on nonsmooth multiobjective fractional programming problems ... - Core

Yugoslav Journal of Operations Research 23 (2013) Number 3, 367-386 DOI: 10.2298/YJOR130131012J

ON NONSMOOTH MULTIOBJECTIVE FRACTIONAL PROGRAMMING PROBLEMS INVOLVING ( p, r ) − ρ − (η , θ ) - INVEX FUNCTIONS Anurag JAYSWAL Department of Applied Mathematics, Indian School of Mines, Dhanbad-826 004, Jharkhand, India [email protected]

Ashish Kumar PRASAD Department of Applied Mathematics, Indian School of Mines, Dhanbad-826 004, Jharkhand, India [email protected]

I. M. STANCU-MINASIAN Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 13 Septembrie Street, No.13, 050711 Bucharest, Romania [email protected] Received: January 2013 / Accepted: April 2013 Abstract: A class of multiobjective fractional programming problems (MFP) is considered where the involved functions are locally Lipschitz. In order to deduce our main results, we introduce the definition of ( p, r ) − ρ − (η , θ ) -invex class about the Clarke generalized gradient. Under the above invexity assumption, sufficient conditions for optimality are given. Finally, three types of dual problems corresponding to (MFP) are formulated, and appropriate dual theorems are proved. Keywords: Multiobjective fractional programming, Clarke gradient, efficiency, sufficient optimality conditions, duality theorems. MSC: 90C32, 90C46, 49N15.

( p, r ) − ρ − (η ,θ ) -invexity,

368

А. Јаyswal, A.K.Prasad, I.M. Stancu-Minasian / On Nonsmooth

1. INTRODUCTION In recent past, optimality conditions and duality results have been of much interest for a class of multiobjective fractional programming problems, where the involved functions are locally Lipschitz and have Clarke differentiability. Many researchers have studied this matter in the presence of various assumptions. See, for example, [4, 5, 10, 11, 13-16] and the references cited therein. Jeyakumar [9] gave the optimality and duality for nondifferentiable nonconvex program under the ρ -invexity assumptions. Chen [4] studied the optimality and duality aspects of a class of nonsmooth multiobjective fractional programming problems expressed in terms of the Clarke generalized gradient under certain generalized ( F , ρ ) convexity assumptions. In particular, using parametric approach, Bector et al. [2] derived Fritz John and Karush-Kuhn-Tucker necessary and sufficient optimality conditions for a class of nonsmooth (nondifferentiable) convex multiobjective fractional optimization problems, and established some duality theorems. Following the approaches of Bector et al. [2], Liu [11] obtained necessary and sufficient optimality conditions and derived duality theorems for a class of nonsmooth multiobjective fractional programming problems involving nonsmooth ( F , ρ ) -convex functions. Recently, Mandal and Nahak [12] introduced a new class of invex functions by combining the concepts of

( p, r ) -invexity

( p, r ) − ρ − (η ,θ ) -

[1] and the notions of

ρ − (η , θ ) -invex functions [17] and they established symmetric duality results. Jayswal et

al. [8] considered a class of functions called

( p, r ) − ρ − (η ,θ ) -invex

functions for a

multiobjective fractional programming problem with inequality constraints in the differentiable case and derived sufficient conditions and duality theorems. However, the corresponding conclusions cannot be obtained for nondifferentiable programming with the help of generalized ( p, r ) − ρ − (η , θ ) -invex functions because the derivative is required in the definitions of such functions. There exists a generalization of convexity to locally Lipschitz functions, with derivative replaced by the Clarke generalized gradient. Consequently, in the present paper, we are concerned with the following nondifferentiable multiobjective fractional programming problem: (MFP)

⎛ f1 ( x ) f ( x) ⎞ ,..., k Minimize ⎜ ⎟ ⎜ g1 ( x ) g k ( x ) ⎟⎠ ⎝

subject to

{

}

Ω = x ∈ X : h ( x ) = ( h1 , h2 ,..., hm )( x ) ∈ − R+m , x ∈ X ⊂ Rn ,

(1)


369

n

where X is a separable reflexive Banach space in the n-dimensional Euclidean space R . f i , gi : X → R, i = 1, 2,..., k and h : X → R m are locally Lipschitz functions on X.

Without loss of generality, we can assume that f i ( x ) > 0, gi ( x ) > 0 , for all i = 1, 2,..., k and x ∈ X .

