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Duality For Multiobjective Programming Involving (, , ) -Invexity ( (, , ) -Invexity) Deo Brat Ojha
Department of Mathematics R.K.G.Institute of Technology Ghaziabad, U.P.,INDIA ojhdb@yahoo.co.in
Keywords(, , ) -(pseudo/quasi)-convexity/ multiobjective programming; duality theorem.
invexity;
can be proved under (, ) -invexity, even if Hanson’s invexity is not satisfied, Puglisi[21]. Therefore, the results of this paper are real extensions of the similar results known in the literature. In Section 2 we define the (, , ) -univexity . In Section 3 we consider a class of Multiobjective programming problems and for the dual model we prove a weak duality result and strong duality.
T
Abstract— The concepts of ( , ρ)-invexity have been given by Carsiti,Ferrara and Stefanescu[20], which is broader class than (F,ρ)convexity. We consider a dual model associated to a multiobjective programming problem involving support functions and a weak duality result is established under appropriate (, , ) -invexity conditions.
.
I.
INTRODUCTION
For nonlinear programming problems, a number of duals have been suggested among which the Wolfe dual [6,23] is well known. While studying duality under generalized convexity, Mond and Weir [29] proposed a number of deferent duals for nonlinear programming problems with nonnegative variables and proved various duality theorems under appropriate pseudo-convexity/quasi-convexity assumptions. For
( x, a,( y, r )) F ( x, a; y) rd 2 ( x, a)
,
where
IJ A
F ( x, a;.) is sublinear on R n , the definition of (, ) - invexity reduces to the definition of ( F , ) -convexity introduced by Preda[17], which in turn Jeyakumar[18] generalizes the concepts of F-convexity and -invexity.For more details reader may consult [1,2,3,4,5,7,9,17,18,19,24,27,29]. The more recent literature, Mishra[22], Xu[11], Ojha [12], Ojha and Mukherjee [15] for duality under generalized ( F , ) convexity, Liang et al. [13] and Hachimi[14] for optimality criteria and duality involving ( F , , , d ) -convexity or generalized {F , , , d ) -type functions. The ( F , ) -convexity was recently generalized to (, ) -invexity by Caristi , Ferrara and Stefanescu [20],and here we will use this concept to extend some theoretical results of multiobjective programming. Whenever the objective function and all active restriction functions satisfy simultaneously the same generalized invexity at a Kuhn-Tucker point which is an optimum condition, then all these functions should satisfy the usual invexity, too. This is not the case in multiobjective programming ; Ferrara and Stefanescu[16] showed that sufficiency Kuhn-Tucker condition
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NOTATIONS AND PRELIMINARIES
ES
II.
We consider
f : R n R p , g : Rn Rq ,are differential
functions and X Rn is an open set. We define the following multiobjective programming problem: (P) minimize f ( x) f1 ( x).......... f p ( x) subject to g ( x) 0 x, x X
Let X 0 be the set of all feasible solutions of (P) that is, X 0 {x X g ( x) 0} .
We quote some definitions and also give some new ones. 2.1 Definition A
vector a X 0 is said to be an efficient solution of problem (P) if there exit no x X 0 such that
f (a) f ( x) Rp \{0} i.e., fi ( x) fi (a) for all i {1,.,.,., p} , and for at least one j {1,.,.,., p} we have fi ( x) fi (a) . 2.2 Definition A point a X is said to be a weak efficient solution of problem (P) if there is no x X such that f ( x) f (a). 0
2.3 Definition A point a X 0 is said to be a properly efficient solution of (P) if it is efficient and there exist a positive constant K such that for each x X and for each i 1, 2...... p satisfying
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0
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fi ( x) fi (a) , there exist at least one i 1, 2...... p such that f ( a ) f ( x) j j
and f (a) f ( x) K f ( x) f (a) . i
j
i
j
A point a X is said to be a weak efficient solution of problem (P) if there is no x X such that f ( x) f (a).
(i) If we consider the case ( x, u; ( x, u)(f (u), )) F ( x, u; f (u)) (with F is sublinear in third argument, and ( x, u) 1 then the above definition reduce to F-convexity.
0
We denote by f (a) the gradient vector at a of a differentiable function f : R p R , and by 2 f (a) the Hessian matrix of f at a . For a real valued twice differentiable function ( x, y) defined on an open set (a, b) in
R p Rq , we denote by x (a, b) the gradient vector of with respect to x , and by xx (a, b) the Hessian matrix with respect to x at (a, b) . Similarly, we may define , xy (a, b) and yy (a, b) . For convenience, let us write the definition of , invexity from [20], Let : X 0 R be a differentiable function dimensional Euclidean Space R n 1 is represented as the ordered pair ( z, r ) with z Rn and r R, is a real number and is real valued function defined on
X 0 X 0 Rn 1 , suchthat x, a,. is
Rn 1 and x, a, 0, r 0, for every
rR .
convex
on
x, a X 0 X 0 and
IJ A
A real-valued twice differentiable function f (., y) : X X R is said to be (, , ) -invex at
u X with respect to p R n , if for all : X X Rn 1 R , : X X R \{0} , i is a real number, we have { fi ( x, y) fi (u, y)} ( x, u; ( x, u)(fi (u, y), i ))
(2.1)
2.5 Definition
A real-valued twice differentiable function f (., y) : X X R is said to be (, , ) -pseudoinvex at
a X with respect to p R n ,if for all : X X Rn 1 R , : X X R \{0} i is a real number, we have { fi ( x, y ) fi (u, y)} 0
(2.2)
2.6 Definition
A real valued twice differentiable function f is (, , ) pseudoconcave if f is (, , ) -pseudoconvex.
