Journal of Applied Mathematics

Volume 2013, Article ID 527183, 5 pages

http://dx.doi.org/10.1155/2013/527183

## New Optimality Conditions for a Nondifferentiable Fractional Semipreinvex Programming Problem

^{1}Department of General Education, National Army Academy, Taoyuan 320, Taiwan^{2}Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

Received 29 July 2012; Accepted 1 January 2013

Academic Editor: Jen Chih Yao

Copyright © 2013 Yi-Chou Chen and Wei-Shih Du. 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.

#### Abstract

We study a nondifferentiable fractional programming problem as follows: subject to , where is a semiconnected subset in a locally convex topological vector space , , and , . If , , and , , are arc-directionally differentiable, semipreinvex maps with respect to a continuous map satisfying and , then the necessary and sufficient conditions for optimality of are established.

#### 1. Introduction

In recent years, there has been an increasing interest in studying the develpoment of optimality conditions for nondifferentiable multiobjective programming problems. Many authors established and employed some different Kuhn and Tucker type necessary conditions or other type necessary conditions to research optimal solutions; see [1–27] and references therein. In [7], Lai and Ho used the Pareto optimality condition to investigate multiobjective programming problems for semipreinvex functions. Lai [6] had obtained the necessary and sufficient conditions for optimality programming problems with semipreinvex assumptions. Some Pareto optimality conditions are established by Lai and Lin in [8]. Lai and Szilágyi [9] studied the programming with convex set functions and proved that the alternative theorem is valid for convex set functions defined on convex subfamily of measurable subsets in and showed that if the system has on solution, where stands for zero vector in a topological vector space, then there exists a nonzero continuous linear function such that In this paper, we study the following optimization problem: where is a semiconnected subset in a locally convex topological vector space , , and , , are functions satisfying some suitable conditions. The purpose of this study is dealt with such constrained fractional semipreinvex programming problem. Finally, we established the Fritz John type necessary and sufficient conditions for the optimality of a fractional semipreinvex programming problem.

#### 2. Preliminaries

Throughout this paper, we let be a locally convex topological vector space over the real field . Denote by the space of all linear operators from into .

Let be a nonempty convex subset of . Let be differentiable at . Then there is a linear operator , such that Recall that a function is called convex on , if or If is convex and differentiable at , then by (3) and (5), we have In 1981, Hanson [13, 14] introduced a generalized convexity on , so-called invexity; that is, is replaced by a vector in (6), or So an invex function is indeed a generalization of a convex differentiable function.

*Definition 1 (see [6]). *(1) A set is said to be semiconnected with respect to a given if

(2) A map is said to be semipreinvex on a semiconnected subset if each corresponds a vector such that
where stands for the zero vector of .

The following is an example of a bounded semiconnected set in , which is semiconnected with respect to a nontrivial .

*Example 2. *Let , and be bounded sets. Let be defined by
Then is a bound semiconnected set with respect to .

Theorem 3 (see [6, Theorem 2.2]). *Let be a semiconnected subset and a semipreinvex map. Then any local minimum of is also a global minimum of over .*

From the assumption in problem (9), there exists a positive number such that Consequently, we can reduce the problem (9) to an equivalent nonfractional parametric problem: where is a parameter.

We will prove that the problem is equivalent to the problem () for the optimal value . The following result is our main technique to derive the necessary and sufficient optimality conditions for problem .

Theorem 4. *Problem has an optimal solution with optimal value if and only if and is an optimal solution of . *

*Proof. *If is an optimal solution of with optimal value , that is,
It follows from (12) that
Thus, we have
Then, by (14), we get
Therefore, is an optimal solution of () and .

Conversely, if is an optimal solution of () with optimal value , then
So
It follows from (17) that
and hence
Therefore,
and we know is an optimal solution of with optimal value .