Definition 1.1 A feasible solution x* of (MFP) is said to be an efficient solution of (MFP) if and only if there exist no other feasible solution x ∈ Ω such that

( ) g ( x) g ( x ) fi ( x )

f i x*

ξ T η ( x , u ) + ρ θ ( x , u ) r ( > w ith x ≠ u ) for p = 0, r ≠ 0,

f ( x ) − f (u ) >

1 T pη ( x , u ) ξ (e − 1) + ρ θ ( x , u ) p

(> f

(x )−

f (u ) > ξ T η ( x , u ) + ρ θ

2

w ith x ≠ u ) fo r p ≠ 0, r = 0,

(x,u ) ( > w ith 2

x ≠ u ) fo r p = 0 , r = 0 ,

371


∀ξ ∈ ∂f ( u ) , then f is said to be (strictly)

( p, r ) − ρ − (η ,θ ) -invex at the point

u∈ X

with respect to η and θ . Remark 2.1 If the above inequalities are satisfied at any point u ∈ X , then f is said to

be (strictly) ( p, r ) − ρ − (η , θ ) -invex on X with respect to η and θ . Remark 2.2 It should be noted that the exponentials appearing on the right-hand sides of inequalities above are understood to be taken componentwise and 1 = (1,1,...,1) ∈ R n .

In the sequel, we need the following results. Lemma 2.1 [3] The point x is an optimal solution to problem (MFP) if and only if x solves (SFPi), where (SFPi) is given as the following problems: (SFPi)

Minimize

fi ( x )

gi ( x )

subject to

⎧⎪ f p ( x) f p ( x ) notation ϕ p ( x ) , p ≠ i, p = 1, 2,..., k , h ( x ) ∈ − R+m x ∈ Mi = ⎨x ∈ X : < ≡≡≡ g x g x ( ) ( ) p p ⎩⎪

{

⎫⎪ ⎬ ⎭⎪

}

= x ∈ X : f p ( x ) − ϕ p ( x ) g p ( x ) < 0, p ≠ i, p = 1, 2,..., k , h ( x ) ∈ − R+m . Theorem 2.1 (Necessary optimality conditions) [11]. If x * is an optimal solution of (MFP) and satisfies a constraint qualification [7, Theorem 12] for (SFPi), i = 1, 2,..., k , then there exist y* ∈ R+k , z* ∈ Rm , such that k

( )

( )

( )

( )

m

0 ∈ ∑ yi* ⎡∂f i x* + ϕi x* ∂ ( − gi ) x* ⎤ + ∑ z *j ∂h j x* , ⎣ ⎦ j =1 i =1

( )

(2)

( ) ( )

(3)

z *j h j x* = 0, for all j = 1, 2,..., m,

(4)

f i x* − ϕi x* gi x* = 0, for all i = 1, 2,..., k ,

( )

( )

h j x* < 0, for all j = 1, 2,..., m,

(5)

y* ∈ I , z* ∈ R+m ,

(6)

{

(

)

where I = y* ∈ R k : y* = y1* , y2* ,..., yk* > 0 and

∑

k i =1

}

yi* = 1 .

Remark 2.3 All the theorems in the subsequent parts of this paper will be proved only in the case when p ≠ 0, r ≠ 0 . The proofs in other cases are easier than in this one, since the differences arise only from the form of inequality. Moreover, without loss of generality,

372


we assume that p > 0 and r > 0 . Furthermore, we assume that ρ , ρ 1 and ρ 2 are all elements of R .

3. SUFFICIENT OPTIMALITY CONDITIONS Now we establish sufficient optimality conditions under introduced classes of functions defined in the previous section. Theorem 3.1 (Sufficiency). Let x* ∈ Ω be a feasible solution for (MFP) to which conditions (2) to (6) are satisfied. Moreover, we assume that any one of the following conditions holds: k m a) ρ > 0 and A (.) = ∑ i =1 yi* ⎡ f i (.) − ϕi x* gi (.) ⎤ + ∑ j =1 z*j h j (.) is ⎣ ⎦

( )

( p, r ) − ρ b)

− (η , θ ) - invex at x with respect to η and *

θ;

ρ 1+ ρ 2 > 0 and B (.) = ∑ i =1 yi* ⎡ fi (.) − ϕi ( x* ) gi (.) ⎤ is ⎣ ⎦ k

invex at

x*

with respect to η

( p, r ) − ρ 2 − (η ,θ ) -invex at

and θ

and

( p, r ) − ρ 1 − (η ,θ ) -

C (.) = ∑ j =1 z *j h j (.) m

is

x* with respect to same η and θ .

Then x* is an efficient solution to (MFP). Proof. Let x* ∈ Ω be a feasible solution to (MFP). By the relation (2), there exist

ξi ∈ ∂fi ( x* ) , ζ i ∈ ∂ ( − gi ) ( x* ) , i = 1, 2,..., k and ψ j ∈ ∂h j ( x* ) , j = 1, 2,..., m , such that yi* ⎡ξi + ϕi ( x* ) ζ i ⎤ + ∑ z *jψ j = 0 . ∑ ⎣ ⎦ j =1 i =1 k

m

(7)

If condition (a) holds, then m ⎤ ⎛ pη ( x , x* ) 1 ⎛ r ( A( x ) − A( x* ) ) ⎞ 1 ⎡ k * ⎡ ⎞ − 1⎟ > ⎢ ∑ yi ξi + ϕi x* ζ i ⎤ + ∑ z *jψ j ⎥ ⎜ e − 1 ⎟ + ρ θ x, x* ⎜e ⎣ ⎦ r⎝ ⎝ ⎠ j =1 ⎠ p ⎣ i =1 ⎦

( )

(

)

2

The above inequality together with (7) and the assumption that ρ > 0 , imply 1 ⎛ r ( A( x ) − A( x* ) ) ⎞ − 1 ⎟ > 0. ⎜e r⎝ ⎠

Using fundamental property of the exponential function, we get

( )

A ( x ) > A x* .

