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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 383692, 8 pages
Univex Interval-Valued Mapping with Differentiability and Its Application in Nonlinear Programming
1Department of Mathematics, School of Science, Xidian University, Xi'an 710126, China
2Department of Mathematics, School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
Received 25 November 2012; Accepted 26 May 2013
Academic Editor: Lotfollah Najjar
Copyright © 2013 Lifeng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Interval-valued univex functions are introduced for differentiable programming problems. Optimality and duality results are derived for a class of generalized convex optimization problems with interval-valued univex functions.
Imposing the uncertainty upon the optimization problems is an interesting research topic. The uncertainty may be interpreted as randomness, fuzziness, or interval-valued fuzziness. The randomness occurring in the optimization problems is categorized as the stochastic optimization problems, and the imprecision (fuzziness) occurring in the optimization problems is categorized as the fuzzy optimization problems. In order to perfectly match the real situations, interval-valued optimization problems may provide an alternative choice for considering the uncertainty into the optimization problems. That is to say, the coefficients in the interval-valued optimization problems are assumed as closed intervals. Many approaches for interval-valued optimization problems have been explored in considerable details; see, for example, [1–3]. Recently, Wu has extended the concept of convexity for real-valued functions to LU-convexity for interval-valued functions, then he has established the Karush-Tucker conditions [4–6] for an optimization problem with interval-valued objective functions under the assumption of LU-convexity. Similar to the concept of nondominated solution in vector optimization problems, Wu has proposed a solution concept in optimization problems with interval-valued objective functions based on a partial ordering on the set of all closed intervals, then the interval-valued Wolfe duality theory  and Lagrangian duality theory  for interval-valued optimization problems have been proposed. Recently, Wu  has studied the duality theory for interval-valued linear programming problems.
In 1981, Hanson  introduced the concept of invexity and established Karush-Tucker type sufficient optimality conditions for a nonlinear programming problem. In , Kaul et al. considered a differentiable multiobjective programming problem involving generalized type I functions. They investigated Karush-Tucker type necessary and sufficient conditions and obtained duality results under generalized type I functions. The class of B-vex functions has been introduced by Bector and Singh  as a generalization of convex functions, and duality results are established for vector valued B-invex programming in . Bector et al.  introduced the concept of univex functions as a generalization of B-vex functions introduced by Bector et al. . Combining the concepts of type I and univex functions, Rueda et al.  gave optimality conditions and duality results for several mathematical programming problems. Aghezzaf and Hachimi  introduced classes of generalized type I functions for a differentiable multiobjective programming problem and derived some Mond-Weir type duality results under the above generalized type I assumptions. Gulati et al.  introduced the concept of --type I functions and also studied sufficiency optimality conditions and duality multiobjective programming problems.
This paper aims at extending the Karush-Tucker optimality conditions to nonconvex optimization problem with interval-valued functions. First, we extend the concept of univexity for a real-valued function to an interval-valued function and present the concept of interval-valued univex functions. Then, the Karush-Tucker optimality conditions are proposed for an interval-valued function under the assumption of interval-valued univexity.
Let one denotes by the class of all closed intervals in . denotes a closed interval, where and mean the lower and upper bounds of , respectively. For every , we denote .
Definition 1. Let and be in ; one has (i) and ; (ii) ; (iii), where and .
Then, we can see that where is a real number.
By using Hausdorff metric, Neumaier  has proposed Hausdorff metric between the two closed intervals and as follows:
Definition 2. Let and be two closed intervals in . One writes if and only if and , if and only if and , that is, the following (a1), (a2), or (a3) is satisfied:(a1) and ;(a2) and ;(a3) and .
Definition 3 (see ). Let and be two closed intervals, the gH-difference of and is defined by
For example, . And .
Proposition 4. (i) For every , always exists and .(ii) if and only if .
3. Interval-Valued Univex Functions
Definition 5 (interval-valued function). The function is called an interval-valued function, where . Then, is a closed interval in for each , and can be also written as , where and are two real-valued functions defined on and satisfy for every .
Definition 6 (continuity of an interval-valued function). The function is said to be continuous at if both and are continuous functions of .
The concept of gH-derivative of a function is defined in .
Definition 7. Let and be such that , then the gH-derivative of a function at is defined as
If exists, then we say that is generalized Hukuhara differentiable (gH-differentiable, for short) at .
