On Interval-Valued Pseudolinear Functions and Interval-Valued Pseudolinear Optimization Problems

Abstract

Some basic characterizations of an interval-valued pseudolinear function are derived. By means of the properties of interval-valued pseudolinearity, a class of interval-valued pseudolinear programs is considered, and the solution set of the interval-valued pseudolinear optimization problem is characterized.

1. Introduction

In recent years, many approaches for interval-valued optimization problems have been explored in considerable details; see examples in [1–3]. Several authors have been interested in the optimality conditions and duality results for the interval-valued optimization problems. Wu has extended the concept of convexity for real-valued functions to LU-convexity for interval-valued functions, and then he has established the Karush-Kuhn-Tucker (KKT) optimality conditions [4] and duality theory [5] for an optimization problem with an interval-valued objective function under the assumption of LU-convexity. Sun and Wang [6] have derived the Fritz John type and Kuhn-Tucker type necessary and sufficient optimality conditions for nondifferentiable interval-valued programming. Jayswal et al. [7] have discussed Mond-Weir and Wolfe type duality theorems for the interval-valued programming problems under the conditions of generalized convexity. Chalco-Cano et al. [8] have obtained KKT optimality conditions for interval-valued programming problems by using a more general concept of gH-derivative. Bhurjee and Panda [9, 10] have introduced the parametric form of interval-valued functions and studied the solution of convex interval-valued programming problems. Jana and Panda [11] have studied the preferable efficient solutions of the problem of interval-valued vector optimization. Zhang et al. [12] have extended the concepts of preinvexity and invexity to interval-valued functions and derived the KKT optimality conditions for LU-preinvex and invex optimization problems with an interval-valued objective function.

Interval-valued linear optimization [13, 14] is a class of important and simple interval-valued optimization problems. Recently, Hladík [15] has proposed a method for testing basis stability of interval-valued linear optimization problems; Hladík’s studies have shown that if some basis stability criterion holds true, then the problems become much more easy to solve. In [16], Hladík has discussed lower and upper bound approximation for the best case optimal value of the interval-valued linear optimization problems and proposed an algorithm for computing the best case optimal value. Li et al. [17, 18] and Luo et al. [19] have established some necessary and sufficient conditions for checking weak and strong optimality of given feasible solutions for the interval-valued linear optimization problems.

In this paper, we consider and give some characterizations for a class of generalized interval-valued linear function, which is interval-valued pseudolinear function. Then, by means of the basic properties of interval-valued pseudolinearity, a class of interval-valued pseudolinear programs is considered, and the solution set of the interval-valued pseudolinear optimization problem is characterized. It can be shown that the interval-valued linear optimization problems [13, 14] and the interval-valued linear fractional programs are included in interval-valued pseudolinear programs. These programs arise in many practical applications.

2. Preliminaries

In this section, we recall some basic concepts with regard to interval-valued functions.

Let be -dimensional Euclidean space, and let be its nonnegative orthant. Let us denote by the class of all closed intervals in . denotes a closed interval, where and mean the lower and upper bounds of , respectively. The closed interval also can be expressed in terms of a parameter in several ways. Throughout this paper, we consider specific parametric representations of the interval as .

Let and be in , and the interval operations can be performed with respect to parameters [9, 10] as follows:(a);(b);(c), and ;(d).

Let be dimensional column whose elements are intervals. That is, , , and , , . The interval-valued function in parametric form is as follows.

Definition 1 (see [9]). For , let . For a given interval vector , an interval-valued function is defined as

For every fixed , if is continuous in , then and exist. In that case

If is linear in , then and exist in the set of vertices of . If is monotonically increasing in , then .

The interval-valued function is differentiable at if is differentiable at for every . The partial derivatives of at may be calculated as follows: If is continuous in , then The gradient of at is an interval vector:

In this paper, we denote the point , , by . For the two interval-valued functions of and , we say that or , which means for all , and may or may not be equal to .

However, we say that or , which means for all ; that is to say, and are the same in both sides.

