Second order duality in multiobjective programming involving generalized type I functions

Numerical Functional Analysis and Optimization

Publication details, including instructions for authors and subscription information:

http://www.tandfonline.com/loi/lnfa20

Second Order Duality in Multiobjective Programming Involving Generalized Type I Functions

M. Hachimi

& B. Aghezzaf

a

Faculté des Sciences Juriduques Economiques et Sociales , Université Ibn Zohr , Agadir, Morocco

b

Département de Mathématiques et d'Informatique, Faculté des Sciences , Université Hassan II Aïn Chock , Casablanca, Morocco

c

Faculté des Sciences Juriduques Economiques et Sociales , Université Ibn Zohr , B.P.

32/S, Agadir, Morocco

Published online: 31 Aug 2006.

To cite this article: M. Hachimi & B. Aghezzaf (2005) Second Order Duality in Multiobjective Programming Involving Generalized Type I Functions, Numerical Functional Analysis and Optimization, 25:7-8, 725-736, DOI: 10.1081/

NFA-200045804

To link to this article: http://dx.doi.org/10.1081/NFA-200045804

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no

representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any

form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://

www.tandfonline.com/page/terms-and-conditions

Second Order Duality in Multiobjective Programming Involving Generalized Type I Functions

M. Hachimi

^* and B. Aghezzaf

1

Faculté des Sciences Juriduques Economiques et Sociales, Université Ibn Zohr, Agadir, Morocco

2

Département de Mathématiques et d’Informatique, Faculté des Sciences, Université Hassan II Aïn Chock, Casablanca, Morocco

ABSTRACT

Recently Hachimi and Aghezzaf introduced the notion of F d-type I functions, a new class of functions that unifies several concepts of generalized type I functions. In this paper, we extend the notion of F d-type I functions to second order and establish several mixed duality results under second order generalized F p d-type I functions. Our results generalize the duality results recently given by Aghezzaf [Aghezzaf, B. (2003). Second order mixed type duality in multiobjective programming problems. J. Math. Anal. Appl.

285:97–106] and Hachimi and Aghezzaf [Hachimi, M., Aghezzaf, B. (2004).

Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions. J. Math. Anal. Appl. 296:382–392].

Key Words: Multiobjective programming problem; Efficient solution; Second order duality; Type I functions; Second order generalized convexity; Mixed duality.

Mathematics Subject Classification: Primary 90C29; Secondary 90C25.

∗

Correspondence: M. Hachimi, Faculté des Sciences Juriduques Economiques et Sociales, Université Ibn Zohr, B.P. 32/S, Agadir, Morocco; E-mail: b.aghezzaf@fsac.ac.ma

725

DOI: 10.1081/NFA-200045804 0163-0563 (Print); 1532-2467 (Online)

Copyright © 2004 by Marcel Dekker, Inc. www.dekker.com

Downloaded by [Umeå University Library] at 13:51 12 November 2014

1. INTRODUCTION

The concept of type I functions was introduced by Hanson and Mond (1982) as a generalization of convexity. Subsequently, Rueda and Hanson (1988) defined pseudo-type I and quasi-type I functions and obtained sufficient optimality conditions involving these functions. Later, Aghezzaf and Hachimi (2000) introduced generalized type I functions, for multiobjective programming problems, which are different from those defined in Kaul et al. (1994) and obtained some duality results. In recent paper, Hachimi and Aghezzaf (2004) defined generalized F d-type I functions, a new class of functions that unifies several concepts of generalized type I functions. They obtained sufficient optimality conditions and duality for multiobjective programming problems.

On other hand, by introducing an additional vector p, Zhang and Mond (1997) introduced a concept of second order F -convexity as a generalization of F - convexity (Preda, 1992) and used the concept to establish some duality results.

Recently, Aghezzaf (2003) introduced a second order mixed type dual and obtained various duality results involving new classes of second order generalized F - convex functions which are different from those defined in Zhang and Mond (1997).

Consider the following nonlinear multiobjective programming problem:

(MOP) minimize fx = f

x f

x subject to x ∈ A = x ∈ X gx 0

where X is an open subset of

and f X →

g X →

are twice differentiable functions at y ∈ A.

In this paper, we propose new classes of functions, called second order F p d-type I functions as a generalizations of F d-type I functions and establish various mixed duality results involving second order generalized F p d-type I functions by associating a second order mixed type dual problem Aghezzaf (2003) with the problem (MOP).

