Sufficient optimality conditions and duality in multiobjective optimization involving generalized convexity

Numerical Functional Analysis and Optimization

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SUFFICIENT OPTIMALITY CONDITIONS AND DUALITY IN MULTIOBJECTIVE OPTIMIZATION INVOLVING

GENERALIZED CONVEXITY

B. Aghezzaf ^a & M. Hachimi ^a

a Département de Mathématiques et d'Informatique, Faculté des Sciences Aïn chock , Université Hassan II , B.P. 5366, Marif, Casablanca, Morocco

Published online: 17 Aug 2006.

To cite this article: B. Aghezzaf & M. Hachimi (2001) SUFFICIENT OPTIMALITY CONDITIONS AND DUALITY IN

MULTIOBJECTIVE OPTIMIZATION INVOLVING GENERALIZED CONVEXITY, Numerical Functional Analysis and Optimization, 22:7-8, 775-788, DOI: 10.1081/NFA-100108308

To link to this article: http://dx.doi.org/10.1081/NFA-100108308

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SUFFICIENT OPTIMALITY CONDITIONS AND DUALITY IN MULTIOBJECTIVE

OPTIMIZATION INVOLVING GENERALIZED CONVEXITY

B. Aghezzaf* and M. Hachimi

De´partement de Mathe´matiques et d’Informatique, Faculte´ des Sciences Aı¨n chock, Universite´ Hassan II,

B.P. 5366 Marif, Casablanca, Morocco

ABSTRACT

We are concerned with nonlinear multiobjective programming problem with inequality constraints. We introduce some classes of nonconvex functions by relaxing the definitions of invex and preinvex functions.

Examples are given to shows relationship between them. Various first and second order sufficient optimality theorems are obtained.

Moreover, we prove duality and converse duality results for Mond- Weir type dual.

Key Words: Multiobjective programming; Pareto minimal solutions;

Sufficient conditions; Convexity; Invexity; Duality; Converse duality

INTRODUCTION

Several classes of generalized convex functions have been defined for the purpose of weakening the limitations of convexity in mathematical program- ming. In 1981, Hanson [7] introduced the concept of invexity as a general- ization of convexity for scalar constrained optimization problems, and he shows that weak duality and sufficiency of the Kuhn-Tucker conditions

775

Copyright^#2001 by Marcel Dekker, Inc. www.dekker.com

*Corresponding author. E-mail: aghezzaf@facsc-achok.ac.ma

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hold when invexity is required instead of the usual requirement of convexity.

Later, a further generalization, called preinvex, was considered by Hanson and Mond [9], and first sufficient optimality conditions and duality for multi- objective problems were given by Weir and Mond [16]. Nevertheless the study of generalized convexity of a vector function is not yet sufficiently explored and some classes of generalized convexity have been suggested by using componentwise generalized convexity.

Based on various generalized convexity assumptions, we establish first order sufficient conditions to gaurantee that a given solution that satisfies the first order optimality conditions is efficient. However, for more weaker con- vexity assumptions (on the objective function), the result may not hold true.

In order to drop them, additional sufficient optimality conditions, involving second order derivatives of the given functions, can be developed. Thus, we use weaker convexity conditions to refine a second order sufficient optimality conditions, which has been obtained in a recent paper [2].

Consider the following multiobjective optimization problem:

minimize f ðxÞ ¼ ð f

ðxÞ,

, f

ðxÞÞ subjectto gðxÞ

0,

x 2 X ð

Þ, X open,

ðMOPÞ

where f

X !

and g: X !

are differentiable functions at x x 2 X , a Pareto minimal solution or a candidate for a Pareto minimal solution.

Minimization means obtaining Pareto minimal solutions of (MOP).