Moreover,  also proved the following theorem.
Theorem 8. Let be such that . The function is gH-differentiable if and only if and are differentiable real-valued functions. Furthermore,
Definition 9 (gradient of an interval-valued function). Let be an interval-valued function defined on , where is an open subset of . Let stand for the partial differentiation with respect to the th variable . Assume that and have continuous partial derivatives so that and are continuous. For , define We will say that is differentiable at , and we write We call the gradient of the interval-valued univex function at .
Example 10. Let defined by . So and . . Thus,
Remark 11. If , then of interval-valued functions is the extension of , where .
The concept of convexity plays an important role in the optimization theory. In recent years, the concept of convexity has been generalized in several directions by using novel and innovative techniques. An important generalization of convex functions is the introduction of univex functions, which was introduced by Bector et al. .
Let be a nonempty open set in , and let , , , and , . If the function is differentiable, then does not depend on ; see  or .
Definition 12. A differentiable real-valued function is said to be univex at with respect to if for all
Let be a nonempty open set in , and let be an interval-valued function, , and .
Definition 13 (interval-valued univex function). A differentiable interval-valued function is said to be univex at with respect to , , if for all
Remark 14. (i) An interval-valued univex function is the extension of a univex function by .
(ii) could be deduced from by .
Example 15. Consider the real-valued function given by , then we can obtain . If . Then
Example 16. Let , , , , then is univex with respect to , , and .
Example 17. Let , Let be defined by , then is univex with respect to and .
4. Optimality Criteria
Let . We say is an optimal solution of if for all -feasible . In this section, we obtain sufficient optimality conditions for a feasible solution to be efficient or properly efficient for in the form of the following theorems.
Theorem 18. Let be -feasible. Suppose that(i) there exist , , , , , such that for all feasible ;(ii) there exist such that Further, suppose that for all feasible . Then, is an optimal solution of ().
Proof. Let be -feasible. Then, From (20), we conclude that Thus, By (15) and Definition 2, we have From (17), It follows from Definition 2 that It is equivalent to From (16), we have From Definition 1, we have Thus, So, From (21), it follows that By (19), From Proposition 4, it follows that Therefore, is an optimal solution of .
Theorem 19. Let be -feasible. Suppose that(i)there exist , , , , , such that for all feasible ;(ii)there exist such that Further, suppose that for all feasible . Then, is an optimal solution of ().
Proof. Let be -feasible. Then, , from (41), we obtain that So, By (37), Thus, Then, we have From (38) and Definition 2, it follows that It is equivalent to Therefore, From Definition 2, we obtain that By (36), Then, from (40) and (42), we have From Proposition 4, it follows that Therefore, is an optimal solution of .
Consider the following:
Theorem 20 (Weak duality). Let be -feasible, and let be -feasible. Assume that there exist , , , , , such that at ; and . Then, .
Proof. It is similar to the proof of Theorem 18.
Theorem 21 (weak duality). Let be -feasible, and let be -feasible. Assume that there exist , , , , such that at ; Then, .
Proof. Since is -feasible, then , from (60) and (61),
Then, we have
Since is -feasible we can obtain that,
By Definition 2, it follows that Therefore, Then, By (57), From (59) and (61), thus,
Theorem 22 (strong duality). If is -optimal and a constraint qualification is satisfied at , then there exists such that is -feasible and the values of the objective functions for () and () are equal at and , respectively. Furthermore, if for all -feasible and -feasible , the hypotheses of Theorem 19 are satisfied, then is -optimal.
Proof. Since a constraint qualification is satisfied at , there exists such that the following Kuhn-Tucker conditions are satisfied:
Therefore, is -feasible.
Suppose that is not -optimal. Then, there exists a -feasible such that . This contradicts Theorem 20. Therefore, is -optimal.
6. Numerical Example
Consider the following example:
Note that the interval-valued objective function is univex with respect to , , , and every is univex with respect to , where and .
It is easy to see that the problem satisfies the assumptions of Theorem 18. Then,
After some algebraic calculations, we obtain that and . Therefore, is a solution.
This work is supported by the National Natural Science Foundation of China (Grant no. 60974082), and the Science Plan Foundation of the Education Bureau of Shaanxi Province (nos. 11JK1051, 2013JK1098, 2013JK1130, and 2013JK1182).
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