Notation. Throughout this paper we use the following Notation. The index sets M = 1 2 m and Q = 1 2 q. If x and y ∈

, then x y ⇔ x

y

i = 1 n x ≤ y ⇔ x y and x = y x < y ⇔ x

< y

i = 1 n xy or x

y denote the inner product.

For the multiobjective programming problem (MOP), the solution is defined in terms of a (weak) efficient solution in the following sense:

Definition 1.1. We say that y ∈ A is an efficient solution for problem (MOP) if and only if there exists no x ∈ A such that fx ≤ fy.

Definition 1.2. We say that y ∈ A is a weak efficient solution for problem (MOP) if and only if there exists no x ∈ A such that fx < fy.

Weak efficient solutions are often useful, since they are completely characterized by scalarization (Sawaragi et al., 1985).

Downloaded by [Umeå University Library] at 13:51 12 November 2014

2. SECOND ORDER GENERALIZED F p d -TYPE I FUNCTIONS

Hachimi and Aghezzaf (2004) introduced a generalization of the class of type I functions to a class called F d-type I functions for use in multiobjective programming. Here, by introducing an additional vector p, we generalize further to second order F p d-type I functions.

Definition 2.1. A functional F X × X ×

−→ is sublinear if for any x y ∈ X , Fx y a

+ a

x y a

+ Fx y a

∀ a

a

∈

(1a) Fx y a = Fx y a ∀ ∈ 0 ∀ a ∈

(1b) Let F be a sublinear functional and suppose the functions f = f

f

X −→

and h = h

h

X −→

are twice differentiable at y ∈ X. Let =

where

=

∈

=

∈

. Let =

where

X × X −→

\0

X × X −→

\0, and let d X × X −→ .

For the sake of simplicity, we will use the following notation. If f X −→

and X × X →

\0, then

f x y p = x yfy +

fyp

If f X −→

, then the symbol Fx y f x y p denotes the vector of components Fx y f

x y p Fx y f

x y p.

Definition 2.2. f h is said to be second order F p d-type I at y, if for all x ∈ A we have

fx − fy + 1

2 p

fyp Fx y f x y

p +

d

x y (2a)

−hy + 1

2 p

hyp Fx y h x y

p +

d

x y (2b) Definition 2.3. f h is said to be second order quasi F p d-type I at y, if for all x ∈ A we have

fx fy − 1

2 p

fyp ⇒ Fx y f x y

p −p

d

x y (3a)

−hy + 1

2 p

hyp 0 ⇒ Fx y h x y

p −

d

x y (3b) If in the above definition, x = y and inequality (3b) is satisfied as

−hy + 1

2 p

hyp 0 ⇒ Fx y h x y

p < −

d

x y (3c) then we say that f h is second order quasistrictly-pseudo F p d-type I at y.

Downloaded by [Umeå University Library] at 13:51 12 November 2014

Definition 2.4. f h is said to be second order pseudoquasi F p d-type I at y, if for all x ∈ A we have

fx < fy − 1

2 p

fyp ⇒ Fx y f x y

p < −

d

x y (4a)

−hy + 1

2 p

hyp 0 ⇒ Fx y h x y

p −

d

x y (4b) If in the above definition, x = y and inequality (4a) is satisfied as

fx fy − 1

2 p

fyp ⇒ Fx y f x y

p < −

d

x y (4c) then we say that f h is second order strictly pseudoquasi F p d-type I at y.

Definition 2.5. f h is said to be second order weak strictly-pseudoquasi F p d-type I at y, if for all x ∈ A we have

fx ≤ fy − 1

2 p

fyp ⇒ Fx y f x y

p < −

d

x y (5a)

−hy + 1

2 p

hyp 0 ⇒ Fx y h x y

p −

d

x y (5b) Definition 2.6. f h is said to be second order strong pseudoquasi F p d- type I at y, if for all x ∈ A we have

fx ≤ fy − 1

2 p

fyp ⇒ Fx y f x y

p ≤ −

d

x y (6a)

−hy + 1

2 p

hyP 0 ⇒ Fx y f x y

p −

d

x y (6b) If in the above definition, inequality (6a) is satisfied as

fx < fy − 1

2 p

fyp ⇒ Fx y f x y

p ≤ −

d

x y (6c) then we say that f h is second order weak pseudoquasi F p d-type I at y.

