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Symmetric duality for a higherorder nondifferentiable multiobjective programming problem
Journal of Inequalities and Applications volume 2015, Article number: 3 (2015)
Abstract
In this paper, a pair of Wolfe type higherorder nondifferentiable symmetric dual programs over arbitrary cones has been studied and then wellsuited duality relations have been established considering KF convexity assumptions. An example which satisfies the weak duality relation has also been depicted.
MSC:90C29, 90C30, 49N15.
1 Introduction
Consider the following multiobjective programming problem:
where be open, , , K, and C are closed convex pointed cones with nonempty interiors in and , respectively.
Several researchers have studied the duality relations for different dual problems of (P) under various generalized convexity assumptions. Chen [1] considered a pair of symmetric higherorder MondWeir type nondifferentiable multiobjecive programming problems and established duality relations under higherorder Fconvexity assumptions. Later on, Agarwal et al. [2] have filled some of the gap in the work of Chen [1] and proved a strong duality theorem for a MondWeir type multiobjective higherorder nondifferentiable symmetric dual program. Khurana [3] considered a pair of MondWeir type symmetric dual multiobjective programs over arbitrary cones and established duality results under conepseudoinvex and strongly conepseudoinvex assumptions. Later on, Kim and Kim [4] extended the results in Khurana [3] to the nondifferentiable multiobjective symmetric dual problem. Gupta and Jayswal [5] studied the higherorder MondWeir type multiobjective symmetric duality over cones using higherorder conepreinvex and conepseudoinvex functions, which further extends some of the results in [3, 6, 7].
Agarwal et al. [8] formulated a pair of MondWeir type nondifferentiable multiobjective higherorder symmetric dual programs over arbitrary cones and established duality theorems under higherorder KF convexity assumptions. In the recent work of Suneja and Louhan [9], the authors have considered Wolfe and MondWeir type differentiable symmetric higherorder dual pairs. The MondWeir type model studied in [9] is similar to the problem considered in Gupta and Jayswal [5]. However, the strong duality result in [9] is for arbitrary cones in instead of only those cones which contain the nonnegative orthant of as considered in [5].
In the present paper, a pair of Wolfe type higherorder multiobjective nondifferentiable symmetric dual program have been formulated and we established weak, strong, and converse duality theorems under KF convexity assumptions. We also illustrate a nontrivial example of a function which satisfies the weak duality relation.
2 Definitions and preliminaries
Let and be closed convex cones with nonempty interiors and let and be nonempty open sets in and , respectively such that . For a real valued twice differentiable function defined on , denotes the gradient vector of f with respect to x at , denotes the Hessian matrix with respect to x at . Similarly, , , and are also defined.
Definition 2.1 [8]
A point is a weak efficient solution of (P) if there exists no such that
Definition 2.2 [5]
A point is an efficient solution of (P) if there exists no such that
Definition 2.3 The positive dual cone of K is defined by
Definition 2.4 For all , a functional is said to be sublinear with respect to the third variable, if

(i)
for all ,

(ii)
, for all and for all .
For convenience, we write .
Definition 2.5 [8]
Let be a sublinear functional with respect to the third variable. Also, let , be a differentiable function. Then the function is said to be higherorder KF convex in the first variable at for fixed with respect to h, such that for , , ,
Definition 2.6 [10]
Let φ be a compact convex set in . The support function of φ is defined by
The subdifferentiable of is given by
For any set , the normal cone to S at a point is defined by
For each , let , and be differentiable functions. and , for and , . and are the positive dual cones of and , respectively. D and E are the compact convex sets in and , respectively. Also, we use the following notations:
3 Problem formulation
Consider the following pair of Wolfe type higherorder nondifferentiable multiobjective symmetric dual programs:
where is fixed.
Remark 3.1 If and , then our problems (WHP) and (WHD) become the problem studied in Suneja and Louhan [9].
Next, we will prove weak, strong, and converse duality results between (WHP) and (WHD).
Theorem 3.1 (Weak duality)
Letandbe feasible solutions for (WHP) and (WHD), respectively. Assume the following conditions hold:

