Non-differentiable minimax fractional programming with higher-order type Ifunctions
© Muhiuddin et al.; licensee Springer. 2013
Received: 25 May 2013
Accepted: 9 September 2013
Published: 7 November 2013
In this article, we are concerned with a class of non-differentiable minimaxfractional programming problems and their higher-order dual model. Weak, strongand converse duality theorems are discussed involving generalized higher-ordertype I functions. The presented results extend some previously known results onnon-differentiable minimax fractional programming.
In nonlinear optimization, problems, where minimization and maximization process areperformed together, are called minimax (minmax) problems. Frequently, problems ofthis type arise in many areas like game theory, Chebychev approximation, economics,financial planning and facility location .
The optimization problems in which the objective function is a ratio of two functionsare commonly known as fractional programming problems. In the past few years, manyauthors have shown interest in the field of minimax fractional programming problems.Schmittendorf  first developed necessary and sufficient optimality conditions for aminimax programming problem. Tanimoto  applied the necessary conditions in  to formulate a dual problem and discussed the duality results, which wereextended to a fractional analogue of the problem considered in [2, 3] by several authors [4–10]. Liu  proposed the second-order duality theorems for a minimax programmingproblem under generalized second-order B-invex functions. Husain etal. formulated two types of second-order dual models for minimax fractionalprogramming and derived weak, strong and converse duality theorems underη-convexity assumptions.
where Y is a compact subset of , and are twice differentiable functions. B is an positive semidefinite symmetric matrices. Ahmadet al. formulated a unified higher-order dual of (P) and established appropriateduality theorems under higher-order -type I assumptions. Recently, Jayswal andStancu-Minasian  obtained higher-order duality results for (P).
In this paper, we formulate a higher-order dual for a non-differentiable minimaxfractional programming problem and establish weak, strong and strict converseduality theorems under generalized higher-order -type I assumptions. This paper generalizes severalresults that have appeared in the literature [11, 12, 14–22] and references therein.
where Y is a compact subset of , and are differentiable functions. B andC are positive semidefinite symmetric matrices. It isassumed that for each in , and .
Lemma 2.1 (Generalized Schwarz inequality)
Let ℱ be a sublinear functional, and let . Let , where and , and let . Let , , and , , be differentiable functions at.
, , , and .
From (ii), it is clear that .
then we say that is higher-order -strictly pseudoquasi-type I at.
If the functions f, g and h in problem (NP) arecontinuously differentiable with respect to , then Liu  derived the following necessary conditions for optimalityof (NP).
Theorem 2.1 (Necessary conditions)
If in addition, we insert , then the results of Theorem 2.1 still hold.
3 Higher-order non-differentiable fractional duality
where , , , with and if . If for a triplet , the set , then we define the supremum over it to be∞.
Theorem 3.1 (Weak duality)
which contradicts (3.1), as . □
Theorem 3.2 (Strong duality)
Then there existandsuch thatis a feasible solution of (ND) and the two objectives have the samevalues. Furthermore, if the assumptions of weak duality(Theorem 3.1) hold for all feasible solutions of (NP)and (ND), thenis an optimal solution of (ND).
Theorem 3.3 (Strict converse duality)
and that, are linearly independent. Then; that is, is an optimal solution of (NP).
which contradicts (3.11). Hence the result. □
4 Special cases
The notion of higher-order -pseudoquasi-type I is adopted, which includes manyother generalized convexity concepts in mathematical programming as special cases.This concept is appropriate to discuss the weak, strong and strict converse dualitytheorems for a higher-order dual (ND) of a non-differentiable minimax fractionalprogramming problem (NP). The results of this paper can be discussed by formulatinga unified higher-order dual involving support functions on the lines of Ahmad .
This work was partially supported by the Deanship of Scientific Research Unit,University of Tabuk, Tabuk, Kingdom of Saudi Arabia. The authors are grateful tothe anonymous referee for a careful checking of the details and for helpfulcomments that improved this paper.