Remark 2.7. Note that for the scalar objective functions the class of second order pseudoquasi F p d-type I, the class of second order weak strictly- pseudoquasi F p d-type I, and the class of second order strong pseudoquasi F p d-type I functions coincide.

Definition 2.8. f h is said to be second order sub-strictly-pseudoquasi F p d-type I at y, if for all x ∈ A, x = y, we have

fx fy − 1

2 p

fyp ⇒ Fx y f x y

p ≤ −

d

x y (7a)

−hy + 1

2 p

hyP 0 ⇒ Fx y h x y

p −

d

x y (7b)

Downloaded by [Umeå University Library] at 13:51 12 November 2014

Definition 2.9. f h is said to be second order weak quasistrictly-pseudo F p d-type I at y, if for all x ∈ A we have

fx ≤ fy − 1

2 p

fyp ⇒ Fx y f x y

p −

d

x y (8a)

−hy + 1

2 p

hyp 0 ⇒ Fx y h x y

p ≤ −

d

x y (8b) Definition 2.10. f h is said to be second order weak quasisemi-pseudo F p d-type I at y, if for all x ∈ A we have

fx ≤ fy − 1

2 p

fyp ⇒ Fx y f x y

p −

d

x y (9a)

−hy + 1

2 p

hyp 0 ⇒ Fx y h x y

p < −

d

x y (9b) Definition 2.11. f h is said to be second order weak strictly-pseudo F p d- type I at y, if for all x ∈ A, x = y, we have

fx ≤ fy − 1

2 p

fyp ⇒ Fx y f x y

p < −

d

x y (10a)

−hy + 1

2 p

hyp 0 ⇒ Fx y h x y

p < −

d

x y (10b) Remark 2.12. Note that when p = 0, the concepts of second order generalized F p d-type I functions reduce to those of F d-type I functions defined in Hachimi and Aghezzaf (2004).

3. MIXED TYPE DUALITY

Let J

be a subset of Q and J

= Q/J

, and let e be the vector of

whose components are all ones. We consider the following mixed type dual of (MOP) formulated by Aghezzaf (2003).

(XMOP) maximize fy + v

g

ye − 1

2 p

fy + v

g

yep

subject to ufy + u

fyp + vgy + v

gyp = 0 (11a) v

g

y − 1

2 p

v

g

yp 0 (11b)

v 0 (11c)

u 0 u

e = 1 (11d)

Note that we get a second order Mond-Weir dual (Zhang and Mond, 1997) for J

= Ø and a second order Mangasarian (1975) dual for J

= Ø in (XMOP), respectively, while in (2GVD) in Sec. 4 of Zhang and Mond (1997) a second order Mangasarian dual cannot be obtained by specifying I

there. Besides, the dual there has more constraints, in general.

Downloaded by [Umeå University Library] at 13:51 12 November 2014

Theorem 3.1 (Weak Duality). Assume that for all feasible x for (MOP) and all feasible y u v p for (XMOP), any of the following holds:

(a) u > 0, and f· + v

g

·e v

g

· is second order strong pseudoquasi F p d-type I at y with u

· y

+

· y

0;

(b) u > 0, and uf· + v

g

· v

g

· is second order pseudoquasi F d-type I at y with

· y

+

· y

0.

Then the following cannot hold:

fx < fy + v

g

ye − 1

2 p

fy + v

g

yep (12)

Proof. Suppose contrary to the result of the theorem that (12) holds. Since x is feasible for (MOP) and v 0, (12) implies

fx + v

g

xe ≤ fy + v

g

ye − 1

2 p

fy + v

g

yep (13a) hold. Since y u v p is feasible for (XMOP), it follows that

−v

g

y + 1

2 p

v

g

yp 0 (13b)

By hypothesis (a) and (13), we have

Fx y f· + v

g

·e x y

p +

d

x y ≤ 0 (14a) Fx y v

g

· x y

p +

d

x y 0 (14b) Since

x y > 0,

x y > 0 and u > 0, the inequalities (14) give

Fx y uf· + v

g

· x y 1 p < −

x y

u

d

x y (15a) Fx y v

g

· x y 1 p −

x y

d

x y (15b) By sublinearity of F , we obtain

Fx y ufy + u

fyp + vgy + v

gyp

Fx y uf· + v

g

· x y 1 p + Fx y v

g

· x y 1 p

< −u

x y

+

x y

d

x y (16) Since u

x y

+

x y

0, we have

Fx y ufy + u

fyp + vgy + v

gyp < 0 (17) which contradicts the duality constraint (11a) because Fx y 0 = 0. Hence, (12) cannot hold.