(I)
is higherorderKFconvex atuwith respect tofor fixedv,

(II)
is higherorderKGconvex atywith respect tofor fixed x,

(III)
,
whereandare the sublinear functionals with respect to the third variable and satisfy the following conditions:
Then
Proof We shall obtain the proof by contradiction. Let (5) not hold. Then
It follows from and that
Now, since is higherorder KF convex at u with respect to for fixed v, we get
Using and , it follows that
Since (by hypothesis (III)), hence . Therefore, using (2) and sublinearity of F in the above expression, we obtain
It follows from (A) and the dual constraint (3) that
for .
Similarly, using hypothesis (II), (B), , (1), (2), and sublinearity of G, we obtain
for .
Now, adding (7) and (8), we have
Finally, it follows from and that
which contradicts (6). Hence the result. □
Example 3.1 Let , . Let , , and .
Then and . Obviously, .
Let , and be defined as
Let and . Then and . Suppose . Also, suppose the sublinear functionals F and G are defined as
Now, substituting the above defined expressions in the problems (WHP) and (WHD), we get
Now, we shall show that for the primaldual pair (EP) and (ED), the hypotheses of Theorem 3.1 hold.
(A.1) is higherorder KF convex at with respect to for fixed v and for all , , and we have
(A.2) is higherorder KG convex at with respect to for fixed x and for all , , and we have
(A.3)
The points and are feasible for the problems (EP) and (ED), respectively. These feasible points do satisfy the result of the weak duality theorem since
Theorem 3.2 (Strong duality)
Letbe a weak efficient solution of (WHD). Let

(I)
the Hessian matrixfor allbe positive or negative definite;

(II)
, for someimply thatfor all;

(III)
, for all;

(IV)
the set of vectorsbe linearly independent;

(V)
, , , for all.
Then

(I)
there existssuch thatis feasible for (WHD) and

(II)
the objective values of (WHP) and (WHD) are equal.
Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of (WHP) and (WHD), thenis an efficient solution for (WHD).
Proof Since is a weak efficient solution for (WHP), by the Fritz John necessary optimality conditions [11], there exist , , and such that
Now, hypothesis (I) and (12) imply that
Using (18) in (10), we have
which yields
Now, we claim that for all . On the contrary, suppose that for some , , then using hypothesis (II), we have
This contradicts hypothesis (III) (by (20) and (21)). Hence,
Using (22) in (18), we have , .
Since , for at least one i,
It follows from (11) and (23) that , , which from implies .
From (19), (22), and hypothesis (V), we get
which from hypothesis (IV) yields
Now, if , then . Therefore, from (23), we get and hence, . This contradicts (17). Thus . Since and , we have
From (23) and (25), we obtain
Further, using inequalities (18), (23)(25) in (9), we obtain
For , it follows from (22) and hypothesis (V) that
Let . Then and hence from (26), we have
Therefore, .
Thus, is a feasible solution for the dual problem.
Consider and in (26), we get
which implies that
Now, (15) and (23) yield . Since , .
Again as E is a compact convex set in , .
Further, (13), (23), and (25) yield
By hypothesis (V) for , (22), (27)(28), we obtain
Hence, the two objective values are equal.
Now, let be not an efficient solution of (WHD), then there exists a point feasible for (WHD) such that
From (27), (28), and hypothesis (V) for and , we obtain
which contradicts Theorem 3.1. Hence, is the efficient solution of (WHD). □
Theorem 3.3 (Converse duality)
Letbe a weak efficient solution of (WHP). Let

(I)
the Hessian matrixfor allbe positive or negative definite;

(II)
, for someimplies thatfor all;

(III)
, for all;

(IV)
the set of vectorsbe linearly independent;

(V)
, , , for all.
Then

(I)
there existssuch thatis feasible for (WHP) and

(II)
the objective values of (WHP) and (WHD) are equal.
Also, if the hypotheses of Theorem 3.1 are satisfied for all feasible solutions of (WHP) and (WHD), thenis an efficient solution for (WHP).
Proof The proof follows along the lines of Theorem 3.2. □
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Acknowledgements
The authors wish to thank the reviewer for her/his valuable and constructive suggestions, which have considerably improved the presentation of the paper. The first author is also grateful to the Ministry of Human Resource and Development, India for financial support to carry out this work.
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Debnath, I.P., Gupta, S.K. & Kumar, S. Symmetric duality for a higherorder nondifferentiable multiobjective programming problem. J Inequal Appl 2015, 3 (2015). https://doi.org/10.1186/1029242X20153
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DOI: https://doi.org/10.1186/1029242X20153
Keywords
 symmetric duality
 higherorder KF convexity
 multiobjective programming
 support function
 efficient solutions