(8)

On the other hand, suppose contrary to the result that x* is not an efficient solution of (MFP). Then, there exists a feasible solution x of (MFP) such that

373


( ) g ( x) g ( x )

for all i = 1, 2,..., k ,

( ) g ( x) g ( x )

for some t ∈ {1, 2,..., k } ,

fi ( x )

0 , we obtain

(12)

374


⎤ ⎛ pη ( x , x* ) 1⎡ k * ⎞ yi ⎡⎣ξi + ϕi ( x* )ζ i ⎤⎦ ⎥ ⎜ e − 1 ⎟ + ρ 1 θ x, x* ∑ ⎢ p ⎣ i =1 ⎝ ⎠ ⎦

(

)

2

>0.

(13)

From the ( p, r ) − ρ 1 − (η ,θ ) -invexity of B (.) , we have ⎤ ⎛ pη ( x , x* ) 1 ⎛ r ( B ( x ) − B ( x* ) ) ⎞ 1 ⎡ k * ⎞ − 1 ⎟ > ⎢ ∑ yi ⎣⎡ξi + ϕi ( x* )ζ i ⎦⎤ ⎥ ⎜ e − 1 ⎟ + ρ 1 θ x , x* ⎜e r⎝ p ⎠ ⎠ ⎣ i =1 ⎦⎝

(

)

2

. (14)

From (13) and (14), we obtain

B ( x ) > B ( x* ) , i.e., k

∑y i =1

If x

*

* i

⎡ f i ( x ) − ϕi ( x* ) g i ( x )⎤ > ⎣ ⎦

k

∑y i =1

* i

⎡ f i ( x* ) − ϕi ( x* ) g i ( x* ) ⎤ . ⎣ ⎦

(15)

is not an efficient solution of (MFP) then, we get (9) in the same way. But (9) *

contradicts (15). Therefore, x is an efficient solution of (MFP). This completes the proof. □

4. PARAMETRIC DUALITY We consider the following form of parametric dual as follows: (PD) Maximize v = ( v1 , v2 ,..., vk ) subject to k

m

i =1

j =1

0 ∈ ∑ yi ⎡⎣∂f i ( u ) + vi ∂ ( − gi )( u ) ⎤⎦ + ∑ z j ∂h j ( u ) ,

(16)

f i ( u ) − vi gi ( u ) = 0, for all i = 1, 2,..., k ,

(17)

m

z j h j ( u ) = 0, ∑ j =1

(18)

u ∈ X , y ∈ I , z ∈ R+m , v > 0.

(19)

k m ⎧⎪ Let Γ = ⎨( u , y, z , v ) ∈ X × I × R+m × R k : 0 ∈ ∑ yi ⎡⎣∂f i ( u ) + vi ∂ ( − gi )( u ) ⎤⎦ + ∑ z j ∂h j ( u ), ⎪⎩ i =1 j =1 m ⎪⎫ f i ( u ) − vi gi ( u ) = 0, for all i = 1, 2,..., k , ∑ z j h j ( u ) = 0⎬ j =1 ⎪⎭


375

denote the set of all feasible solutions of (PD). Moreover, we denote by prX Γ the

{

}

projection of the set Γ on X , i.e. prX Γ = u ∈ X : ( u, y, z, v ) ∈ Γ . Theorem 4.1 (Weak duality). Let

x be a feasible solution for (MFP), and let ( u, y, z, v )

be a feasible solution for (PD). Moreover, we assume that any of the following condition holds: a)

ρ > 0 and O (.) = ∑ i =1 yi ⎡⎣ fi (.) − vi gi ( .)⎤⎦ + ∑ j =1 z j h j (.) is ( p, r ) − ρ − (η , θ ) k

m

invex at u ∈ Ω ∪ prX Γ with respect to η and θ ;

b) ρ 1+ ρ 2 > 0 and P (.) = ∑ i =1 yi ⎡⎣ fi (.) − vi gi (.)⎤⎦ is ( p, r ) − ρ 1 − (η ,θ ) ‐invex k

and Q (.) = ∑ j =1 z j h j (.) is m

( p, r ) − ρ 2 − (η ,θ ) ‐invex at

u ∈ Ω ∪ prX Γ with

respect to same η and θ .