Downloaded by [Umeå University Library] at 13:51 12 November 2014

On the other hand, multiplying (13a) with u > 0, we get ufx + v

g

x < ufy + v

g

y − 1

2 p

ufy + v

g

yp (18) When hypothesis (b) holds, inequalities (13b) and (18) implies

Fx y uf· + v

g

· x y

p +

d

x y < 0 (19a) Fx y v

g

· x y

p +

d

x y 0 (19b) Since

x y > 0 and

x y > 0, the inequalities (19) give

Fx y uf· + v

g

· x y 1 p < −

x y

d

x y (20a) Fx y v

g

· x y 1 p −

x y

d

x y (20b) By sublinearity of F, we obtain

Fx y ufy + u

fyp + vgy + v

gyp

Fx y uf· v

g

· x y 1 p + Fx y v

g

· x y 1 p

< −

x y

+

x y

d

x y (21) So we also have (17) which contradicts the duality constraint (11a).

We need the condition u > 0 in Theorem 3.1. In order to get the results without the condition u > 0, then other convexity assumption should be enforced, which leads to the following theorem.

Theorem 3.2 (Weak Duality). Assume that for all feasible x for (MOP) and all feasible y u v p for (XMOP), any of the following holds:

(a) f· + v

g

·e v

g

· is second order weak strictly-pseudoquasi F p d-type I at y with u

· y

+

· y

0;

(b) uf· + v

g

· v

g

· is second order strictly pseudoquasi F p d-type I at y with

· y

+

· y

0.

(c) f· + v

g

·e v

g

· is second order weak quasisemi-pseudo F p d-type I at y with u

· y

+

· y

0;

(d) uf· + v

g

· v

g

· is second order quasistrictly-pseudo F p d-type I at y with

· y

+

· y

0.

Then the inequality (12) cannot hold.

Proof. Suppose contrary to the result of the theorem that (12) holds. Since x is feasible for (MOP) and y u v p is feasible for (XMOP), we have system (13).

Multiplying (13a) by u ≥ 0, we have ufx + v

g

x ufy + v

g

y − 1

2 p

ufy + v

g

yp (22)

Downloaded by [Umeå University Library] at 13:51 12 November 2014

Now, by hypothesis (a) and (13), we have

Fx y f· + v

g

·e x y

p +

d

x y < 0 (23a) Fx y v

g

· x y

p +

d

x y 0 (23b) Since

x y > 0,

x y > 0 and u ≥ 0, the inequalities (23) give

Fx y uf· + v

g

· x y 1 p < −

x y

up

d

x y (24a) Fx y v

g

· x y 1 p −

x y

d

x y (24b) By sublinearity of F , we obtain (16) which contradicts (11a).

When hypothesis (b) holds, inequalities (22) and (13b) imply system (19) and now the proof follows similar lines as that of part (b) of Theorem 3.1.

Now, if hypothesis (c) holds, system (13) lead to

Fx y f· + v

g

·e x y

p +

d

x y 0 (25a) Fx y v

g

· x y

p +

d

x y < 0 (25b) Since

x y > 0,

x y > 0 and u ≥ 0, the system (25) implies

Fx y uf· + v

g

· x y 1 p −

x y

u

d

x y (26a) Fx y v

g

· x y 1 p < −

x y

d

x y (26b) By sublinearity of F , we obtain (16) which contradicts again (11a).

Finally, if hypothesis (d) holds, inequalities (22) and (13b) lead to

Fx y uf· + v

g

· x y

p +

d

x y 0 (27a) Fx y v

g

· x y

p +

d

x y < 0 (27b) By sublinearity of F , we obtain (21) which also contradicts (11a).

It is obvious that Theorems 3.1 and 3.2 still hold when (12) is strict inequality.

However, it is important to know that the convexity assumptions of Theorems 3.1 and 3.2 can be weakened if we replace the sign ≤ by < in (12).

