Symmetric duality for a higher-order nondifferentiable multiobjective programming problem

Abstract

In this paper, a pair of Wolfe type higher-order nondifferentiable symmetric dual programs over arbitrary cones has been studied and then well-suited duality relations have been established considering K-F 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 higher-order Mond-Weir type nondifferentiable multiobjecive programming problems and established duality relations under higher-order F-convexity 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 Mond-Weir type multiobjective higher-order nondifferentiable symmetric dual program. Khurana [3] considered a pair of Mond-Weir type symmetric dual multiobjective programs over arbitrary cones and established duality results under cone-pseudoinvex and strongly cone-pseudoinvex 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 higher-order Mond-Weir type multiobjective symmetric duality over cones using higher-order cone-preinvex and cone-pseudoinvex functions, which further extends some of the results in [3, 6, 7].

Agarwal et al. [8] formulated a pair of Mond-Weir type nondifferentiable multiobjective higher-order symmetric dual programs over arbitrary cones and established duality theorems under higher-order K-F convexity assumptions. In the recent work of Suneja and Louhan [9], the authors have considered Wolfe and Mond-Weir type differentiable symmetric higher-order dual pairs. The Mond-Weir 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 higher-order multiobjective nondifferentiable symmetric dual program have been formulated and we established weak, strong, and converse duality theorems under K-F 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 higher-order K-F 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 higher-order nondifferentiable multiobjective symmetric dual programs:

(1)

(2)

(3)

(4)

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 higher-orderK-Fconvex atuwith respect tofor fixedv,

• (II) is higher-orderK-Gconvex atywith respect tofor fixed x,

• (III) ,

whereandare the sublinear functionals with respect to the third variable and satisfy the following conditions:

(A)

(B)

Then

(5)

Proof We shall obtain the proof by contradiction. Let (5) not hold. Then

It follows from and that

(6)

Now, since is higher-order K-F 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

(7)

for .

Similarly, using hypothesis (II), (B), , (1), (2), and sublinearity of G, we obtain

(8)

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 primal-dual pair (EP) and (ED), the hypotheses of Theorem 3.1 hold.

(A.1) is higher-order K-F convex at with respect to for fixed v and for all , , and we have

(A.2) is higher-order K-G 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

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

Now, hypothesis (I) and (12) imply that

(18)

Using (18) in (10), we have

(19)

which yields

(20)

Now, we claim that for all . On the contrary, suppose that for some , , then using hypothesis (II), we have

(21)

This contradicts hypothesis (III) (by (20) and (21)). Hence,

(22)

Using (22) in (18), we have , .

Since , for at least one i,

(23)

It follows from (11) and (23) that , , which from implies .

From (19), (22), and hypothesis (V), we get

which from hypothesis (IV) yields

(24)

Now, if , then . Therefore, from (23), we get and hence, . This contradicts (17). Thus . Since and , we have

(25)

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

(26)

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

(27)

Now, (15) and (23) yield . Since , .

Again as E is a compact convex set in , .

Further, (13), (23), and (25) yield

(28)

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. □