Then ϕ ( x ) ≤/ y . Proof. Let

x and ( u, y, z, v ) be feasible solution to (MFP) and (PD), respectively. Then

there exist ξi ∈ ∂f i ( u ) , ζ i ∈ ∂ ( − gi ) ( u ) , i = 1, 2,..., k

and ψ j ∈ ∂h j ( u ) , j = 1, 2,..., m ,

such that k

m

yi ⎣⎡ξi + viζ i ⎦⎤ + ∑ z jψ j = 0 . ∑ i =1 j =1

(20)

If condition (a) holds, then

)

(

m ⎤ pη x ,u 2 1 r (O( x )−O(u )) 1⎡ k − 1 > ⎢ ∑ yi ⎣⎡ξi + viζ i ⎦⎤ + ∑ z jψ j ⎥ e ( ) − 1 + ρ θ ( x, u ) . e r p ⎣ i =1 j =1 ⎦

(

)

The above inequality together with (20) and the assumption that ρ > 0 , imply

)

(

1 r (O( x )−O(u )) e − 1 > 0. r Using fundamental property of the exponential function, we get O ( x ) > O (u ). On the other hand, suppose contrary to the result that ϕ ( x ) ≤ v . Then fi ( x )

gi ( x ) ft ( x )

gt ( x )

< vi for all i = 1, 2,..., k , < vt for some t ∈ {1, 2,..., k} ,

(21)

376


that is, fi ( x ) − v i gi ( x ) < 0 = f i ( u ) − vi gi ( u ) for all i = 1, 2,..., k ,

ft ( x ) − vt gt ( x ) < 0 = ft ( u ) − vt gt ( u ) for some t ∈ {1, 2,..., k} . By (19) and the above inequalities, we have k

∑y i =1

i

⎡⎣ fi ( x ) − v i g i ( x )⎤⎦
⎢ ∑ z jψ j ⎥ e ( ) − 1 + ρ 2 θ ( x, u ) , ∀ψ j ∈ ∂h j ( u ) . e r p ⎣ j =1 ⎦

(

)

(24)

From (23) and (24) and the fundamental property of the exponential function, we obtain ⎤ pη ( x,u ) 2 1⎡m − 1 + ρ 2 θ ( x, u ) < 0, ∀ψ j ∈ ∂h j ( u ) . ⎢ ∑ z jψ j ⎥ e p ⎣ j =1 ⎦

(

)

(25)

By (20), (25) and the assumption that ρ 1+ ρ 2 > 0 , we obtain

(

)

2 ⎤ pη x ,u 1⎡ k yi ⎡⎣ξi + viζ i ⎤⎦⎥ e ( ) − 1 + ρ 1 θ ( x, u ) > 0 . ∑ ⎢ p ⎣ i =1 ⎦

(26)

From the ( p, r ) − ρ 1 − (η , θ ) -invexity of P (.) , we have

(

)

(

)

2 ⎤ pη x ,u 1 r ( P ( x )− P (u )) 1⎡ k e − 1 > ⎢ ∑ yi ⎣⎡ξi + vi ζ i ⎦⎤⎥ e ( ) − 1 + ρ 1 θ ( x, u ) . r p ⎣ i =1 ⎦


(27)

377


P ( x ) > P (u ) , i.e., k

k

yi ⎡⎣ f i ( x ) − v i g i ( x )⎤⎦ > ∑ yi ⎡⎣ fi (u ) − v i g i (u )⎤⎦ . ∑ i =1 i =1

(28)

Again if ϕ ( x ) ≤ v , then we get (22) in the same way. But (22) contradicts (28). Therefore, ϕ ( x ) ≤/ v . This completes the proof.

□

*

Theorem 4.2 (Strong duality). Let x be an efficient solution for (MFP) and let h satisfy the constraints qualification at x* . Then there exist y* ∈ I , z* ∈ R m and v* ∈ R k such

(

that x* , y* , z * , v*

)

is feasible for (PD). Also, if the weak duality theorem 4.1 holds for

(

)

all feasible solutions of the problems (MFP) and (PD), then x* , y* , z * , v* is an efficient solution for (PD). Proof. Since x* is an efficient solution for (MFP) and h satisfy the constraints qualification at x* , by Theorem 2.1, there exist y* ∈ I , z* ∈ R m and v* ∈ R k such that

( x , y , z , v ) satisfies (2) - (6). This, in turn, implies that ( x , y , z , v ) is feasible for *

*

*

*

*

*

( )

solution to (PD). This completes the proof. Theorem 4.3 (Strict converse duality). Let x for (MFP) and (PD), respectively with vi* =

*

(

*

*

( x, y , z , v )

(PD). From the weak duality theorem, for any feasible points inequality ϕ x* ≥ v holds. Hence we conclude that

*

(x , y , z ,v ) *

*

*

*

*

*

*

)

to (PD), the

is an efficient □

and u , y , z , v be efficient solutions

( ) g (x ) f i x*

*

for all i = 1, 2,..., k . Assume that

i

ρ > 0 and A (.) = ∑ i =1 yi* ⎡⎣ f i (.) − vi* gi (.) ⎤⎦ + ∑ j =1 z *j h j (.) k

m

is strictly ( p, r ) − ρ − (η , θ ) -invex at u* ∈ Ω ∪ prX Γ with respect to η and θ . Then x* = u * ; that is, u* is an efficient solution for (MFP).

