Variational Inequalities and Vector OptimizationView this Special Issue
Research Article | Open Access
New Optimality Conditions for a Nondifferentiable Fractional Semipreinvex Programming Problem
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.
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 , Lai and Ho used the Pareto optimality condition to investigate multiobjective programming problems for semipreinvex functions. Lai  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 . Lai and Szilágyi  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.
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 ). (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 .
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 .
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 ). 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.
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,
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 .
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.
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.
- B. D. Craven, “Invex functions and constrained local minima,” Bulletin of the Australian Mathematical Society, vol. 24, no. 3, pp. 357–366, 1981.
- B. D. Craven and B. M. Glover, “Invex functions and duality,” Australian Mathematical Society, vol. 39, no. 1, pp. 1–20, 1985.
- F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983.
- 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.
- H.-C. Lai, “Optimality conditions for semi-preinvex programming,” Taiwanese Journal of Mathematics, vol. 1, no. 4, pp. 389–404, 1997.
- 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.
- 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.
- 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.
- 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.
- J.-C. Chen and H.-C. Lai, “Fractional programming for variational problem with -invexity,” Journal of Nonlinear and Convex Analysis, vol. 4, no. 1, pp. 25–41, 2003.
- J. C. Liu, “Optimality and duality for generalized fractional programming involving nonsmooth ()-convex functions,” Computers and Mathematics with Applications, vol. 42, pp. 437–446, 1990.
- M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545–550, 1981.
- M. A. Hanson and B. Mond, “Necessary and sufficient conditions in constrained optimization,” Mathematical Programming, vol. 37, no. 1, pp. 51–58, 1987.
- S. K. Mishra and G. Giorgi, Invexity and Optimization, vol. 88 of Nonconvex Optimization and Its Applications, Springer, Berlin, Germnay, 2008.
- 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.
- 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.
- S. K. Mishra and V. Laha, “On approximately star-shaped functions and approximate vector variational inequalities,” Journal of Optimization Theory and Applications. In press.
- S. K. Mishra and V. Laha, “On V-r-invexity and vector variational Inequalities,” Filomat, vol. 26, no. 5, pp. 1065–1073, 2012.
- S. K. Mishra and B. B. Upadhyay, “Nonsmooth minimax fractional programming involving -pseudolinear functions,” Optimization. In press.
- 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.
- 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.
- T. W. Reiland, “Nonsmooth invexity,” Bulletin of the Australian Mathematical Society, vol. 12, pp. 437–446, 1990.
- T. Weir, “Programming with semilocally convex functions,” Journal of Mathematical Analysis and Applications, vol. 168, no. 1, pp. 1–12, 1992.
- 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.
- T. Antczak, “(p, r)-invex sets and functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 355–379, 2001.
- Y. L. Ye, “D-invexity and optimality conditions,” Journal of Mathematical Analysis and Applications, vol. 162, no. 1, pp. 242–249, 1991.
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.