- Du D, Pardalos PM, Wu WZ: Minimax and Applications. Kluwer Academic, Dordrecht; 1995.MATHView ArticleGoogle Scholar
- Schmitendorff WE: Necessary conditions and sufficient conditions for static minimaxproblems. J. Math. Anal. Appl. 1977, 57: 683–693. 10.1016/0022-247X(77)90255-4MathSciNetView ArticleGoogle Scholar
- Tanimoto S: Duality for a class of nondifferentiable mathematical programmingproblems. J. Math. Anal. Appl. 1981, 79: 283–294.MathSciNetView ArticleGoogle Scholar
- Ahmad I, Husain Z: Optimality conditions and duality in nondifferentiable minimax fractionalprogramming with generalized convexity. J. Optim. Theory Appl. 2006, 129: 255–275. 10.1007/s10957-006-9057-0MATHMathSciNetView ArticleGoogle Scholar
- Chandra S, Kumar V: Duality in fractional minimax programming. J. Aust. Math. Soc. A 1995, 58: 376–386. 10.1017/S1446788700038362MATHMathSciNetView ArticleGoogle Scholar
- Lai HC, Liu JC, Tanaka K: Necessary and sufficient conditions for minimax fractional programming. J. Math. Anal. Appl. 1999, 230: 311–328. 10.1006/jmaa.1998.6204MATHMathSciNetView ArticleGoogle Scholar
- Liu JC, Wu CS: On minimax fractional optimality conditions and invexity. J. Math. Anal. Appl. 1998, 219: 21–35. 10.1006/jmaa.1997.5786MATHMathSciNetView ArticleGoogle Scholar
- Liu JC, Wu CS:On minimax fractional optimality conditions with -convexity. J. Math. Anal. Appl. 1998, 219: 36–51. 10.1006/jmaa.1997.5785MATHMathSciNetView ArticleGoogle Scholar
- Yadav SR, Mukherjee RN: Duality in fractional minimax programming problems. J. Aust. Math. Soc. A 1990, 31: 484–492.MATHMathSciNetView ArticleGoogle Scholar
- Yang XM, Hou SH: On minimax fractional optimality conditions and duality with generalizedconvexity. J. Glob. Optim. 2005, 31: 235–252. 10.1007/s10898-004-5698-4MATHMathSciNetView ArticleGoogle Scholar
- Liu JC: Second-order duality for minimax programming. Util. Math. 1999, 56: 53–63.MATHMathSciNetGoogle Scholar
- Husain Z, Ahmad I, Sharma S: Second order duality for minimax fractional programming. Optim. Lett. 2009, 3: 277–286. 10.1007/s11590-008-0107-4MATHMathSciNetView ArticleGoogle Scholar
- Ahmad I, Husain Z, Sharma S: Second-order duality in nondifferentiable minmax programming involving type Ifunctions. J. Comput. Appl. Math. 2008, 215: 91–102. 10.1016/j.cam.2007.03.022MATHMathSciNetView ArticleGoogle Scholar
- Husain Z, Jayswal A, Ahmad I: Second-order duality for nondifferentiable minimax programming withgeneralized convexity. J. Glob. Optim. 2009, 44: 509–608. 10.1007/s10898-008-9354-2MathSciNetView ArticleGoogle Scholar
- Ahmad I, Husain Z, Sharma S: Higher-order duality in nondifferentiable minimax programming withgeneralized type I functions. J. Optim. Theory Appl. 2009, 141: 1–12. 10.1007/s10957-008-9474-3MATHMathSciNetView ArticleGoogle Scholar
- Jayswal A, Stancu-Minasian I: Higher-order duality in nondifferentiable minimax programming problem withgeneralized convexity. Nonlinear Anal., Theory Methods Appl. 2011, 74: 616–625. 10.1016/j.na.2010.09.016MATHMathSciNetView ArticleGoogle Scholar
- Ahmad I: Second order nondifferentiable minimax fractional programming with squareroot terms. Filomat 2013, 27: 126–133.Google Scholar
- Ahmad I: Higher-order duality in nondifferentiable minimax fractional programminginvolving generalized convexity. J. Inequal. Appl. 2012., 2012: Article ID 306Google Scholar
- Ahmad I, Husain Z: Duality in nondifferentiable minimax fractional programming with generalizedconvexity. Appl. Math. Comput. 2006, 176: 545–551. 10.1016/j.amc.2005.10.002MATHMathSciNetView ArticleGoogle Scholar
- Ahmad I, Gupta SK, Kailey N, Agarwal RP: Duality in nondifferentiable minimax fractional programming with B - -invexity. J. Inequal. Appl. 2011., 2011: Article ID 75Google Scholar
- Gupta SK, Dangar D: On second-order duality for nondifferentiable minimax fractionalprogramming. J. Comput. Appl. Math. 2014, 255: 878–886.MATHMathSciNetView ArticleGoogle Scholar
- Hu Q, Chen Y, Jian J: Second-order duality for nondifferentiable minimax fractional programming. Int. J. Comput. Math. 2012, 89: 11–16. 10.1080/00207160.2011.631529MATHMathSciNetView ArticleGoogle Scholar
- Ahmad I: Unified higher order duality in nondifferentiable multiobjectiveprogramming. Math. Comput. Model. 2012, 55: 419–425. 10.1016/j.mcm.2011.08.020MATHView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.