For any x ¼ ðx

, x

,

, x

Þ

, y ¼ ð y

, y

,

, y

Þ

2

, we denote:

x ¼ y implying x

¼ y

, i ¼ 1,

, n;

x

y implying x

y

, i ¼ 1,

, n;

x y implying x

y, and x 6¼ y;

x

y implying x

y

, i ¼ 1,

, n:

Let

A ¼ fx 2 X, gðxÞ

0 g, I ¼ fj

g

ð x xÞ ¼ 0g, card I ¼ jI j ¼

P ¼ f1,

, pg, M ¼ f1,

, mgl

For such multicriterion optimization problems, the solution is defined in terms of a (weak) Pareto minimal solution in the following sense [13].

Definition 1.1. We say that x x 2 A is a Pareto minimal point of f on A if and only if there exists no x 2 A such that f ðxÞ f ð x xÞ.

Definition 1.2. We say that x x 2 A is a weak Pareto minimal point of f on A if and only if there exists no x 2 A such that f ðxÞ

f ð x xÞ.

In this paper, a generalization of convexity, called weak strictly pseudoinvexity, strong pseudoinvexity, weak quasiinvexity, weak strictly

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pseudoinvexity, and prequasiinvexity, are introduced. In the first half of this paper, we get various first and second order sufficient optimality conditions involving the above classes of functions. In the second half, we prove weak, strong, and converse duality theorems for Mond-Weir type dual under the above generalized convexity assumptions.

PRELIMINARIES

It will be assumed throughout that f is the vector objective function and g is the constraint vector function in Problem (MOP).

The definition of invexity for a real valued function can be generalized easily to a vector function [17].

Definition 2.1. f is said to be invex with respect to at x x 2 X if there exists a vector function

x xÞ defined on X X such that, for all x 2 X,

f ðxÞ f ð x xÞ

rf ð x xÞðx, x xÞ: ð1Þ

If in the above definition, (1) is strict inequality, then we say that f is strictly invex at x x.

We now define and introduce the notions of weak strictly pseudoinvex, strong pseudoinvex, weak quasiinvex and strong quasiinex functions for (MOP).

Definition 2.2. f is said to be weak strictly pseudoinvex with respect to at x

x 2 X if there exists a vector function

x xÞ defined on X X such that, for all x 2 X,

f ðxÞ f ð x xÞ ) rf ð x xÞðx, x xÞ

0: ð2Þ This definition is a slight extension of that of the pseudoinvex functions. This class of functions does not contain the class of invex functions, but does contain the class of strictly pseudoinvex functions.

Every strictly pseudoinvex function is weak strictly pseudoinvex with respect to the same

seen from the following example.

Example 2.1. The function f ¼ ð f

, f

Þ defined on X ¼

, by f

ðxÞ ¼ xðx þ 2Þ and f

ðxÞ ¼ xðx þ 2Þ

is weak strictly pseudoinvex function with respect to

x xÞ ¼ x þ 2 at x x ¼ 0 but it is not strictly pseudoinvex with respect to the same

x xÞ at x x because for x ¼ 2

f ðxÞ

f ð x xÞ but rf ð x xÞðx, x xÞ ¼ 0 6< 0

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Definition 2.3. f is said to be strong pseudoinvex with respect to at x x 2 X if there exists a vector function

x xÞ defined on X X such that, for all x 2 X,

f ðxÞ f ð x xÞ ) rf ð x xÞðx, x xÞ 0: ð3Þ Instead of the class of weak strictly pseudoinvex, the class of strong pseu- doinvex functions does contain the class of invex functions. Also, every weak strictly pseudoinvex function is strong pseudoinvex with respect to the same

Example 2.2. The function f ¼ ð f

, f

Þ defined on X ¼

, by f

ðxÞ ¼ x

and f

ðxÞ ¼ xðx þ 2Þ

is strong pseudoinvex function with respect to

x xÞ ¼ x at

x

x ¼ 0 but it is not weak strictly pseudoconvex with respect to the same at x x because for x ¼ 1

f ðxÞ f ð x xÞ but rf ð x xÞðx, x xÞ ¼ ð0, 4Þ

6< 0,

also f is not invex with respect to the same at x x, as can be seen by taking x ¼ 2.