III.
A real-valued twice differentiable function f (., y) : X X R is said to be (, , ) -quasiinvex at
a X with respect to p R ,if for all : X X R
n 1
R,
: X X R \{0} , i is a real number, we have { fi ( x, y) fi (u, y)} 0 ( x, u; ( x, u )(fi (u, y), i )) 0
WOLFE TYPE SYMMETRIC DUALITY
We consider here the following pair of multiobjective mathematical programs and establish weak, strong duality theorems. (MP)
Minimize fi ( x, y)
subject to r
f ( x, y ) 0
i
i 1
(3.1)
y i
0 , T e 1
(3.2) (3.3)
x0
(MD)
Maximize fi (u, v)
subject to i
i 1
f (u, v) 0
(3.4)
T e 1
(3.5) (3.6)
u i
0
,
v0
i R, i 1, 2,.,., r Here, , and e(1,1,1,.,.,1)T R f i , i 1, 2,.,., r are twice differentiable function from Rn Rn R , and : X X R \{0} , above. Theorem 3.1 (Weak duality)
i as discussed
Let ( x, y, , z1 , z2 ,.,., zr ) be a feasible solution of (MP) and (u, v, , w1 , w2 ,.,., wr ) a feasible solution of (MD) and (i)
r
f (., v) i 1
(ii)
n
, where
(pseudo/quasi)-convexity.
( x, u; ( x, u )(fi (u, y ), i )) 0
F ( x, u; f (u )) ( x, u )T f (u )
: X X Rn , the above definition reduce to -
r
2.4 Definition
( x, u; ( x, u)(f (u ), ))
ES
( X R n ) , X X , and a X . An element of all (n+1)0 0 0
When
T
Denoting by WE(P), E(P) and PE(P) the sets of all weakly efficient, efficient and properly efficient solutions of (VP), we have PE(P) E(P) WE(P).
(ii)
i
is (0 , 0i , ) -invex at u for fixed v ,
r
f ( x,.) is i 1
(iii)
i
i
i
(1 , 1i , ) -incave at y for fixed x ,
0 ( x, u; ( x, u)( , 0i )) uT 0
,
r
i u fi (u, v) ,and i 1
(2.3)
2.7 Remark
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where
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,
for
i y fi ( x, y ) , i 1
Then, fi ( x, y) fi (u, v) . Proof r
f (., v) i
i 1
i
be (0 , 0i , ) -invex at u for fixed
r
v for 0 ,
{ f ( x, v ) i 1
i
i
f i (u, v)}
Let ( x, y, , z1 , z2 ,.,., zr ) be a feasible solution of (MP) and (u, v, , w1 , w2 ,.,., wr ) a feasible solution of (MD) and (i)
we get facilitate,with the help of hypothesis (iii) and (3.4), r
f ( x, v) ( f (u, v) i
i 1
i
i 1
i
Now, fi ( x,.) be (1 , 1i ) -pseudoincavity assumption at y for fixed for 0 , we have, x ,
[ f ( x, v ) i
i 1
i
i 1
i
r
f ( x, y )
i
i 1
i
Combining (3.7) and (3.8), we get r
r
f ( x, y) f (u, v) i 1
i
i
i 1
i
r
f ( x,.)
fixed x ,
i
i 1
(1 , 1i , ) -pseudoincave at y for
is
i
0 ( x, u; ( x, u)( , 0i )) uT 0
(iii) i 1
,
i
1 (v, y; (v, y)( , 1i )) yT 0
(iv)
and hypothesis (iv),(3.1), it implies that i
is (0 , 0i , ) -pseudoinvex at u for fixed
i
where
i u fi (u, v) ,and
1 ( x, y; ( x, y)( y f i ( x, y), 1i ))
f ( x, v )
(ii)
r
fi ( x, y)]
r
i
i 1
ES
r
r
f (., v)
v,
(3.7)
i
(3.15) (3.16)
Here, e(1,1,1,.,.,1)T R , i R, i 1, 2,.,., r and f i , i 1, 2,.,., r are twice differentiable function from Rn Rn R and : X X R \{0} , i as discussed above. Theorem 3.1 (Weak duality)
0 ( x, u; ( x, u )(u f i (u, v), 0i ))
r
, T e 1
v0
r
Since
0
T
1 (v, y; (v, y)( , 1i )) yT 0
(iv)
,
for
r
i y fi ( x, y ) , i 1
(3.8)
Then, fi ( x, y) fi (u, v) .