Definition 2 (see [9]). Suppose that is a convex set and for given , . For , .(i) is said to be convex with respect to if  which means  for all , ; may or may not be equal to .(ii) is said to be convex with respect to if  which means  for all ; is same in both sides.

It can be shown that is convex [9] with respect to if and only if is a convex function on for every . Similar result does not hold for convexity with respect to .

Mangasarian [20] has introduced the following pseudoconvex real-valued functions.

Definition 3 (see [20]). The differentiable real-valued function on the open set is pseudoconvex on if or equivalently, Furthermore, we say that is strictly pseudoconvex on if

We can also define the following pseudoconvex interval-valued function.

Definition 4. A differentiable interval-valued function , on the convex set for given , is pseudoconvex with respect to on if or equivalently, An interval-valued function is pseudoconcave if is pseudoconvex with respect to on .

Remark 5. It is worth pointing out that the definition of pseudoconvex interval-valued function in [4] has some limitations, which define the pseudoconvex interval-valued function by using the endpoint functions and of the interval-valued function . However, in some case, if the two endpoint functions and are all real-valued pseudoconvex functions, the interval-valued function may be not a pseudoconvex interval-valued function, which can be seen in the following example.

Example 6. The interval-valued function , , , and , where and .
It can be shown that is strictly increasing function, being , for all ; therefore, . Thus, . So is pseudoconvex.
The function is strictly decreasing on , as . We have ; , so is pseudoconvex.
The sum is and , . We have , . Therefore, has a maximum point at , so it is not pseudoconvex.

For a real-valued differentiable function, which is said to be a pseudolinear function [21–23], if it is both pseudoconvex and pseudoconcave, similar to the case of real-valued functions, we can also define the following interval-valued pseudolinear functions.

Definition 7. A differentiable interval-valued function , on the convex set for given , is with respect to on if it is both pseudoconvex and pseudoconcave on .

This class of pseudolinear interval-valued functions includes many useful functions. For example, the following function is a pseudolinear interval-valued function.

Example 8. The interval-valued function , where , , , and . It can be shown that this interval-valued function is a pseudolinear interval-valued function.

3. Characterizations of PseudolinearInterval-Valued Functions

In this section, we provide some characterizations of the pseudolinear interval-valued function.

Theorem 9. Let a differentiable interval-valued function , on the convex set for given . Then, the following statements (i)–(iii) are equivalent.(i) is pseudolinear over .(ii)For any and in if and only if .(iii)There exists an interval-valued function defined on such that for given and for any and in .

Proof. (i)⇒(ii) Suppose that is pseudolinear interval-valued function for given on . It can be shown that implies .
Let and be two points of such that . For any , we denote the point by , and it can be shown that .
If , then since is pseudoconvex for given . But , and then . According to the pseudoconvexity of , then , which contradicts the assumption of . So the assumption of is false. Similarly, it can be shown that is false by the pseudoconcavity of the interval-valued function . Then, we can show that for given on and all in . So (ii)⇒(iii) Let the interval-valued function : defined on , if for , in and given on , and we define for given on ; if for , in and given on , we define We can show that for given .
Suppose that for given . For some , if , using the continuity of , there exists such that and . By (ii), which contradicts the assumption of for in and given on . So for given and all ; we also have and since for in and given on , holds. It can be shown that for given on .
Similarly, we can show that if for given .
(iii)⇒(i) It is obvious that for any and in , which implies if and only if .

Theorem 10. Let be an interval-valued function defined on the convex set for given , which is once continuously differentiable on . Then is pseudolinear on if and only if implies for any and in and any .

Proof. Suppose that is pseudolinear on . Let and in be such that for given . Then, for any , By (ii) of Theorem 9, for any and in and any .
Conversely, suppose that implies for any and in and any . If is not pseudolinear on , then there exist and in such that , but , which means that and cannot occur. Suppose that , which means for . We define , which is once continuously differentiable function. According to the assumption, and . So assumes a local maximum at some point . Then So . By the assumption, for all , and then , which is a contradiction.