Theorem 3.3 (Weak Duality). Assume that for all feasible x for (MOP) and all feasible y u v p for (XMOP), any of the following holds:

(a) u > 0, and f· + v

g

·e v

g

· is second order weak pseudoquasi F p d-type I at y with u

· y

+

· y

0;

(b) uf· + v

g

· v

g

· is second order pseudoquasi F p d-type I at y with

· y

+

· y

0.

(c) f· + v

g

·e v

g

· is second order pseudoquasi F p d-type I at y with u

· y

+

· y

0;

(d) uf· + v

g

· v

g

· is second order quasistrictly-pseudo F p d-type I at y with

· y

+

· y

0.

Downloaded by [Umeå University Library] at 13:51 12 November 2014

Then the following cannot hold:

fx < fy + v

g

ye − 1

2 p

fy + v

g

yep (28)

Proof. Suppose contrary to the result of the theorem that (28) holds. Since x is feasible for (MOP) and v 0, (28) implies

fx + v

g

xe < fy + v

g

ye − 1

2 p

fy + v

g

yep (29a) hold. Since y u v p is feasible for (XMOP), it follows that

−v

g

y + 1

2 p

v

g

yp 0 (29b)

Multiplying (29a) with u ≥ 0, we get ufx + v

g

x < ufy + v

g

y − 1

2 p

ufy + v

g

yp (29c) and now we proceed along similar lines as in Theorems 3.1 and 3.2.

Remark 3.4. It may be noted that, by taking p = 0, second order weak duality results (Theorems 3.1, 3.2 and 3.3) reduce to first order weak duality results (see for example Hachimi and Aghezzaf, 2004).

Corollary 3.5. Let ¯ y u ¯ v ¯ p ¯ be feasible solution for (XMOP) such that

¯

v

g

¯ ye − 1

2 p ¯

f¯ y + ¯ v

g

¯ ye p ¯ 0 (30) and assume that y ¯ is feasible for (MOP). If weak duality (any of Theorem 3.1 or 3.2) holds between (MOP) and (XMOP), then y ¯ is efficient for (MOP) and ¯ y u ¯ v ¯ p ¯ is efficient for (XMOP).

Proof. Suppose that y ¯ is not efficient for (MOP), then there exists a feasible x for (MOP) such that fx ≤ f¯ y. On using hypothesis (30), then

fx ≤ f y ¯ + ¯ v

g

¯ ye − 1

2 p ¯

f¯ y + ¯ v

g

¯ ye p ¯ (31) Since ¯ y u ¯ v ¯ p ¯ is feasible for (XMOP) and x is feasible for (MOP), inequality (31) contradicts weak duality (Theorem 3.1 or 3.2).

Also suppose that ¯ y u ¯ v ¯ p ¯ is not efficient for (XMOP), then there exists a feasible y u v p for (XMOP) such that

f y ¯ + ¯ v

g

¯ ye − 1

2 p ¯

f¯ y + ¯ v

g

¯ ye p ¯

≤ fy + v

g

ye − 1

2 p

fy + v

g

yep (32)

Downloaded by [Umeå University Library] at 13:51 12 November 2014

Taking into account (30), inequality (32) reduces to f¯ y ≤ fy + v

g

ye − 1

2 p

fy + v

g

yep (33)

Since y u v p is feasible for (XMOP) and y ¯ is feasible for (MOP), inequality (33)

contradicts weak duality (Theorem 3.1 or 3.2).

Corollary 3.6. Let ¯ y u ¯ v ¯ p ¯ be feasible solution for (XMOP) such that inequality (30) hold and assume that y ¯ is feasible for (MOP). If weak duality (Theorem 3.3) holds between (MOP) and (XMOP), then y ¯ is weak efficient for (MOP) and ¯ y u ¯ v ¯ p ¯ is weak efficient for (XMOP).

Proof. The proof of this theorem is similar to that of Corollary 3.5.

Theorem 3.7 (Strong Duality). Let x ¯ be an efficient solution for (MOP) and assume that the (generalized) constraint qualification (Abadie, 1967; Maeda, 1994) holds at

¯

x. Then there exist u ¯ ∈

¯ v ∈

and p ¯ ∈

such that ¯ x u ¯ v ¯ p ¯ is feasible for (XMOP) and

¯

v

g

¯ x − 1

2 p ¯

f¯ x + ¯ v

g

¯ xe¯ p 0

If also weak duality (Theorem 3.1 or 3.2) holds between (MOP) and (XMOP) then

¯ x u ¯ v ¯ p ¯ is efficient for (XMOP).