( )

Proof. Suppose on contrary that x* ≠ u * . By relation (16), there exist ξi ∈ ∂f i x* ,

ζ i ∈ ∂ ( − gi ) ( x* ) , i = 1, 2,..., k and ψ j ∈ ∂h j ( x* ) , j = 1, 2,..., m , such that k

m

yi* ⎡⎣ξi + vi*ζ i ⎤⎦ + ∑ z *jψ j = 0 . ∑ i =1 j =1

(29)

378


From (17), (18) and (19), we get

( )

( )

k

( )

m

( )

A u* = ∑ yi* ⎡ fi u * − vi* gi u* ⎤ + ∑ z *j h j u * = 0 . ⎣ ⎦ j =1 i =1

(30)

From the strict ( p, r ) − ρ − (η , θ ) -invexity of A (.) at u* , we have m ⎤ ⎛ pη ( x* ,u* ) ⎞ 1 ⎛ r ( A( x* ) − A( u* ) ) ⎞ 1 ⎡ k * * * ⎡ ⎤ − > + + − 1 ⎟ + ρ θ x* , u * e y ξ v ζ 1 ⎢∑ i ⎣ i i i ⎦ ∑ z jψ j ⎥ ⎜ e ⎜ ⎟ r⎝ p ⎝ ⎠ ⎠ j =1 ⎣ i =1 ⎦

(

)

2

. (31)

The above inequality together with (29) and the assumption that ρ > 0 imply 1 ⎛ r ( A( x* ) − A( u* ) ) ⎞ − 1 ⎟ > 0. ⎜e r⎝ ⎠

Using fundamental property of the exponential function, we get

( )

( )

A x* > A u * . Since vi* =

( ) g (x ) f i x*

(32)

for all i = 1, 2,..., k , we have

*

i

( )

( )

fi x* − vi* gi x* = 0 for all i = 1, 2,..., k .

(33)

From the relation h ( x ) ∈ − R+m and (19), we have z *j h j ( x* ) < 0 . ∑ j =1 m

(34)

Therefore, from (33) and (34), we conclude that

( )

k

( )

( )

m

( )

A x* = ∑ yi* ⎡ f i x* − vi* gi x* ⎤ + ∑ z *j h j x* < 0 . ⎣ ⎦ j =1 i =1

(35)

( )

Hence from (32) and (35), we have A u* < 0 which contradicts (30). Hence x* = u * . This completes the proof. □ Remark 4.1 The function A (.) in Theorem 4.3 is expressed by the sum of the modified objective part B (.) of (MFP) and its constraint part C (.) . If B (.) is strictly

( p, r ) − ρ still holds.

− (η , θ ) -invex and C (.) is

( p, r ) − ρ

− (η , θ ) -invex then the Theorem 4.3

379


5. WOLFE DUALITY In what follows, we take the following form of theorem 2.1: Theorem 5.1 Let x* be an efficient solution to (MFP). Assume that

h satisfies the

constraint qualification at x . Then there exist y ∈ R and z ∈ R , such that *

*

k +

*

m

k m ⎡ ⎤ 0 ∈ ∑ yi* gi x* ⎢∂f i x* + ∑ z *j ∂h j x* ⎥ i =1 j =1 ⎣ ⎦

( )

( )

( )

k m ⎡ ⎤ + ∑ yi* ⎢ f i x* + ∑ z *j h j x* ⎥ −∂gi x* , i =1 j =1 ⎣ ⎦

( )(

( )

( ))

( )

z *j h j x* = 0, for all j = 1, 2,..., m,

( )

(36) (37)

h j x* < 0, for all j = 1, 2,..., m,

(38)

y* ∈ I , z* ∈ R+m .

(39)

Now we consider the following Wolfe type dual problem to (FP): m m ⎛ ⎞ fk (u ) + ∑ z j h j ( u ) ⎟ ⎜ f1 ( u ) + ∑ z j h j ( u ) j =1 j =1 ⎟ ,..., (WD) Maximize ⎜ ⎜ ⎟ g1 ( u ) gk ( u ) ⎜ ⎟ ⎝ ⎠ subject to k m ⎡ ⎤ 0 ∈ ∑ yi gi ( u ) ⎢ ∂f i ( u ) + ∑ z j ∂h j ( u ) ⎥ i =1 j =1 ⎣ ⎦ k m ⎡ ⎤ + ∑ yi ⎢ f i ( u ) + ∑ z j h j ( u ) ⎥ ( −∂gi ( u ) ) , i =1 j =1 ⎣ ⎦

u ∈ X , y ∈ I , z ∈ R+m .

(40)

(41)

k m ⎧⎪ ⎡ ⎤ Let Γ% = ⎨( u , y, z ) ∈ X × I × R+m : 0 ∈ ∑ yi g i ( u ) ⎢ ∂f i ( u ) + ∑ z j ∂h j ( u ) ⎥ i =1 j =1 ⎪⎩ ⎣ ⎦ k m ⎫⎪ ⎡ ⎤ + ∑ yi ⎢ fi ( u ) + ∑ z j h j ( u ) ⎥ ( −∂g i ( u ) ) ⎬ i =1 j =1 ⎣ ⎦ ⎭⎪

denote the set of all feasible solutions of (WD). Moreover, we denote by prX Γ% the projection of the set Γ% on X .

