Second-order duality for nondifferentiable minimax fractional programming problems with generalized convexity
© Khan; licensee Springer. 2013
Received: 29 June 2013
Accepted: 12 September 2013
Published: 8 November 2013
In the present paper, we are concerned with second-order duality for nondifferentiable minimax fractional programming under the second-order generalized convexity type assumptions. The weak, strong and converse duality theorems are proved. Results obtained in this paper extend some previously known results on nondifferentiable minimax fractional programming in the literature.
MSC:90C32, 49K35, 49N15.
Keywordsminimax programming fractional programming duality generalized convexity
It is well known that the minimax fractional programming has wide applications. These types of problems arise in the design of electronic circuits; moreover, minimax fractional programming problems appear in formulation of discrete and continuous rational approximation problems with respect to the Chebyshev norm , continuous rational games , multiobjective programming  and engineering design as well as some portfolio solution problems discussed by Bajaona-Xandari and Martinez-Legaz .
In the last few years, much attention has been paid to optimality conditions and duality theorems for the minimax fractional programming problems. For the case of convex differentiable minimax fractional programming, Yadav and Mukherjee  formulated two dual models for the primal problem and derived a duality theorem for convex differentiable minimax fractional programming. A step forward was taken by Chandra and Kumar  who improved the dual formulation of Yadav and Mukherjee. They provided two modified dual problems for minimax fractional programming and proved duality results. Liu and Wu [7, 8] and Ahmad  obtained sufficient optimality conditions and duality for minimax fractional programming under generalized convex type assumptions.
Mangasarian  introduced the notion of second-order duality for nonlinear programs and obtained second-order duality results under certain inequalities. Mond  modified the second-order duality results assuming rather simple inequalities. In this continuation, Bector and Chandra  formulated a second-order dual for a fractional programming problem and obtained usual duality results under the assumptions  by naming these as convex/concave functions. Recently, Ahmad  has formulated two types of second-order dualities for minimax fractional programming problems and derived weak, strong and strict converse duality theorems under generalized convexity type assumptions. He raised a question as to whether the second-order duality results developed in  hold for nondifferentiable minimax fractional programming problems. In the present paper, a positive answer is given to the question of Ahmad  and a second-order duality for nondifferentiable minimax fractional programming is formulated. The weak, strong and strict converse duality theorems are proved for these programs under the second-order generalized convexity type assumptions.
where Y is a compact subset of , , are functions. C and D are positive semidefinite symmetric matrices. Throughout this paper, we assume that and for all .
2 Notations and preliminaries
Generalized Schwarz inequality
, , .
Definition 2.2 A point is said to be an optimal solution of (NFP) if for each .
In the case where the functions f, g and h in problem (NFP) are continuously differentiable with respect to , Lai et al.  proved the following first-order necessary conditions for optimality of (NFP), which will be required to prove the strong duality theorem.
Theorem 1 (Necessary condition)
Throughout the paper, we assume that ℱ is a sublinear functional. For , let , , ρ, , be real numbers, and let .
3 Parametric nondifferentiable fractional duality
If, for a triplet , the set , then we define the supremum over it to be −∞.
Theorem 2 (Weak duality)
for all .
which contradicts (7), since . □
Theorem 3 (Strong duality)
Let be an optimal solution of (NFP), and let , be linearly independent. Then there exist and such that is a feasible solution of (FD). In addition, if the hypotheses of the weak duality theorem are satisfied for all feasible solutions of (FD), then is an optimal solution of (FD), and the two objectives have the same optimal values.
Proof Since is an optimal solution of (NFP) and , are linearly independent then, by Theorem 1, there exist and such that is a feasible solution of (FD) and the two objectives have the same values. Optimality of for (FD) thus follows from the weak duality theorem (Theorem 2). □
Theorem 4 (Strict converse duality)
Then , that is, is an optimal solution of (NFP).
for all , .
which contradicts (7) since . □
4 Conclusion and further development
In this paper, weak, strong and strict converse duality theorems have been discussed for nondifferentiable minimax fractional programming problems in the framework of generalized convexity type assumptions. This paper has generalized the results of Ahmad .
where , for , , are analytic with respect to W, W is a specified compact subset in , S is a polyhedral cone in , and is analytic. Also, are positive semidefinite Hermitian matrices.
The work is supported by the Deanship of Scientific Research, University of Tabuk, K.S.A.
- Barrodale I: Best rational approximation and strict quasiconvexity. SIAM J. Numer. Anal. 1973, 10: 8–12. 10.1137/0710002MathSciNetView ArticleMATHGoogle Scholar
- Schroeder RG: Linear programming solutions to ratio games. Oper. Res. 1970, 18: 300–305. 10.1287/opre.18.2.300MathSciNetView ArticleMATHGoogle Scholar
- Soyster A, Lev B, Loof D: Conservative linear programming with mixed multiple objectives. Omega 1977, 5: 193–205. 10.1016/0305-0483(77)90102-5View ArticleGoogle Scholar
- Bajona-Xandri C, Martinez-Legaz JE: Lower subdifferentiability in minimax programming involving type-I functions. J. Comput. Appl. Math. 2008, 215: 91–102. 10.1016/j.cam.2007.03.022MathSciNetView ArticleMATHGoogle Scholar
- Yadav SR, Mukherjee RN: Duality for fractional minimax programming problems. J. Aust. Math. Soc. Ser. B, Appl. Math 1990, 31: 484–492. 10.1017/S0334270000006809MathSciNetView ArticleMATHGoogle Scholar
- Chandra S, Kumar V: Duality in fractional minimax programming. J. Aust. Math. Soc. A 1995, 58: 376–386. 10.1017/S1446788700038362MathSciNetView ArticleMATHGoogle Scholar
- Liu JC, Wu CS: On minimax fractional optimality conditions with invexity. J. Math. Anal. Appl. 1998, 219: 21–35. 10.1006/jmaa.1997.5786MathSciNetView ArticleMATHGoogle Scholar
- Liu JC, Wu CS:On minimax fractional optimality conditions -invexity. J. Math. Anal. Appl. 1998, 219: 36–51. 10.1006/jmaa.1997.5785MathSciNetView ArticleGoogle Scholar
- Ahmad I: Optimality conditions and duality in fractional minimax programming involving generalized ρ -invexity. Int. J. Stat. Manag. Syst. 2003, 19: 165–180.Google Scholar
- Mangasarian OL: Second and higher order duality in nonlinear programming. J. Math. Anal. Appl. 1975, 51: 607–620. 10.1016/0022-247X(75)90111-0MathSciNetView ArticleGoogle Scholar
- Mond B: Second order duality for nonlinear programs. Opsearch 1974, 11(2–3):90–99.MathSciNetGoogle Scholar
- Bector CR, Chandra S: Generalized convexity and higher order duality for fractional programming. Opsearch 1975, 24(3):143–154.MathSciNetMATHGoogle Scholar
- Ahmad I: On second-order duality for minimax fractional programming problems with generalized convexity. Abstr. Appl. Anal. 2011. 10.1155/2011/563924Google Scholar
- Lai HC, Liu JC, Tanaka K: Necessary and sufficient conditions for minimax fractional programming. J. Math. Anal. Appl. 1999, 230(2):311–328. 10.1006/jmaa.1998.6204MathSciNetView ArticleMATHGoogle Scholar
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