Proof. This follows on the lines of Mond and Zhang (1995, Theorem 3).

We now turn our attention to strict converse duality.

Theorem 3.8. Let x ¯ be feasible solution for (MOP) and ¯ y u ¯ v ¯ p ¯ be feasible solution for (XMOP) such that

¯

uf¯ x uf ¯ y ¯ + ¯ v

g

¯ y − 1

2 p ¯

¯ uf¯ y + ¯ v

g

¯ y p ¯ (34) If, condition (b) or (d) of Theorem 3.2 is satisfied for x ¯ and ¯ y u ¯ v ¯ p, then ¯ x ¯ = ¯ y.

Proof. We assume x ¯ = ¯ y and exhibit a contradiction. Since x ¯ and ¯ y u ¯ v ¯ p ¯ are feasible for (MOP) and (XMOP) respectively, then v ¯ 0 g¯ x 0 and, (34) and (11b) yield

¯

uf¯ x + ¯ v

g

¯ x uf¯ ¯ y + ¯ v

g

¯ y − 1

2 p ¯

uf ¯ y ¯ + ¯ v

g

¯ y¯ p (35a)

−¯ v

g

¯ y + 1

2 p ¯

v ¯

g

¯ y p ¯ 0 (35b)

Condition (b) of Theorem 3.2 with (35), gives

F¯ x y ¯ ¯ uf· + ¯ v

g

· x ¯ y ¯

p ¯ +

d

¯ x y < ¯ 0 (36a) F¯ x y ¯ ¯ v

g

· x ¯ y ¯

p ¯ +

d

¯ x y ¯ 0 (36b)

Downloaded by [Umeå University Library] at 13:51 12 November 2014

The same condition (d) of Theorem 3.2 with (35), gives

F¯ x y ¯ ¯ uf· + ¯ v

g

· x ¯ y ¯

p ¯ +

d

x ¯ y ¯ 0 (37a) F¯ x y ¯ ¯ v

g

· x ¯ y ¯

p ¯ +

d

¯ x y < ¯ 0 (37b) By sublinearity of F, both systems (36) and (37) gives

F¯ x y ¯ uf¯ ¯ y + ¯ u

f¯ y p ¯ + ¯ vg¯ y + ¯ v

g¯ y p ¯

F¯ x y ¯ ¯ uf· + ¯ v

g

· x ¯ y ¯ 1 p ¯ + F¯ x y ¯ ¯ v

g

· x ¯ y ¯ 1 p ¯

< −

¯ x y ¯

+

¯ x y ¯

d

¯ x y ¯ (38)

which contradicts the duality constraint (11a)

Corollary 3.9. Let assumptions of Theorem 3.8 be verified, and let x ¯ be (weak) efficient for (MOP) and ¯ y u ¯ v ¯ p ¯ be (weak) efficient for (XMOP). Then x ¯ = ¯ y, i.e., y ¯ is (weak) efficient for (MOP).

Proof. The proof follows from Theorem 3.8

Theorem 3.10. Let x ¯ be feasible solution for (MOP) and ¯ y u ¯ v ¯ p ¯ be feasible solution for (XMOP) such that

f x ¯ f¯ y + ¯ v

g

¯ ye − 1

2 p ¯

f¯ y + ¯ v

g

¯ ye p ¯ (39) For each feasible x for (MOP) and y u v p for (XMOP), if weak duality (any of Theorem 3.1 or 3.2) holds

(a) at y, then ¯ x ¯ is efficient for (MOP).

(b) at y, then ¯ y u ¯ v ¯ p ¯ is efficient for (XMOP).

Proof. (a) Suppose that x ¯ is not an efficient solution for (MOP). Then, there exist a feasible x for (MOP) such that

fx ≤ f x ¯

using condition (39) we get fx ≤ f y ¯ + ¯ v

g

¯ ye − 1

2 p ¯

f¯ y + ¯ v

g

¯ ye p ¯ (40) which contradicts the weak duality for feasible solutions x for (MOP) and

¯ y u ¯ v ¯ p ¯ for (XMOP). Thus, x ¯ is efficient for (MOP).

