# Non-differentiable minimax fractional programming with higher-order type Ifunctions

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## Abstract

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.

## Keywords

Feasible Solution Duality Theorem Duality Result Sufficient Optimality Condition Weak Duality
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## 1 Introduction

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 [1].

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 [2] first developed necessary and sufficient optimality conditions for aminimax programming problem. Tanimoto [3] applied the necessary conditions in [2] 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 [11] proposed the second-order duality theorems for a minimax programmingproblem under generalized second-order B-invex functions. Husain etal.[12] formulated two types of second-order dual models for minimax fractionalprogramming and derived weak, strong and converse duality theorems underη-convexity assumptions.

Ahmad et al.[13] and Husain et al.[14] discussed the second-order duality results for the followingnon-differentiable minimax programming problem:
(P)

where Y is a compact subset of , and are twice differentiable functions. B is an positive semidefinite symmetric matrices. Ahmadet al.[15] formulated a unified higher-order dual of (P) and established appropriateduality theorems under higher-order -type I assumptions. Recently, Jayswal andStancu-Minasian [16] 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.

## 2 Preliminaries

The problem to be considered in the present analysis is the followingnon-differentiable minimax fractional problem:
(NP)

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 .

Let denote the set of all feasible solutions of (NP). Anypoint is called the feasible point of (NP). For each, we define
such that for each ,
For each , we define
where
Since f and g are continuously differentiable and Y iscompact in , it follows that for each , , and for any , we have

Lemma 2.1 (Generalized Schwarz inequality)

Let A be a positive-semidefinite matrix of order n. Then, for all,
(2.1)
The equalityholds for some. Clearly, if, we have

Let ℱ be a sublinear functional, and let . Let , where and , and let . Let , , and , , be differentiable functions at.

Definition 2.1 A functional is said to be sublinear in its third argument if forall ,
1. (i)

, ;

2. (ii)

, , , and .

From (ii), it is clear that .

Definition 2.2[15]

For each , is said to be higher-order -pseudoquasi-type I at if for all , and ,
In the above definition, if

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 [11] derived the following necessary conditions for optimalityof (NP).

Theorem 2.1 (Necessary conditions)

Ifis a solution of (NP) satisfying, , and, are linearly independent, then there exist, , , andsuch that
In the above theorem, both matrices B and C are positivesemidefinite. If either or is zero, then the functions involved in the objectivefunction of problem (NP) are not differentiable. To derive these necessaryconditions under this situation, for , we define
1. (i)
,

2. (ii)
,

3. (iii)
,

4. (iv)
,

If in addition, we insert , then the results of Theorem 2.1 still hold.

## 3 Higher-order non-differentiable fractional duality

In this section, we consider the following dual problem to (NP):
(ND)
where denotes the set of all satisfying
(3.1)
(3.2)
(3.3)
(3.4)

where , , , with and if . If for a triplet , the set , then we define the supremum over it to be∞.

Theorem 3.1 (Weak duality)

Let x andbe feasible solutions of (NP) and (ND), respectively.Suppose that
is higher-order-pseudoquasi-type I at z and
Then
Proof Suppose to the contrary that
Then we have
It follows from , , that
with at least one strict inequality, since . Taking summation over i and using, we have
It follows from the generalized Schwarz inequality and (3.4) that
(3.5)
By the feasibility of x for (NP) and , we obtain
(3.6)
The above inequality with (3.5) gives
(3.7)
From (3.2) and (3.7), we have
(3.8)
Also, from (3.3), we have
(3.9)
The higher second-order -pseudoquasi-type I assumption on
at z, with (3.8) and (3.9), implies
By using , , and the sublinearity of ℱ in the aboveinequalities, we summarize to get
Since , therefore

which contradicts (3.1), as . □

Theorem 3.2 (Strong duality)

Letbe an optimal solution of (NP) and let, be linearly independent. Assume that

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).

Proof Since is an optimal solution of (NP) and, are linearly independent, by Theorem 2.1, thereexist and such that is a feasible solution of (ND) and problems (NP) and(ND) have the same objectives values and

□

Theorem 3.3 (Strict converse duality)

Letandbe the optimal solutions of (NP) and (ND), respectively.Suppose that
is higher-order-strictly pseudoquasi-type I atwith

and that, are linearly independent. Then; that is, is an optimal solution of (NP).

Proof We assume that and reach a contradiction. From the strong dualitytheorem (Theorem 3.2), it follows that
(3.10)
Now, proceeding as in Theorem 3.1, we get
(3.11)
The feasibility of for (NP), and (3.3) imply
which along with the second part of higher-order -strictly pseudoquasi-type I assumption on
at gives
As and as ℱ is sublinear, it follows that
(3.12)
From (3.1), (3.12) and the sublinearity of ℱ, we have
In view of , and the sublinearity of ℱ, the above inequalitybecomes
By using the first part of the said assumption imposed on
at , it follows that

which contradicts (3.11). Hence the result. □

## 4 Special cases

Let , , , and , . Then (ND) becomes the second-order dual studied in [17, 22]. If, in addition, , then we obtain the dual formulated by Ahmad etal.[20].

## 5 Conclusion

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 [23].

## Notes

### Acknowledgements

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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