A fuzzy semi-infinite optimization problem
© Megahed; licensee Springer. 2014
Received: 11 July 2014
Accepted: 5 November 2014
Published: 19 November 2014
In this paper, we present a fuzzy semi-infinite optimization problem. Moreover, we will deduce the Fritz-John and Kuhn-Tucker necessary conditions of this problem. Finally, a numerical example is given to illustrate the results.
MSC: 90C34, 90C05, 90C70, 90C30, 90C46.
In many practical problems, we might have information containing some uncertainty, which is treated in this paper as fuzzy information; the considered semi-infinite optimization problem with fuzzy information is a fuzzy semi-infinite optimization problem.
Fuzzy set theory was introduced into conventional linear programming by Zimmermann , fuzzy mathematical programming was presented in , and fuzzy programming and linear programming with several objective functions were presented by .
Optimality conditions of a nonlinear programming problem with fuzzy parameters are established, and a fuzzy function is defined, with its differentiability, convexity, and some important properties being studied in ; the fuzzy solution of optimization problems and the incentive solution of the optimization problems are presented which explain that the solution of optimization problems is a generalization of the solutions in the case of crisp optimization problems . As regards fuzzy mathematical programming: theory, application, and an extension are presented in .
This paper is organized as follows: In Section 2, the formulation of the problem of semi-infinite optimization is considered. In Section 3, the main section of the paper, we will study a fuzzy semi-infinite optimization problem, and the Fritz-John and Kuhn-Tucker necessary conditions. Finally, the conclusion is drawn in Section 5.
2 A semi-infinite programming problem
I, K, and L are finite index sets with and (where denotes the cardinality), all appearing functions are real valued and continuously differentiable, and the set is compact for each , and the set-valued mapping is upper semi-continuous at each .
For the special case that the set does not depend on the variable x, this problem is a common semi-infinite problem (SIP). The generalized semi-infinite and bi-level optimization problem are presented by Stein and Still . Bi-level problems are of the following form.
The generalized semi-infinite programming on generic local minimizers was introduced by Gunzel et al. . The feasible set in generalized semi-infinite optimization is presented by Jongen et al. in . Furthermore, the linear and linearized generalized semi-infinite optimization problems were introduced by Rukmann . In , a first-order optimality condition in generalized semi-infinite programming is introduced. Still discussed the optimality conditions for generalized semi-infinite programming problems in .
3 A fuzzy semi-infinite programming problem
3.1 Problem formulation
3.2 Lower level problem
If and , then is a minimizer of the problem (3.5).
In other words, for and the set is nonempty and, furthermore, is also compact.
4 Fritz-John conditions
Firstly, we will give some lemmas, definitions, and a proposition which will be used in the proof of the Fritz-John conditions.
Lemma 2 
Let , and . Then the set is bounded, and whenever .
Lemma 3 
Let , and for let . Then is an interior point of M.
Lemma 5 
Let . Then the set is compact.
Definition 6 
i.e. consists of all finite convex combinations of the elements of V.
Lemma 7 
Let be a nonempty compact set. Then there exists a with for all if and only if .
Lemma 8 
- (i)There are real numbers , , , , satisfying(4.3)
- (ii)The set is linearly independent and there exists a with(4.4)
Proposition 9 
The set () is linearly independent.
- (ii)There is a satisfying(4.5)
Proof Let be a local minimizer of the problem (4.7). We distinguish three cases.
Case 1: The set is linearly dependent. Then we are done by choosing , , in (4.9) (if ), or , in (4.10) (if ) as well as a linear combination with .
Case 3: Neither Case 1 nor Case 2 holds. Then the proof is similar to Theorem 1.1 in . □
5 A constraint qualification
the set is linearly independent and
- (2)there exists a such that(5.1)
The proof is similar to the proof of Theorem 10 if we choose .
The optimal solution of the crisp problem is , , and ; and , .
In this work, we discussed a fuzzy semi-infinite optimization problem, by considering that the minimum of the objective function is fuzzy (). The Fritz-John conditions and the constraint qualification are discussed for this problem. Finally, an illustrative example is given to clarify the results.
The author wants to express his deep thanks and his respect to his faculty, colleagues, the Journal, and Prof. Dr. Rachel M Bernales of the Journal of Editorial office. Also the author wants to express his thanks and respect to Jane Doe who provided medical writing services on behalf of XYZ pharmaceuticals Ltd.
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