Proof Since
r
f (., v) i 1
i
be (0 , 0i , ) -pseudoinvex at u for
i
fixed v , for 0 , we get facilitate with the help of hypothesis (iii) and (3.14) ,
MOND-WEIR TYPE SYMMETRIC DUALITY
IJ A
IV.
r
We consider here the following pair of multiobjective mathematical programs and establish weak, strong and converse duality theorems. (MP)
i 1
r
i
i 1
y
f i ( x, y ) 0
r
(3.9)
yT i [ y fi ( x, y )] 0
(3.10)
0 , T e 1
(3.11) (3.12)
i
i
i 1
r
r
i
i 1
i
i
i 1
Now, fi ( x,.) be (1 , 1i , ) pseudoincavity assumption at y for fixed x , for 0 , we have, we have, hypothesis (iv)and (3.10), it implies that r
r
f ( x, v) [ f ( x, y) 0 i 1
i
i
r
i 1
i
i
subject to
(3.20) Combining (3.18) and (3.20), we get
i 1
r
i
i 1
u
fi (u, v) 0
r
uT i u fi (u, v) 0
(3.13)
r
(3.14)
i i
i fi ( x, y) i 1
(3.19)
r
(MD)
(3.18)
i
f ( x, v) ( f ( x, y)
Maximize fi (u, v)
(3.17)
i
f ( x, v) f (u, v)
i 1
x0
i
for all i 1, 2,.,., r
Minimize fi ( x, y)
subject to
r
f ( x, v) [ f (u, v) 0
i 1
i
i
r
f (u, v) i 1
i i
Theorem 3.2(Strong duality)
i 1
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fixed r
[ i 1
i
f]
yy i
r – ive definite for all
i 1, 2,.,., r ;
(ii) and the set [ y f1 , y f 2 ,.,., y f r ] for all i 1, 2,.,., r ; is linearly independent, such that ( x, y, ) is a feasible solution of (MD), and the two objectives have the same values. Also, if the hypothesis of Theorem 3.1 are satisfied for all feasible solutions of (MP) and (MD), then ( x, y, ) is an efficient solution for (MD). Proof Let ( x, y, ) is a weak efficient solution for (MP), then it is weakly efficient solution. Hence, there exist R r , R r ,
R r , R r and n R not all zero, i 1, 2,.,., r such that the following Fritz john optimality condition (28) are satisfied at ( x, y, ) .
i x fi ( ny) i ( yy fi ) s
( n ) i
i
y fi
(3.21)
( ny )T ( yy fi ) 0 ,
(3.22)
i 1
( ny) y fi i 0 ,
(3.23)
T
for all i 1, 2,.,., r . r
y fi
0
ny T
r
y fi
i 1
i fi (u, v)
r
f ( x, y)
(3.25)
0
IJ A
i 1
T 0 (3.26) ( , , s, , , n) 0, (3.27) ( , , s, , , n) 0, 0 (3.28) and 0 ,(3.26) implies 0 . Consequently, (3.23) gives ( ny)T y fi 0
(3.29)
( ny)T [ yy fi ]( ny) 0
(3.30)
hence, in the view of (i), ny,
(3.31)
r
From (3.22) and (3.31)
( n ) i
i
y fi
0.
i 1
assumption (ii) and (3.32), gives i ni for all i 1, 2,.,., r .
(3.33)
If, n 0, i 0, 0, i 0, s 0, for all i 1, 2,.,., r , ( , , s, , , n) 0, n 0 . Then we obtain ( , , s, , , n) 0 , which contradicts(3.28), hence n 0 ,from (3.33) 0 we have i 0 , i 1, 2,.,., r . From (3.21), (3.31)and (3.33)we get , i x fi s / n 0 (3.34) By (3.27) and (3.31)
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(3.42)
i i
i 1
Which contradicts the weak duality theorem 3.2.
(3.24)
i 1
(3.40)
(3.41) Whenever fi (u, v) fi ( x, y) It implies that fi (u, v) fi ( x, y) can be made arbitrarily large and hence for with i 0 , we have r
for all i 1, 2,.,., r .
T
for all j satisfying f j ( x, y) f j (u, v)
ES
r
T
Since, 0 ,we have, y / n 0 (3.35) From (3.34) ,it follows that i x fi 0 (3.36) From ((3.34)- (3.36)),we know that ( x, y, ) is feasible for (MD). fi ( x, y) fi ( x, y) (3.37) and the objective values of (MD) and (MP) are equal. We claim that ( x, y, ) is an efficient solution for (MD) for if it is not true , then ,there would exist (u, v, ) feasible for (MD) such that fi (u, v) fi ( x, y) , i 1, 2,.,., r (3.38) Using equality(3.38) a contradiction( Weak Duality Theorem3.2 )is obtained. If ( x, y, ) is improperly efficient, then for every scalar M > 0, there exist a feasible solution (u , v, ) in (MD) and an index i such that (3.39) fi (u, v) fi ( x, y) M [ f j ( x, y) f j (u, v)]
T
Let ( x, y, ) be a weak efficient solution for (MP) for
V.
CONCLUSION
Our paper shows the real extension of results obtained previously on multiobjective optimization through convexity. Specially, results obtained in [20,21] will become special case , if we take ( x, u) 1 .
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ISSN: 2230-7818
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