380

А. Јаyswal, A.K.Prasad, I.M. Stancu-Minasian / On Nonsmooth m

Denote

fi ( u ) + ∑ z j h j ( u )

ψ i ( u, z ) =

j =1

and

gi ( u )

ψ ( u , z ) = (ψ 1 ( u , z ) ,ψ 2 ( u , z ) ,...,ψ k ( u , z ) ) . m

Throughout this section, we assume that fi ( u ) + ∑ z j h j ( u ) > 0 and gi ( u ) > 0, for all j =1

i = 1, 2,..., k .

x be a feasible solution for (MFP), and let ( u, y, z ) be

Theorem 5.2 (Weak duality). Let

a feasible solution for (WD). Assume that ρ > 0 and

is

k m m ⎡ ⎤ k ⎡ ⎤ S (.) = ∑ yi gi ( u ) ⎢ fi (.) + ∑ z j h j (.) ⎥ − ∑ yi gi (.) ⎢ fi ( u ) + ∑ z j h j ( u ) ⎥ i =1 j =1 j =1 ⎣ ⎦ i =1 ⎣ ⎦ % ( p, r ) − ρ − (η ,θ ) -invex at u ∈ Ω ∪ prX Γ with respect to η and θ . Then

ϕ ( x ) ≤/ ψ ( u, z ) .

x and ( u, y, z ) be feasible solution to (MFP) and (WD), respectively. By the

Proof. Let

relation (40) there exist ξi ∈ ∂f i ( u ) , ζ i ∈ ∂ ( − gi ) ( u ) , i = 1, 2,..., k and ψ j ∈ ∂h j ( u ) ,

j = 1, 2,..., m such that ⎡

k

m

⎤

k

⎡

m

⎤

yi gi ( u ) ⎢ξi + ∑ z jψ j ⎥ + ∑ yi ⎢ fi ( u ) + ∑ z j h j ( u ) ⎥ ζ i = 0 . ∑ i =1 j =1 i =1 j =1 ⎣

⎦

⎣

From ( p, r ) − ρ − (η , θ ) -invexity of S (.) at

⎦

(42)

u with respect to η and θ , we have

)

(

m m ⎡ ⎤ k ⎡ ⎤ ⎤ 1 r ( S ( x ) − S (u )) 1⎡ k − 1 > ⎢ ∑ yi g i ( u ) ⎢ξi + ∑ z jψ j ⎥ + ∑ yi ⎢ f i ( u ) + ∑ z j h j ( u ) ⎥ ζ i ⎥ e r p ⎢⎣ i =1 j =1 j =1 ⎣ ⎦ i =1 ⎣ ⎦ ⎦⎥

(e

pη ( x ,u )

)

− 1 + ρ θ ( x, u ) . 2

(43)

The above inequality together with (42) and the assumption that ρ > 0 imply

(

)

1 r ( S ( x ) − S (u )) e − 1 > 0. r Using fundamental property of the exponential function, we get S ( x ) > S ( u ) = 0, i.e., S ( x ) > 0. On the other hand, suppose contrary to the result that ϕ ( x ) ≤ ψ ( u , z ) . Then

(44)

381

А. Јаyswal, A.K.Prasad, I.M. Stancu-Minasian / On Nonsmooth m

fi ( x )

gi ( x )

0 2

and k m m ⎡ ⎤ k ⎡ ⎤ T (.) = ∑ yi* gi u* ⎢ fi (.) + ∑ z*j h j (.) ⎥ − ∑ yi* gi (. ) ⎢ f i u* + ∑ z*j h j u* ⎥ i =1 j =1 j =1 ⎣ ⎦ i =1 ⎣ ⎦

( )

( )

( )

is strictly ( p, r ) − ρ − (η , θ ) -invex at u * ∈ Ω ∪ prX Γ% with respect to η and θ . Then *

x* = u * ; that is, u is an efficient solution for (MFP). *

Proof. Suppose on contrary that x* ≠ u * . Let x and (u * , y* , z * ) be feasible solution to (MFP) and (WD), respectively. By the relation (36) there exist

ξi ∈ ∂fi ( u * ) , ζ i ∈ ∂ ( − gi ) ( u* ) , i = 1, 2,..., k and ψ j ∈ ∂h j ( u * ) , j = 1, 2,..., m, such that ⎡

⎤

⎡

⎤

yi gi ( u ) ⎢ξi + ∑ z jψ j ⎥ + ∑ yi ⎢ fi ( u ) + ∑ z j h j ( u ) ⎥ ζ i = 0 . ∑ i= j= i= j= k

*

*

m

⎣

1

*

k

⎦

1

*

*

m

⎣

1

1

*

*

(46)

⎦

(

)

From Theorem 5.3, we know that there exist y and z such that x* , y , z is an efficient solution for (WD) and

( )

( )

m

f i x* + ∑ z j h j x* j =1

( )

gi x*

=

( )

m

( )

f i u* + ∑ z*j h j u * j =1

( )

gi u*

.