There exist functions f that are pseudoinvex but do not satisfy (3) with respect to the same

f that are strong pseudoinvex, but they are not pseudoinvex with respect to the same

Example 2.3. The function f

!

, defined by f

ðxÞ ¼ xðx 2Þ

and f

ðxÞ ¼ xðx 3Þ, is pseudoinvex with respect to

x xÞ ¼ x x x at x x ¼ 0, but f does not satisfy (3) when x ¼ 2.

Example 2.4. The function f

!

, defined by f

ðxÞ ¼ xðx 2Þ and f

ðxÞ ¼ x

ðx 1Þ, is strong pseudoinvex with respect to

x xÞ ¼ x x x at x

x ¼ 0, but f is not pseudoinvex with respect to the same at that point.

Remark 2.1. If f is both pseudoinvex and quasiinvex with respect to at

x

x 2 X, then it is strong pseudoinvex function with respect to the same at x x.

Definition 2.4. f is said to be weak quasiinvex with respect to at x x 2 X if there exists a vector function

x xÞ defined on X X such that, for all x 2 X,

f ðxÞ f ð x xÞ ) rf ð x xÞðx, x xÞ

0: ð4Þ Every quasiinvex function is weak quasiinvex with respect to the same

However, the converse is not necessarily true.

Example 2.5. Define a function f

!

by f

ðxÞ ¼ xðx 2Þ

and f

ðxÞ ¼ x

ðx 2Þ. f is weak quasiinvex with respect to

x xÞ ¼ x x x at x x ¼ 0 but is

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not quasiinvex with respect to the same at x x ¼ 0, because f ðxÞ

f ð x xÞ but rf ð x xÞðx, x xÞ 6% 0, for x ¼ 2:

Definition 2.5. f is said to be weak pseudoinvex with respect to at x x 2 X if there exists a vector function

x xÞ defined on X X such that, for all x 2 X,

f ðxÞ

f ð x xÞ ) rf ð x xÞðx, x xÞ 0: ð5Þ The class of weak pseudoinvex functions does contain the class of invex, pseudoinvex, strong pseudoinvex, and strong quasiinvex functions.

Remark 2.1. As can be seen from Examples 2.1 to 2.4, the concepts of weak strictly pseudoinvex, strong pseudoinvex, weak pseudoinvex, and pseudoin- vex vector-valued functions are different, in general. However, they coincide in the scalar-valued case.

Definition 2.6. f is said to be strong quasiinvex with respect to at x x 2 X if there exists a vector function

x xÞ defined on X X such that, for all x 2 X,

f ðxÞ

f ð x xÞ ) rf ð x xÞðx, x xÞ 0:

Every strong quasiinvex function is both quasiinvex and strong pseudoinvex with respect to the same

Now, we introduce a class of weak prequasiinvex functions by general- izing the class of preinvex [9] and the class of prequasiinvex functions [14].

Definition 2.7. We say that f is weak prequasiinvex at x x 2 X with respect to if X is invex at x x with respect to and, for each x 2 X,

f ðxÞ f ð x xÞ ) f ð x x þ

x xÞÞ

f ð x xÞ, 0

0: ð7Þ Every prequasiinvex function [14] is weak prequasiinvex with respect to the same

Example 2.5. The function f

!

, defined by f

ðxÞ ¼ xðx 2Þ

and f

ðxÞ ¼ xðx 2Þ, is weak prequasiinvex at x x ¼ 0 with respect to

x xÞ ¼ x x x, but it is not prequasiinvex at x x with respect to the same

f

is not prequasiinvex at x x with respect to same

SUFFICIENT OPTIMALITY CONDITIONS

In establishing sufficiency under weaker convexity assumptions, Singh’s Theorems 3.3 and 3.4 use the scalarization of the objective function.

