FDEs may be feasible, it becomes a complicated task for a general setting such as nonlinear FDEs including unknown functions. (4) Unnatural behavior in modeling (UBM) phenomenon is the other challenge concerning the family of Hukuhara derivatives. This phenomenon represents that different guises of a same structure of a system model may show different behaviors of the system. (5) These methods are unable to give a solution to the zero forms. Specifically, one fails to obtain a fuzzy solution based on any concept belonging to Hukuhara derivatives family. (6) The factorization cannot be applied in fuzzy differential equations if the approaches based on H-derivative, SGH-derivative, gH-derivative, g-derivative, or any concept that has been defined based on (or equivalent to) such derivatives are employed. For specific examples, one can refer to the reference of [30].
To overcome the above limitations, Mazandarani et al. [18] proposed the concept of the granular differentiability (gr-differentiability) of the fuzzy functions. Under the condition of gr-differentiability, Mazandarani et al. studied some optimal control problems of fuzzy linear dynamical systems [19], [20] and Z-differential equations [21]. In [17], Son, Long and Dong studied some theoretical results for the fuzzy delay differential equations under the condition of gr-differentiability. Najariyan, Pariz and Vu [22] introduced some new notions such as fuzzy nilpotent matrix, fuzzy linearly independent vectors, fuzzy eigenvectors, rank, index and fuzzy Jordan canonical form of a fuzzy matrix, and they also studied the solution of fully fuzzy linear singular differential equations. Vu and Hoa [23] proposed the concepts of granular Riemann-Liouville q-fractional integral and granular Caputo q-fractional derivative, and studied the existence and uniqueness of solution for the Caputo q-fractional initial value problem under the condition of granular differentiability. Najariyan and Zhao [24], [25] studied the fuzzy fractional quadratic regulator problem and fuzzy singular differential equations under granular fuzzy fractional derivatives. Mustafa, Gong and Osman [26] derived granular Euler-Lagrange condition for the fuzzy variational problem and necessary conditions of Pontryagin type for fixed and free final state fuzzy optimal control problem based on the concepts of horizontal membership function. Gr-differentiability has two advantages, one is that there is only one fuzzy differential equation that needs to be solved; the other is that fuzzy differential equations will not have multiple solutions.
Motivated by the above results, under the condition of gr-differentiation, we consider the fuzzy optimization problems with the general fuzzy function as the objective function. A list of novel contributions of this paper is as follows:
1. Defining the granular convex fuzzy function, and deducing some properties of the granular convex fuzzy function.
2. Introducing the concept of fuzzy global and local relative optimal solution of fuzzy optimization problems, and proposing the Karush-Kuhn-Tucker type optimality conditions for the constrained fuzzy optimization problems.
3. Finally, the relationships between a class of variational inequality and the fuzzy optimization problems are established.
The rest of paper is organized as follows: Section 2 presents some preliminaries. Section 3 introduces the concepts of granular convexity, and proposes some properties of the granular convex fuzzy function. Section 4 presents the Karush-Kuhn-Tucker type optimality conditions of the fuzzy optimal solution of more general fuzzy programming problems and some test examples. Finally, the relationships between a class of variational inequalities and the fuzzy optimization problems are established.
Section snippets
Preliminaries
In this section, we introduce some definitions and some results that we will use throughout this paper.
Let be n-dimensional Euclidean space, be its nonnegative orthant, and E be the space of all fuzzy numbers on having the parametric form , where are called the left and right end points of , respectively. The following definition introduces the notion of horizontal membership function (HMF).
Definition 2.1 [18] Let be a fuzzy number. The
Granular convex fuzzy functions
In this section, we introduce the concept of granular convex fuzzy function, and establish various properties of the granular convex fuzzy function.
Definition 3.1 Granular convex fuzzy function Let be a fuzzy function defined on a convex set . We say that is granular convex if for all , with regard to the distinct fuzzy numbers , , and each . We say that is granular strict convex if for all
Fuzzy optimization problem
In this section, we consider the fuzzy-valued optimization problem as follows: where is a fuzzy-valued function with regard to distinct fuzzy numbers , , and , are real-valued functions. We denote the feasible set of the problem of (FOP) by
Unlike the solution concept used by Wu in [1], [2], [3] and Chalco-Cano, Arana-Jiménez, and Osuna-Gómez in [11], [12], [13], [14] based on the partial
Optimality conditions for the fuzzy optimization problem based on gr-differentiability
Now, we show that the KKT conditions are necessary and sufficient for optimality under the assumptions of gr-convexity and modified Slater condition are satisfied.
Let denote the set of active constraints at , which is defined by
Definition 5.1 Let , we say that d is a feasible direction of S at if there exists such that We denote the set of the all feasible directions of S at is .
Definition 5.2 Let , we say that d is a linearization
Relationships between a class of variational inequality and the fuzzy optimization problem
In this section, using the concepts of gr-convexity and fuzzy relative optimal solution for the fuzzy optimization problems, we will establish the relationships between a class of variational inequality and the fuzzy optimization problems.
We will use the following variational inequality problems.
(VIP) A Stampacchia variational inequality problem is to find a point , such that
(VWIP) A Stampacchia weak variational inequality problem is to find a point
Conclusion
In the present paper, we have introduced a concept of the gr-convex fuzzy function. Under the hypotheses of gr-differentiability and gr-convexity, we have presented optimality results for the fuzzy optimization problems. Our results are significantly different from those presented in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Thus, the results in this paper improve and generalize the earlier studies in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is supported by Shaanxi Provincial Key Research and Development Program (No. 2021SF-480).
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