(47)

By (37), (39) and (47), we obtain

( )= g (x ) *

fi x

*

i

Hence

( )

m

( )

fi u* + ∑ z *j h j u * j =1

( )

gi u*

.

(48)

383

А. Јаyswal, A.K.Prasad, I.M. Stancu-Minasian / On Nonsmooth m ⎡ ⎤ f i x* gi u* = ⎢ fi u * + ∑ z*j h j u* ⎥ gi x* , j =1 ⎣ ⎦

( ) ( )

( )

( ) ( )

(49)


( )

( )∑ z h (x )

k

m

T x* = ∑ yi* gi u* i =1

j =1

( )

* j

*

j

.

( )

From the relation h x* ∈ − R+m , gi u* > 0 , from (41) and the above inequality, we have

( )

T x* < 0. Therefore,

( )

( )

T x* < 0 = T u * , i.e.,

( )

( )

T x* < T u * .

(50)

From the strict ( p, r ) − ρ − (η , θ ) -invexity of T (.) at u* in Ω ∪ prX Γ% with respect to η and θ , we have m m ⎡ ⎤ k ⎡ ⎤ ⎤ 1 ⎛ r ( T ( x* ) − T ( u * ) ) ⎞ 1 ⎡ k * − 1⎟ > ⎢ ∑ yi g i u * ⎢ξ i + ∑ z *jψ j ⎥ + ∑ yi* ⎢ f i u * + ∑ z *j h j u * ⎥ ζ i ⎥ ⎜e r⎝ ⎠ p ⎢⎣ i =1 j =1 j =1 ⎣ ⎦ i =1 ⎣ ⎦ ⎦⎥

( )

⎛ pη ( x* ,u* ) ⎞ − 1 ⎟ + ρ θ x* , u * ⎜e ⎝ ⎠

(

)

2

( )

( )

.

The above inequality together with (50) imply m m ⎡ ⎤ k *⎡ ⎤ ⎤ ⎛ pη ( x* ,u* ) 1⎡ k * ⎞ * * * * * − 1 ⎟ + ρ θ x* , u * ⎢ ∑ yi gi u ⎢ξ i + ∑ z jψ j ⎥ + ∑ yi ⎢ fi u + ∑ z j h j u ⎥ ζ i ⎥ ⎜ e p ⎣⎢ i =1 ⎝ ⎠ j =1 j =1 ⎣ ⎦ i =1 ⎣ ⎦ ⎦⎥

( )

( )

( )

(

which by virtue of (46) imply

ρ θ ( x* , u * ) < 0 2

,

(

which contradicts the assumptions that ρ θ x* , u *

)

2

> 0 . Hence x* = u * ; that is, u* is

an efficient solution for (FP). This completes the proof.

□

)

2

,

384


6. MOND-WEIR DUALITY In this section, we consider the following Mond-Weir dual to (MFP):

⎛ f1 ( u ) f (u ) ⎞ ,..., k (MWD) Maximize ⎜ ⎟ ⎜ g1 ( u ) g k ( u ) ⎟⎠ ⎝ subject to k m m ⎡ ⎤ k ⎡ ⎤ 0 ∈ ∑ yi gi ( u ) ⎢∂fi ( u ) + ∑ z j ∂h j ( u )⎥ + ∑ yi ( −∂gi ( u ) ) ⎢ f i ( u ) + ∑ z j h j ( u ) ⎥ ,(51) i =1 j =1 j =1 ⎣ ⎦ i =1 ⎣ ⎦ m

z j h j ( u ) > 0, ∑ j =1

(52)

u ∈ X , y ∈ I , z ∈ R+m .

(53)

Let k m ⎧⎪ ⎡ ⎤ Γ = ⎨( u , y , z ) ∈ X × I × R+m : 0 ∈ ∑ yi g i ( u ) ⎢ ∂f i ( u ) + ∑ z j ∂h j ( u ) ⎥ i =1 j =1 ⎪⎩ ⎣ ⎦

k m ⎫⎪ ⎡ ⎤ m + ∑ yi ( −∂gi ( u ) ) ⎢ fi ( u ) + ∑ z j h j ( u ) ⎥ , ∑ z j h j ( u ) > 0 ⎬ , i =1 j =1 ⎪⎭ ⎣ ⎦ j =1