However, Marusciac’s Theorem 3.2 explore the property of convex vector functions. Here, we establish various sufficient theorems in the same sprit as

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Marusciac’s Theorem 3.2, but under weaker generalized convexity conditions that are different from these of Singh and Marusciac.

Theorem 3.1. Suppose that there exists a feasible solution x x for (MOP)and vectors u u

0 and vv

0 such that

u

u

rf ð x xÞ þ vv

rg

ð x xÞ ¼ 0: ð8Þ If f is strong pseudoinvex and g

is quasiinvex with respect to same at

x

x 2 X, then x x is (weak) Pareto minimal for (MOP).

Proof. If x x is not Pareto minimal, then there exists a feasible x for (MOP) such that

f ðxÞ f ð x xÞ 0, ð9Þ

g

ðxÞ

0: ð10Þ

Since f and g

are strong pseudinvex and quasiinvex respectively at x x, (9)–

(10) yields

rf ð x xÞðx, x xÞ 0, ð11Þ

rg

ð x xÞðx, x xÞ

0: ð12Þ

From Tucker’s theorem of the alternative [11] it follows that the system

u

rf ð x xÞ þ v

rg

ð x xÞ ¼ 0, ð13Þ

u

0, v

0, ð14Þ

is inconsistent. But this violates hypothesis (8). Hence, x x is Pareto minimal

for (MOP), and the Proof is complete.

Remark 3.1. It is important to point out here that, in Theorem 3.1, it suffices to assume u u 0, when f is weak strictly pseudoinvex, instead of u u

0. This is the subject of the following theorem.

Theorem 3.2. Let f and g

be a weak strictly pseudoinvex and quasiinvex, respectively, with respect to the same to x x 2 A. If there exist u u 0 and vv

0 such that (8)hold, then x x is (weak)Pareto minimal for (MOP).

Proof. The proof of this theorem follows exactly similar lines as that of Theorem 3.1, except that here Motzkin’s theorem of the alternative [11] is used instead of Tucker’s theorem.

In the following theorem, we invoke the notions of weak quasiinvex and weak strictly pseudoinvex functions.

Theorem 3.3. Let f be weak quasiinvex and g

be strong quasiinvex with respect to the same tp x x 2 A. If there exist u u

0 and vv

0 such that (8)hold, then x x is (weak)Pareto minimal for (MOP).

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Proof. If x x is not Pareto minimal, then there exists an x

2 A such that f ðx

Þ f ð x xÞ 0,

g

ðx

Þ

0:

Since f and g

are weak quasiinvex and strong quasiinvex respectively at x x, we have

rf ð x xÞðx

, x xÞ

0, ð15Þ

rgð x xÞðx

, x xÞ 0: ð16Þ

By the Tucker’s theorem of the alternative [11], there exist no u u

0, vv

0, such that

ðrf ð x xÞÞ

u u þ ðrg

ð x xÞÞ

vv ¼ 0:

Hence, x x is Pareto minimal.

Since any Pareto minimal solution to problem (MOP) is a weakly Pareto minimal solution, the sufficient conditions in Theorems 3.1 to 3.3 are valid to weakly Pareto minimal solution of (MOP). Weakly Pareto mini- mal solutions are often useful, since they are completely characterized by scalarization. We give the following theorem

Theorem 3.4. Suppose that there exists a feasible solution x x for (MOP)and vectors u u

0 and vv

0 such that (8). If f is weak pseudoinvex and g

is quasiinvex with respect to same at x x 2 X, then x x is weak Pareto minimal for (MOP).

Now we turn to discuss second order sufficient conditions. Since there may be weaker convexity conditions for which the result of Theorems 3.1 and 3.2 may not hold true, additional second order optimality conditions can be employed when all the functions are twice differentiable. For each x x 2 A we denote by

C ¼ fy 2

rf ð x xÞy

0, rf

ð x xÞy ¼ 0 for at least one i, rg

ð x xÞy

0g the set of critical directions at x x 2 A [2]. We have the following theorem.