#### 3. The Existence of the Necessary and Sufficient Conditions for Semipreinvex Functions

*Definition 5 (see [6]). *A mapping is said to be arcwise directionally (in short, arc-directionally) differentiable at with respect to a continuous arc if for with
that is, the continuous function is differentiable from right at , and the limit

Note that the arc directional derivative is a mapping from into . Moreover, how can we make to be a semiconnected set? Indeed, we can construct a function concerned with defined as follows.

For any and , we choose a vector then Let , and , , be semipreinvex maps on a semiconnected subset in . Consider a constrained programming problem as .

The following Fritz John type theorem is essential in this section for programming problem .

Theorem 6 (Necessary Optimality Condition). *Suppose that , and , are arc-directionally differentiable at and semipreinvex on with respect to a continuous arc defined as in Definition 5. If minimizes locally for the semipreinvex programming problem , then there exist and such that
**
where and
*

*Proof. *By Theorem 4, the minimum solution to is also a minimum to (). Then is the local minimal solution to (). By Theorem 3, we have is the global minimal solution to (). It follows that the system
has no solution in , then we have
has no solution in for any . Thus for any ,
for some . Putting in (29), we get
Since and , it follows that
So (26) is proved.

As is a semiconnected set, for any and , we have
For , the point does not solve the system (27). So substituting in (29) and using the result (26), we obtain
Since and are arc-directionally differentiable with respect to , choose a vector as (23), so that (24) hold. It follows that if we divide (33) by and take the limit as , then we have
which proves (25) and the proof of theorem is completed.

Theorem 7 (Sufficient Optimality Condition). *Let , and , be arc-directionally differentiable at and semipreinvex on with respect to a continuous arc defined as in Definition 5. If there exist and satisfying
**
with and
**
then is an optimal solution for problem . *

*Proof. *Suppose to the contrary that is not optimal for problem and . Then . Therefore,
thus .

By Theorem 4, was not optimal for problem . Then there is an such that
for . Moreover, we have
for any . Thus
Since the semi-preinvex maps , and , are arc-directionally differentiable, it follows that for there corresponds a vector such that
and so
Letting , we have and the last inequalities imply
Consequently, from (41) and (44), we obtain
which contradicts the fact of (35). Therefore is an optimal solution of problem .

Since any global minimal is a local minimal, applying Theorems 6 and 7, we can obtain the necessary and sufficient conditions for problem .

Theorem 8. * Suppose that , and , are arc-directionally differentiable at at and semi-preinvex on with respect to a continuous arc defined as in Definition 5. If minimizes globally for the semi-preinvex programming problem if and only if there exists , , such that
**
where and
*

*Remark 9. * Our results also hold for preinvex functions.

#### Acknowledgments

The research of Wei-Shih Du was supported partially under Grant no. NSC 101-2115-M-017-001 by the National Science Council of the Republic of China.