How to check for interval pseudolinearity can be shown by the following example.

Example 11. Consider the interval-valued function on : From (ii) of Theorem 9, for any and in , means for all . Since , if and only if , so, for and , we have for all , , which implies .
Therefore, the interval-valued function is a pseudolinear interval-valued function.

The following interval-valued linear fractional function is a class of interval-valued pseudolinear functions: where , and are all in , and and are interval number. We can show that is pseudolinear on , where if and are in .

For interval-valued linear fractional function, we have for any , , and . It can be shown that (29) implies and for any and in . We can show the following conclusion.

Theorem 12. Let a pseudolinear interval-valued function , on the convex set with proportional functional for given . If satisfies condition (29) and is differentiable for each in , then there exists an interval-valued linear fractional function such that and for all in .

Proof. Suppose that is not constant on and contains more than one point. Take and in such that . Then for any and in .
Since is differentiable for each in , then for any and in , where denotes the gradient of with respect to the second variable.
From (30), we have By (31), we have By (32)-(33), we have Since , we have . From (34), we have Multiplying (35) by , we obtain So Substituting (37) into (30) and replacing by , it can be shown that is an interval-valued linear fractional function .

The result of Theorem 12 and (iii) of Theorem 10 can be shown by the following example.

Example 13. Consider the interval-valued linear fractional function where , , and are all in and and are interval number, where , .
We denote that It can be shown that satisfies (29), and the following is satisfied: for any and in . From (iii) of Theorem 9, the interval-valued function is a pseudolinear interval-valued function.
According to Theorem 12, there exists an interval-valued linear fractional function , such that .
For all , , we have Then, we get By (iii) of Theorem 9, we have .

4. The Solution Sets of Interval-Valued Pseudolinear Programs

In this section, we consider the following interval-valued programming: where is an interval-valued function for given and the feasible set is a convex subset of . We denote the solution set of (IVOP) as which is nonempty. If is pseudolinear, then the solution set of (IVOP) is a convex set.

The following characterization of the solution set of the problem of (IVOP) can be obtained.

Theorem 14. Let be a pseudolinear interval-valued function for given and . Then , where

Proof. It is obvious that if and only if , which implies for and . From (ii) of Theorem 9, if and only if Therefore, .
It can be seen that . Let , and then For each , we get Therefore, from (ii) of Theorem 9, we have By the pseudolinearity of and (52), we obtain So and . Thus, .

Corollary 15. Let be a pseudolinear interval-valued function for given and . Then , where

Proof. From (46), it is clear that . Let ; that is, for , Then, from (iii) of Theorem 9 and the pseudolinearity of , we have Then , which implies . It can be also similarly proved that .
We can also prove that as follows. In fact, it is clear that where is defined by (48). Thus, the conclusion follows.

Example 16. Consider the following pseudolinear interval-valued program: where is an interval-valued function, , , , and . By Example 11, is a pseudolinear interval-valued function and .
According to Theorem 14, we get .

5. Conclusions

In this paper, we introduced the concept of the pseudolinear interval-valued functions. We have obtained some basic characterizations of an interval-valued pseudolinear function. By means of the properties of interval-valued pseudolinearity, a class of interval-valued pseudolinear programs was considered, and the solution set of the interval-valued pseudolinear optimization problem was characterized.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by Natural Science Basic Research Plan in Shaanxi Province of China (Program nos. 2013JQ1020, 2013KJXX-29, and 2014 JM8307), National Natural Science Foundation of China (Program nos. 11301415, 61100166, 61303092, 11401469, 11426176, and 11401357), special funds for the construction of key disciplines funded projects in Shaanxi Province, project funded by China Postdoctoral Science Foundation (no. 2014M552453), the National Key Technologies R&D Program of China under Grant no. 2012BAH16F02. Hanzhong Administration of Science and Technology under Grant no. 2013hzzx-39, and the Science Plan Foundations of the Education Bureau of Shaanxi Province (nos. 11JK1051, 2013JK1098, 2013JK1130, 2013JK1182, and 14JK1661).