Theorem 3.4. Suppose that f and g are prequasiinvex with respect to same at x

x 2 A, and are twice continuously differentiable at x x. Further suppose that

yÞ 6¼ 0 for all x 6¼ y. If for each critical direction y 6¼ 0, there exist

u

u 2

and vv 2

satisfying (8)and

i¼1

u

u

r

f

ð x xÞ þ

j2I

vv

r

g

ð x xÞ

!

ð y, yÞ

0, ð17Þ

u

u 0, vv

0, ð18Þ

then x x is (weak) Pareto minimal for (MOP).

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Proof. Assume that, for each critical direction y 6¼ 0, there exit u u 2

and

vv 2

such that (8) and (17–18) hold and that x x is not Pareto minimal for (MOP). Then, there is x 2 A such that

f ðxÞ f ð x xÞ and g

ðxÞ

g

ð x xÞ: ð19Þ Using the prequasiinvexity of f and g and (19), we obtain

rf ð x xÞðx, x xÞ

0 and rg

ð x xÞðx, x xÞ

0:

Two cases are to be considered:

Case 1. If rf ð x xÞðx, x xÞ

0, then y ¼

x xÞ is solution to the following system

rf ð x xÞy

0, rg

ð x xÞy

0:

By the Motzkin theorem of the alternative, the following system

u

u

rf ð x xÞ þ vvrg

ð x xÞ ¼ 0,

u

u 0, vv

0,

is inconsistent. Then, contradicting (8) and (18).

Case 2. If rf

ð x xÞðx, x xÞ ¼ 0, for at lest one r 2 f1,

, pg, then y ¼

x xÞ is a nonzero critical direction. Let 0

t

1, from the prequasiinvexity of f , we get

f ð x x þ tyÞ f ð x xÞ ¼ trf ð x xÞy þ t

2r

f ð x xÞð y, yÞ þ oðt

Þ

0 where oðt

Þ is a vector satisfying koðt

Þkt

, hence

rf ð x xÞy þ t2½r

f ð x xÞð y, yÞ þ oðt

Þt

0: ð20Þ Similarly,

rg

ð x xÞy þ t2½r

g

ð x xÞð y, yÞ þ oðt

Þt

0: ð21Þ Multiplying (20), (21) by u u, vv and summing, we get

u

u

rf ð x xÞ þ vv

rg

ð x xÞ

y

þ t 2

X^p

i¼1

u

u

r

f

ð x xÞ þ

j2I

vv

r

g

ð x xÞ

!

ð y, yÞ þ t

2 oðt

Þt

0:

From expressions (8), (18), and t

0, we get

i¼1

u

u

r

f

ð x xÞ þ

j2I

vv

r

g

ð x xÞ

!

ð y, yÞ þ oðt

Þt

0:

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Letting t ! 0, we obtain

i¼1

u

u

r

f

ð x xÞ þ

j2I

vv

r

g

ð x xÞ

!

ð y, yÞ

0,

contradicting (17). Hence, x x is Pareto minimal for (MOP).

The following theorem gives a second order optimality conditions for (MOP), under weaker convexity assumptions on the objective function than the ones current in Theorem 3.4.

Theorem 3.4. Suppose that f is weak prequasiinvex and g is prequasiinvex with respect to same at x x 2 A, and are twice continuously differentiable at x x.

Further suppose that

yÞ 6¼ 0 for all x 6¼ y. If for each critical direction y 6¼ 0, there exist u u 2

and vv 2

satisfying (8), (17) and (18), then x x is (weak)Pareto minimal for (MOP).

Proof. It follows on the lines of Theorem 3.3.

MOND-WEIR VECTOR DUALITY

We consider the Mond-Weir type dual and generalize the duality results of Egudo [5] under the weaker convexity assumptions.