#### References

- B. D. Craven, “Invex functions and constrained local minima,”
*Bulletin of the Australian Mathematical Society*, vol. 24, no. 3, pp. 357–366, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. D. Craven and B. M. Glover, “Invex functions and duality,”
*Australian Mathematical Society*, vol. 39, no. 1, pp. 1–20, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. H. Clarke,
*Optimization and Nonsmooth Analysis*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983. View at MathSciNet - F. H. Clarke,
*Optimization and Nonsmooth Analysis*, John Wiley & Sons, New York, NY, USA, 2nd edition, 1983. - H. Dietrich, “On the convexification procedure for nonconvex and nonsmooth infinite-dimensional optimization problems,”
*Journal of Mathematical Analysis and Applications*, vol. 161, no. 1, pp. 28–34, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-C. Lai, “Optimality conditions for semi-preinvex programming,”
*Taiwanese Journal of Mathematics*, vol. 1, no. 4, pp. 389–404, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. C. Lai and C. P. Ho, “Duality theorem of nondifferentiable convex multiobjective programming,”
*Journal of Optimization Theory and Applications*, vol. 50, no. 3, pp. 407–420, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-C. Lai and L.-J. Lin, “Moreau-Rockafellar type theorem for convex set functions,”
*Journal of Mathematical Analysis and Applications*, vol. 132, no. 2, pp. 558–571, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. C. Lai and P. Szilágyi, “Alternative theorems and saddlepoint results for convex programming problems of set functions with values in ordered vector spaces,”
*Acta Mathematica Hungarica*, vol. 63, no. 3, pp. 231–241, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. C. Lai and J. C. Liu, “Minimax fractional programming involving generalised invex functions,”
*The ANZIAM Journal*, vol. 44, no. 3, pp. 339–354, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-C. Chen and H.-C. Lai, “Fractional programming for variational problem with $F,\rho ,\theta $-invexity,”
*Journal of Nonlinear and Convex Analysis*, vol. 4, no. 1, pp. 25–41, 2003. View at Google Scholar · View at MathSciNet - J. C. Liu, “Optimality and duality for generalized fractional programming involving nonsmooth ($F,\rho $)-convex functions,”
*Computers and Mathematics with Applications*, vol. 42, pp. 437–446, 1990. View at Google Scholar - M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 80, no. 2, pp. 545–550, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Hanson and B. Mond, “Necessary and sufficient conditions in constrained optimization,”
*Mathematical Programming*, vol. 37, no. 1, pp. 51–58, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. K. Mishra and G. Giorgi,
*Invexity and Optimization*, vol. 88 of*Nonconvex Optimization and Its Applications*, Springer, Berlin, Germnay, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - S. K. Mishra, S. Wang, and K. K. Lai,
*V-Invex Functions and Vector Optimization*, vol. 14 of*Springer Optimization and Its Applications*, Springer, New York, NY, USA, 2008. View at MathSciNet - S. K. Mishra, S.-Y. Wang, and K. K. Lai,
*Generalized Convexity and Vector Optimization*, vol. 90 of*Nonconvex Optimization and Its Applications*, Springer, Berlin, Germnay, 2009. View at MathSciNet - S. K. Mishra and V. Laha, “On approximately star-shaped functions and approximate vector variational inequalities,”
*Journal of Optimization Theory and Applications*. In press. View at Publisher · View at Google Scholar - S. K. Mishra and V. Laha, “On V-r-invexity and vector variational Inequalities,”
*Filomat*, vol. 26, no. 5, pp. 1065–1073, 2012. View at Google Scholar - S. K. Mishra and B. B. Upadhyay, “Nonsmooth minimax fractional programming involving $\eta $-pseudolinear functions,”
*Optimization*. In press. View at Publisher · View at Google Scholar - S. K. Mishra and M. Jaiswal, “Optimality conditions and duality for nondifferentiable multiobjective semi-infinite programming,”
*Vietnam Journal of Mathematics*, vol. 40, pp. 331–343, 2012. View at Google Scholar - N. G. Rueda and M. A. Hanson, “Optimality criteria in mathematical programming involving generalized invexity,”
*Journal of Mathematical Analysis and Applications*, vol. 130, no. 2, pp. 375–385, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. W. Reiland, “Nonsmooth invexity,”
*Bulletin of the Australian Mathematical Society*, vol. 12, pp. 437–446, 1990. View at Google Scholar - T. Weir, “Programming with semilocally convex functions,”
*Journal of Mathematical Analysis and Applications*, vol. 168, no. 1, pp. 1–12, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Weir and V. Jeyakumar, “A class of nonconvex functions and mathematical programming,”
*Bulletin of the Australian Mathematical Society*, vol. 38, no. 2, pp. 177–189, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Antczak, “(
*p, r*)-invex sets and functions,”*Journal of Mathematical Analysis and Applications*, vol. 263, no. 2, pp. 355–379, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. L. Ye, “
*D*-invexity and optimality conditions,”*Journal of Mathematical Analysis and Applications*, vol. 162, no. 1, pp. 242–249, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet