Equivalent Conditions of Generalized Convex Fuzzy Mappings

Abstract

We obtain some equivalent conditions of (strictly) pseudoconvex and quasiconvex fuzzy mappings. These results will be useful to present some characterizations of solutions for fuzzy mathematical programming.

1. Introduction

The occurrence of randomness and fuzziness in the real world is inevitable owing to some unexpected situations. In [1], Zadeh initially introduced the concept of fuzzy number. Since then, theories of fuzzy number and their applications have been extensively and intensively studied by many scholars; one can refer to [2–9]. Mathematical programming under fuzzy environment or which involves fuzziness is called fuzzy mathematical programming. Bellman and Zadeh [10] introduced fuzzy optimization problems and stated that a fuzzy decision can be viewed as the intersection of fuzzy goals and problem constraints.

Nanda and Kar [11] proposed the concept of convex fuzzy mappings and proved that a fuzzy mapping is convex if and only if its epigraph is a convex set. At the same time, some applications to fuzzy mathematical programming problems were studied. The convexity has been playing an important role in fuzzy mathematical programming theory. Some related research work has been carried out; one can refer to [12–21]. But it is obvious that the condition of convex fuzzy mappings is too strict. Recently, different types of generalized convex fuzzy mappings were defined. Some properties and the applications were studied in fuzzy mathematical programming problems. Especially, Panigrahi et al. [22] proposed the concept of quasiconvex fuzzy mappings, which is different from that of Nanda and Kar [11] as well as Syau [14], and derived the Karush-Kuhn-Tucker optimality conditions for the constrained fuzzy minimization problems. Strict inequality relation between fuzzy numbers is used in [22], which is too much restrictive. Syau [15] introduced the concept of generalized convexity such as pseudoconvexity for fuzzy mappings with several variables and studied some basic differentiability properties of fuzzy mappings from the standpoint of convex analysis.

Motivated by the earlier works of Panigrahi et al. [22], Karamardian [23], and Karamardian and Schaible [24], in this paper, we establish some equivalent conditions of (strictly) pseudoconvex and quasiconvex fuzzy mappings.

2. Preliminaries

In this section, we quote some preliminary notations and definitions.

Let be the set of all real numbers. A fuzzy number is a mapping with the following properties:(1)is normal; that is, ,(2) is upper semicontinuous,(3) is convex; that is, for all ,(4)the support of , and its closure cl() is compact.

Let be the set of all fuzzy numbers on . The -level set of a fuzzy number , , denoted by , is defined as

It is clear that the -level set of a fuzzy number is a closed and bounded interval . denotes the left-hand end point of and denotes the right-hand end point of . Also any can be regarded as a fuzzy number defined by In particular, the fuzzy number is defined as if and otherwise . Thus, fuzzy number can be identified by parameterized triples .

For fuzzy numbers and parameterized by respectively, and each nonnegative real number , we define the addition and nonnegative scalar multiplication as follows: Obviously, for each real number , Moreover, define the opposite of a fuzzy number to be the fuzzy number satisfying . In other words, if is represented by the parametric form , then is represented by the corresponding parametric form . We represent a fuzzy number as .

A fuzzy number is said to be a triangular fuzzy number if . Moreover, if and are linear, then we say that    is a linear triangular fuzzy number. We denote .

Definition 1 (see [22]). For , we say that if for each , , . If , , then . We say that , if and there exists such that or . For , if either or , then we say that and are comparable, otherwise noncomparable.

A mapping is said to be a fuzzy mapping. For any and for any , we denote .

Definition 2 (see [22]). Let be a fuzzy mapping. Then, is said to be comparable if for each pair , and are comparable. Otherwise, is said to be noncomparable.

Definition 3 (see [22]). Let be an open set and assume that is a fuzzy mapping. Let and let stand for the partial differentiation with respect to the th variable . Assume that for all , and have continuous partial derivatives. Define If for each defines the -cut of a fuzzy number, then we say that is differentiable at , and we denote We call the gradient of the fuzzy mapping at . A fuzzy mapping is said to be differentiable at if exists and both and for each are differentiable at .

Definition 4 (see [22]). Let be a nonempty open convex set and let be a differentiable fuzzy mapping. is said to be pseudoconvex if for each , implies that .

Definition 5 (see [22]). Let be a nonempty convex set and let be a fuzzy mapping. is said to be quasiconvex if for each and for each , the following implications hold: whenever and are comparable.

Definition 6 (see [25]). Let be a nonempty open convex set and let be a differentiable fuzzy mapping. is said to be strictly pseudoconvex if for each , implies that .

In the following sections, we always assume that be a nonempty open convex set, be a differentiable fuzzy mapping, and be comparable.

3. Pseudoconvexity of Fuzzy Mappings

In this section, we establish the equivalent conditions of pseudoconvex and strictly pseudoconvex fuzzy mappings. We first give some lemmas which will be used in the sequel.

Lemma 7 (see [25]). Assume that is a pseudoconvex fuzzy mapping. Then is a quasiconvex fuzzy mapping.

Lemma 8 (see [22]). is a quasiconvex fuzzy mapping if and only if for each , implies that .

Theorem 9. is a pseudoconvex fuzzy mapping if and only if for each , implies that .

Proof. Suppose that is a pseudoconvex fuzzy mapping. Let be such that We need to show that . Assume to the contrary that By the pseudoconvexity of and (9), we have By Lemmas 7 and 8, it follows from (11) that which contradicts (10).
Conversely, let such that We need to show that . Assume the contrary, that is, . Hence, for each , and there exists an such that Without loss of generality, we assume that From the mean-value theorem, we have where for some . From (16) and (17), we have On the other hand, from (13), it follows that From the comparability assumption of , this implies that Then, for each , we have which contradicts (18).

Remark 10. Theorem 9 generalizes Karamardian’s result (Theorem  3.1 in [23]) to fuzzy mapping case.

Theorem 11. is a strictly pseudoconvex fuzzy mapping if and only if for each , implies that .

Proof. Suppose that is a strictly pseudoconvex fuzzy mapping. Let , , such that We need to show that Assume to the contrary that Combine with (22) and from strict pseudoconvexity of , we have On the other hand, (24) can be written as From strict pseudoconvexity of , it follows that which contradicts (25).
Conversely, let , such that We need to show that . Assume to the contrary that Hence, for each , From the mean-value theorem, we have where for some .
From (30) and (31), we have On the other hand, from (28), it follows that , which implies that . Then, for each , which contradicts (32).

Remark 12. Theorem 11 generalizes Karamardian and Schaible’s result (Proposition  4.1 in [24]) to fuzzy mapping case.

4. Quasiconvexity of Fuzzy Mappings

In this section, we establish an equivalent condition of a differentiable quasiconvex fuzzy mapping.

Theorem 13. is a quasiconvex fuzzy mapping if and only if for each , implies that .

Proof. Suppose that is a quasiconvex fuzzy mapping. Let such that The relation is not possible. Otherwise, it will imply that according to Lemma 8, which contradicts to (34). From the compatibility, From Lemma 8 and (36), it follows that that is,
Conversely, assume that is not a quasiconvex fuzzy mapping. Then, there exists such that and , where . There exists such that or Without loss of generality, we assume that By the mean-value theorem, then there exist , such that where From (41), and (42), it follows that Thus, (43) yields On the other hand, from (45) and the hypothesis, for each , we obtain that . Hence, which contradicts (46).

Remark 14. Theorem 13 generalizes Karamardian and Schaible’s result (Proposition  5.2 in [24]) to fuzzy mapping case.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgment

The research was supported by the National Science Foundation of China (Grant. 11171363, 11001289).