We consider the following Mond-Weir type dual of (MOP) ðDMOPÞ maximize f ð yÞ,

subject to ðrf ð yÞÞ

u þ ðrgð yÞÞ

v ¼ 0, ð22Þ

v

gð yÞ

0, ð23Þ

v

0, ð24Þ

u

0, ð25Þ

u

e ¼ 1; ð26Þ

where e ¼ ð1, 1,

, 1Þ

2

.

We shall prove weak, strong, and converse duality results for (MOP) and (DMOP) under weaker assumptions of vector convexity. Our results do not makes use of Egudo’s idea of scalarization of the objective function.

Theorem 4.1. Let x be feasible for (MOP), and let ð y, u, vÞ be feasible for (DMOP). If also any of the following holds:

(a)u

0, f is strong pseudoinvex and v

g is quasiinvex at y with respect to the same

(b)u 0, f is weak strictly pseudoinvex and v

g is quasiinvex at y with respect to the same

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(c)f is weak quasiinvex and v

g is strictly pseudoinvex at y with respect to the same

then the following cannot hold:

f ðxÞ f ð yÞ: ð27Þ

Proof. Suppose contrary to the result of the theorem that

f ðxÞ f ð yÞ, ð28Þ

hold. Since ð y, u, vÞ is feasible for (DMOP) and x is feasible for (MOP), it follows that

v

gðxÞ

v

gð yÞ: ð29Þ

Therefore, by the above inequalities, we get

rf ð yÞðx, yÞ 0, ð30Þ

rv

gð yÞðx, yÞ

0: ð31Þ

Using hypothesis (a). Since u

0, the above inequalities give

^t

ðx, yÞ½u

rf ð yÞ þ v

rgð yÞ

0, ð32Þ which contradicts (22).

Again from (28) and (29), we have

rf ð yÞðx, yÞ

0, ð33Þ

rv

gð yÞðx, yÞ

0: ð34Þ

Using hypothesis (b). Since u 0, (33) and (34) imply (32), again contra- dicting (22).

On using hypothesis (c), the inequalities (28) and (29) give

rf ð yÞðx, yÞ

0, ð35Þ

rv

gð yÞðx, yÞ

0: ð36Þ

Since u

0, (35) and (36) imply (32), again contradicting (22).

Corollary 4.1. Assume weak duality (Theorem 4.1) holds between (MOP) and (DMOP). If ð y y, u u, vvÞ is feasible for (DMOP) such that y y is feasible for (MOP), then y

is Pareto minimal for (MOP) and ð y y, u u, vvÞ is Pareto minimal for (DMOP).

Proof. The proof of this corollary is the same as that of Corollary 2 of Egudo [5].

Theorem 4.2. Strong Duality. Let x x be Pareto minimal for (MOP)and assume that x x satisfies a constraint qualification [12] for (MOP). Then there exist

u

u 2

and vv 2

such that ð x x, u u, vvÞ is feasible for (DMOP). If also weak

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duality (Theorem 4.1)holds between (MOP)and (DMOP)then ð x x, u u, vvÞ is Pareto minimal for (DMOP).

Proof. Since x x is Pareto minimal for (MOP) and satisfies a constraint quali- fication [12] for (MOP), then from Fritz John necessary conditions [12] we obtain u u 0 and vv

0 such that

u

u

rf ð x xÞ þ vv

rgð x xÞ ¼ 0,

vv

gð x xÞ ¼ 0

The vector u u may be normalized according to u u

e ¼ 1, u u

0, which gives that the triplet ð x x, u u, vvÞ is feasible for (DMOP). The efficiency of ð x x, u u, vvÞ for

(DMOP) now follows from corollary 4.1.

Now we state and prove our converse duality theorem of Mond-Weir vector type duality showing that, under certain assumptions, a Pareto mini- mal solution of the dual (DMOP) is also Pareto minimal solution of the primal (MOP).

Theorem 4.3. Converse Duality. Let ð x x, u u, vvÞ be Pareto minimal for (DMOP), and let the hypotheses of Theorem 4.1 hold. If the n n Hessian matrix r

½ u u

f ð x xÞ þ vv

gð x xÞ is negative definite and if r vv

gð x xÞ 6¼ 0, then x x is Pareto minimal for (MOP).

