Non-differentiable minimax fractional programming with higher-order type Ifunctions
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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.
KeywordsFeasible Solution Duality Theorem Duality Result Sufficient Optimality Condition Weak Duality
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, 5, 6, 7, 8, 9, 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, 15, 16, 17, 18, 19, 20, 21, 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.
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