(b) Let as assume on the contrary that ¯ y u ¯ v ¯ p ¯ is not an efficient solution for (XMOP). Then, there exist a feasible y u v p for (XMOP) such that

f y ¯ + ¯ v

g

¯ ye − 1

2 p ¯

f¯ y + ¯ v

g

¯ ye p ¯

≤ fy + v

g

ye − 1

2 p

fy + v

g

yep (41)

Downloaded by [Umeå University Library] at 13:51 12 November 2014

On using condition (39) we get f¯ x ≤ fy + v

g

ye − 1

2 p

fy + v

g

yep (42)

which contradicts the weak duality for feasible solutions x ¯ for (MOP) and y u v p for (XMOP). Thus, ¯ y u ¯ v ¯ p ¯ is efficient for (XMOP).

REFERENCES

Abadie, J. (1967). On Kuhn-Tucker Theorem, in Nonlinear Programming. Abadie, J., ed. North Holland: Amsterdam, pp. 21–36.

Aghezzaf, B. (2003). Second order mixed type duality in multiobjective programming problems. J. Math. Anal. Appl. 285:97–106.

Aghezzaf, B., Hachimi, M. (2000). Generalized invexity and duality in multiobjective programming problems. J. Global Optim. 18:91–101.

Hachimi, M., Aghezzaf, B. (2004). Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions. J. Math.

Anal. Appl. 296:382–392.

Kaul, R. N., Suneja, S. K., Srivastava, M. K. (1994). Optimality criteria and duality in multiobjective optimization involving generalized invexity. J. Optim. Theory Appl. 80:465–482.

Maeda, T. (1994). Constraint qualification in multiobjective optimization problems:

Differentiable case. J. Optim. Theory Appl. 80:483–500.

Mangasarian, O. L. (1975). Second and higher-order duality in nonlinear programming. J. Math. Anal. Appl. 51:607–620.

Mond, B., Zhang, J. (1995). Duality for multiobjective programming involving second order V-invex functions. In: Golver, B. M., Jeyakumar, V. eds.

Proceedings of Optimization Miniconference. Sydney: University of New South Wales, pp. 89–100.

Preda, V. (1992). On efficiency and duality for multiobjective programs. J. Math.

Anal. Appl. 166:365–377.

Rueda, N. G., Hanson, M. A. (1988). Optimality criteria in mathematical programming involving generalized invexity. J. Math. Anal. Appl. 130:375–385.

Sawaragi, Y., Nakayama, H., Tanino, T. (1985). Theory of Multiobjective Optimization. Orlando, Florida: Academic Press.

Zhang, J., Mond, B. (1997). Second order duality for multiobjective non- linear programming involving generalized convexity. In: Golver, B. M., Craven, B. D., Ralph, D., eds. Proceedings of Optimization Miniconference III. University of Ballarat, pp. 79–95.

Downloaded by [Umeå University Library] at 13:51 12 November 2014

Request Permission/Order Reprints

Reprints of this article can also be ordered at

http://www.dekker.com/servlet/product/DOI/101081NFA200045804

Law requires that you get permission from the article’s rightsholder before using copyrighted content.

All information and materials found in this article, including but not limited to text, trademarks, patents, logos, graphics and images (the "Materials"), are the copyrighted works and other forms of intellectual property of Marcel Dekker, Inc., or its licensors. All rights not expressly granted are reserved.

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly. Simply click on the "Request Permission/

Order Reprints" link below and follow the instructions. Visit the U.S. Copyright Office for information on Fair Use limitations of U.S.

copyright law. Please refer to The Association of American Publishers’

(AAP) website for guidelines on Fair Use in the Classroom.

The Materials are for your personal use only and cannot be reformatted, reposted, resold or distributed by electronic means or otherwise without permission from Marcel Dekker, Inc. Marcel Dekker, Inc. grants you the limited right to display the Materials only on your personal computer or personal wireless device, and to copy and download single copies of such Materials provided that any copyright, trademark or other notice appearing on such Materials is also retained by, displayed, copied or downloaded as part of the Materials and is not removed or obscured, and provided you do not edit, modify, alter or enhance the Materials. Please refer to our Website

User Agreement for more details.

Downloaded by [Umeå University Library] at 13:51 12 November 2014