Proof. Since ð x x, u u, vvÞ is Pareto minimal for (DMOP) then the following Fitz John conditions hold [4]: there exists 2

, 2

, 2

, 2

, and 2

, such that

ðrf ð x xÞÞ

þ r

½ u u

f ð x xÞ þ vv

gð x xÞ

vv

gð x xÞ ¼ 0, ð37Þ

rf ð x xÞ ¼ 0, ð38Þ

rgð x xÞ

x xÞ ¼ 0, ð39Þ

vv

gð x xÞ ¼ 0, ð40Þ

t

vv ¼ 0, ð41Þ

^t

u u ¼ 0, ð42Þ

ð,

0: ð43Þ

ð,

0: ð44Þ

Multiplying (38) by u u

and using (42), multiplying (39) by vv

and using (40) and (41), we obtain

u

u

ðrf ð x xÞÞ ¼ 0, ð45Þ

vv

ðrgð x xÞÞ ¼ 0: ð46Þ

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Premultiplying (37) by

and using (46), we have

ðrf ð x xÞÞ

þ

r

½ u u

f ð x xÞ þ vv

gð x xÞ ¼ 0, ð47Þ or

þ

r

½ u u

f ð x xÞ þ vv

gð x xÞ ¼ 0: ð48Þ We now claim that

6¼ 0: ð49Þ

Otherwise, from (48), we have

^t

r

½ u u

f ð x xÞ þ vv

gð x xÞ ¼ 0: ð50Þ Since r

½ u u

f ð x xÞ þ vv

gð x xÞ is assumed negative definite, ¼ 0. Therefore, from (37) we have

r vv

gð x xÞ ¼ 0: ð51Þ

Using the fact that r vv

gð x xÞ 6¼ 0, from (51) we obtain ¼ 0, from (38), ¼ 0 and from (39), ¼ 0 contradicting (44). Hence, (49) holds.

From (48) and using (43) we obtain

^t

r

½ u u

f ð x xÞ þ vv

gð x xÞ ¼

0: ð52Þ Since r

½ u u

f ð x xÞ þ vv

gð x xÞ is assumed negative definite,

¼ 0. Hence ¼ 0;

therefore, from (37) we have

^t

rf ð x xÞ ¼ vv

rgð x xÞ, ð53Þ

From (22), we have

u u

rf ð x xÞ ¼ vv

rgð x xÞ: ð54Þ

From (53) and (54), we get

ð

u u

Þðrgð x xÞÞ

vv ¼ 0: ð55Þ

Using the hypothesis that, the vector r vv

gð x xÞ 6¼ 0, from (55) we get

¼ u u: ð56Þ

Hence 6¼ 0, because 6¼ 0. Using (56) with ¼ 0 in (39), we obtain

gð x xÞ

0, ð57Þ

which shows that x x is feasible for the primal. Therefore, using Corollary 4.1 and the hypothesis of the theorem, we have that x x is Pareto minimal for

(MOP).

It is very important to observe here, in Theorem 4.3, the Hessian matrix r

½ u u

f ð x xÞ þ vv

gð x xÞ is only assumed negative definite rather than positive or negative definite as in [15]. Of course, to get the desired results we need the condition u u

0, which leads to the following theorem.

786 AGHEZZAF AND HACHIMI

Downloaded by [Univ of Louisiana at Lafayette] at 00:20 28 December 2014

Theorem 4.4. Converse Duality. Let ð x x, u u, vvÞ be Pareto minimal for (DMOP) with u u

0, and let the hypotheses of Theorem 4.1 hold. If the n n Hessian matrix r

½ u u

f ð x xÞ þ vv

gð x xÞ is positive or negative definite and if r vv

gð x xÞ 6¼ 0, then x x is Pareto minimal for (MOP).

Proof. It follows on the lines of Theorem 4.3.

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