denote the set of all feasible solutions to (MWD). Moreover, we denote by prX Γ the projection of the set Γ on X . fi ( u ) Denote Φ i ( u ) = and Φ ( u ) = ( Φ1 ( u ) , Φ 2 ( u ) ,..., Φ k ( u ) ) . gi ( u ) Now we shall state weak, strong and strict converse duality theorems without proof as they can be proved in light of Theorem 5.2, Theorem 5.3 and Theorem 5.4, proved in previous section. Theorem 6.1 (Weak duality). Let x be a feasible solution for (MFP), and let ( u , y , z ) be a feasible solution for (MWD). Assume that k m m ⎡ ⎤ k ⎡ ⎤ ρ > 0 and S (.) = ∑ yi gi ( u ) ⎢ fi (.) + ∑ z j h j (.) ⎥ − ∑ yi gi (.) ⎢ fi ( u ) + ∑ z j h j ( u ) ⎥ i =1 j =1 j =1 ⎣ ⎦ i =1 ⎣ ⎦ is ( p, r ) − ρ − (η , θ ) -invex at u ∈ Ω ∪ prX Γ with respect to η and θ . Then, ϕ ( x ) ≤/ Φ ( u ) . Theorem 6.2 (Strong duality). Let x* be an efficient solution for (MFP) and let h satisfy

the constraints qualification at x* . Then there exist y* ∈ I and z* ∈ R m such that

385


(x , y , z ) *

*

*

is feasible to (MWD). Also, if the weak duality theorem 6.1 holds for all

(

feasible solutions to the problems (MFP) and (MWD), then x* , y* , z* solution for (MWD).

(

)

is an efficient

)

Theorem 6.3 (Strict converse duality). Let x* and u* , y* , z* be efficient solutions for

(

(MFP) and (MWD), respectively. Assume that ρ θ x* , u *

)

2

> 0 and

k m m ⎡ ⎤ k ⎡ T (.) = ∑ yi* gi u * ⎢ f i (.) + ∑ z*j h j (.) ⎥ − ∑ yi* gi (.) ⎢ f i u* + ∑ z *j h j u* i =1 j =1 j =1 ⎣ ⎦ i =1 ⎣ * is strictly ( p, r ) − ρ − (η , θ ) -invex at u ∈ Ω ∪ prX Γ with respect to

( )

( )

⎤

( )⎥ ⎦

η and θ . Then

x = u ; that is, u is an efficient solution for (MFP). *

*

*

Acknowledgement: The research of the first author is financially supported by University Grant Commission, New Delhi, India, through grant no. (F. No. 41801/2012(SR)).

REFERENCES [1]

Antczak, T., “ ( p, r ) -invex sets and functions”, J. Math. Anal. Appl., 263 (2001) 355-379.

[2]

Bector, C. R., Chandra, S., and Husain, I., “Optimality conditions and duality in subdifferentiable multiobjective fractional programming”, J. Optim. Theory Appl., 79 (1993) 105-125. [3] Chankong, V., and Haimes, Y. Y., Multiobjective Decision Making: Theory and Methodology, North-Holland, New York, 1983. [4] Chen, X. H., “Optimality and duality for the multiobjective fractional programming with the generalized ( F , ρ ) -convexity”, J. Math. Anal. Appl., 273 (2002) 190-205. [5]

Chinchuluun, A., Yuan, D., and Pardalos, P. M., “Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity”, Ann. Oper. Res., 154 (2007) 133-147. [6] Clarke, F. H., Nonsmooth Optimization, Wiley-Interscience, New-York 1983. [7] Hiriart-Urruty, J. B., “On optimality conditions in nondifferentiable programming”, Math. Program., 14 (1978) 73-86. [8] Jayswal, A., Kumar, R., and Kumar, D., “Multiobjective fractional programming problems involving ( p, r ) − ρ − (η , θ ) -invex function”, J. Appl. Math. Comput., 39 (2012) 35-51. [9]

Jeyakumar, V., “Equivalence of saddle-points and optima and duality for a class of non smooth non-convex problems”, J. Math. Anal. Appl., 130 (1988) 334-343. [10] Kim, D. S., Kim, M. H., and Lee, G. M., “On optimality and duality for nonsmooth multiobjective fractional optimization problems”, Nonlinear Anal., 63 (2005) e1867-e1876. [11] Liu, J. C., “Optimality and duality for multiobjective fractional programming involving nonsmooth ( F , ρ ) -convex functions”, Optimization, 36 (1996) 333-346. [12] Mandal, P., and Nahak, C., “Symmetric duality with

( p, r ) − ρ − (η , θ ) -invexity”,

Math. Comput., 217 (2011) 8141-8148. [13] Soleimani-damaneh, M., “Optimality for nonsmooth programming”, Nonlinear Anal., 68 (2008) 2873-2878.

fractional

Appl.

multiple objective

386


[14] Stancu-Minasian, I. M., Fractional Programming: Theory, Methods and Applications, Kluwer

Academic Publishers, Dordrecht, The Netherlands (1997). [15] Stancu-Minasian, I. M., “A fifth bibliography of fractional programming”, Optimization, 45

(1999) 343-367. [16] Stancu-Minasian, I. M., “A sixth bibliography of fractional programming”, Optimization, 55

(2006) 405-428. [17] Zalmai, G. J., “Generalized sufficiency criteria in continuous-time programming with

application to a class of variational-type inequalities”, J. Math. Anal. Appl., 153 (1990) 331355.