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Higher-order duality in nondifferentiable minimax fractional programming involving generalized convexity
Journal of Inequalities and Applications volume 2012, Article number: 306 (2012)
A higher-order dual for a non-differentiable minimax fractional programming problem is formulated. Using the generalized higher-order η-convexity assumptions on the functions involved, weak, strong and strict converse duality theorems are established in order to relate the primal and dual problems. Results obtained in this paper naturally unify and extend some previously known results on non-differentiable minimax fractional programming in the literature.
MSC:26A51, 90C32, 49N15.
The problem to be considered in the present analysis is the following nondifferentiable minimax fractional problem:
where Y is a compact subset of , , , and are continuously differentiable functions. B and C are positive semidefinite symmetric matrices. It is assumed that for each in , and .
In an earlier work, Schmittendorf  established necessary and sufficient optimality conditions for the following minimax programming problem:
where Y is a compact subset of , the functions , and are in .
Tanimoto  applied the necessary conditions in  to formulate a dual problem and discussed the duality results, which were extended to fractional analogue of (P1) by several authors [3–12]. Liu  proposed the second-order duality theorems for (P1) under generalized second-order B-invex functions. Mishra and Rueda  and Ahmad, Husain and Sarita  discussed the second-order duality results for the following nondifferentiable minimax programming problems:
where Y is a compact subset of , , and are twice differentiable functions. B is an positive semidefinite symmetric matrix. Ahmad, Husain and Sharma  formulated a unified higher-order dual of (P2) and established weak, strong and strict converse duality theorems under higher-order -Type I assumptions. Recently, Jayswal and Stancu-Minasian  obtained higher-order duality results for (P2).
In this paper, we formulate a higher-order dual of (P) and establish weak, strong and strict converse duality theorems under generalized higher-order η-convexity assumptions. More precisely, this paper is an extension of second-order duality results of Hu, Chen and Jian  to a class of higher-order duality and it also presents an answer to a question raised in .
2 Notation and preliminaries
Let denote the set of all feasible solutions of (NFP). Any point is called a feasible point of (NFP). For each , we define
such that for each ,
For each , we define
Since f and g are continuously differentiable and Y is compact in , it follows that for each , , and for any , we have a positive constant
Lemma 2.1 (Generalized Schwarz inequality)
Let A be a positive-semidefinite matrix of order n. Then, for all ,
The equality holds for some . Clearly, if , we have
We will use the following definitions.
Let and be differentiable functions.
Definition 2.1 
A function ϕ is said to be higher-order η-convex if there exists a certain mapping such that for all , we have
Definition 2.2 
A function ϕ is said to be higher-order (strictly) η-pseudoconvex if there exists a certain mapping such that for all , we have
Definition 2.3 
A function ϕ is said to be higher-order (strictly) η-quasiconvex if there exists a certain mapping such that for all , we have
3 Higher-order duality model
In this section, we formulate the higher-order dual for (NFP) and derive duality results.
where denotes the set of all subject to
where , with and if . If for a triplet , the set , then we define the supremum over it to be ∞.
(ii) If , then the above dual becomes the dual formulated in .
Theorem 3.1 (Weak duality)
Let x and be feasible solutions of (NFP) and (NMD) respectively. Assume that
is higher-order η-pseudoconvex at z,
, , is higher-order η-quasiconvex at z.
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
By the feasibility of x for (NFP)and , we obtain
The above inequality with (3.5) gives
Now, the feasibility of x for (NFP), and (3.3) yields
The inequality (3.8) and Hypothesis (ii) gives
From (3.1) and (3.9), we have
which by virtue of Hypothesis (i) yields
This contradicts (3.7). □
Theorem 3.2 (Strong duality)
Let be an optimal solution of (NFP) and let , be linearly independent. Assume that
Then there exist and such that is a feasible solution of (NMD) and the two objectives have the same values. Furthermore, if the assumptions of weak duality (Theorem 3.1) hold for all feasible solutions of (NFP) and (NMD), then is an optimal solution of (NMD).
Proof Since is an optimal solution of (NFP) and , are linearly independent, by the necessary conditions obtained in , there exist and such that is a feasible solution of (NMD) and the problems (NFP) and (NMD) have the same objectives values and
Theorem 3.3 (Strict converse duality)
Let and be the optimal solutions of (NFP) and (NMD), respectively. Assume that
, are linearly independent,
is higher-order strictly η-pseudoconvex at ,
, , is higher-order η-quasiconvex at .
Then, ; that is, is an optimal solution of (NFP).
Proof We will assume that and reach a contradiction. From the strong duality theorem (Theorem 3.2), it follows that
Now, proceeding as in Theorem 3.1, we get
By Hypothesis (ii), (3.2) and (3.13), we have
which contradicts (3.12). Hence the results. □
4 Concluding remarks
The notion of higher-order invexity is adopted, which includes many other generalized convexity concepts in mathematical programming as special cases. If we take , , , and , in Theorems 3.1-3.3, then we get Theorems 3.1-3.3 in .
The presented results in this paper can be further extended to the following related class of nondifferentiable minimax fractional programming problems:
where , for , . and are analytic with respect to ξ, W is a specified compact subset in , is a polyhedral cone in and is analytic. Also B, are positive semidefinite Hermitian matrices.
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The research of the author is supported by the Internal Project No. IN111015 of King Fahd University of Petroleum and Minerals, Dhahran-31261, Saudi Arabia.
The author declares that he has no competing interest.
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Ahmad, I. Higher-order duality in nondifferentiable minimax fractional programming involving generalized convexity. J Inequal Appl 2012, 306 (2012). https://doi.org/10.1186/1029-242X-2012-306
- fractional programming
- nondifferentiable programming
- higher-order duality
- optimal solutions
- generalized higher-order convexity