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Lagrangian Duality for Preinvex Set-Valued Functions
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
214, 599]612 Ž1997.
AY975599
Lagrangian Duality for Preinvex Set-Valued Functions Davinder Bhatia Department of Mathematics, S.G.T.B. Khalsa College, Uni¨ ersity of Delhi, Delhi, 110007, India
and Aparna Mehra Department of Mathematics, Acharya Narendra De¨ College, Gobindpuri, Kalkaji, New Delhi, 110019, India Submitted by Koichi Mizukami Received July 31, 1996
In this paper, generalizing the concept of cone convexity, we have defined cone preinvexity for set-valued functions and given an example in support of this generalization. A Farkas]Minkowski type theorem has been proved for these functions. A Lagrangian type dual has been defined for a fractional programming problem involving preinvex set-valued functions and duality results are established. Q 1997 Academic Press
1. INTRODUCTION AND PRELIMINARIES Problems involving set-valued functions have a wide range of applications in economics w8x, nonlinear programming w17x, differential inclusion w4x, and many more. Tanino and Sawaragi w12x and Corley w5x developed the duality theories for multiobjective programming problems which implicitly involve set-valued functions. Later, Corley w6x considered maximization of concave set-valued functions, in possibly infinite dimensions, and established an existence result and developed Lagrangian duality theory for a programming problem involving such functions. More recently, Lin w9x generalized the Moreau]Rockafeller type theorem and the Farkas]Minkowski type theorem for convex set-valued functions and obtained necessary and sufficient optimality conditions for the existence of a Geoffrion 599 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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BHATIA AND MEHRA
efficient solution for a set-valued problem. He further defined Mond]Weir and Wolfe type duals and established duality results. The notion of preinvexity for scalar-valued functions was introduced into literature by Weir and Jeyakumar w16x and Weir w15x by relaxing the convexity assumption on the domain set of the functions. The advantage with preinvex functions is that their nonnegative linear combination is also preinvex. Several contributions have been made in the past Žsee w2, 3, 7, 10, 11x and the references therein . in developing the duality theory for nonlinear fractional programming problems by using a parameterization technique given by Bector w1x. Motivated by these ideas, in the present paper, we have extended the class of cone-convex set-valued functions to the class of cone-preinvex set-valued functions. A fractional programming problem involving set-valued functions has been considered. A weakly efficient solution of this problem has been related to the weakly efficient solution of a multiobjective set-valued programming problem obtained by using a parameterization technique. Also a Lagrangian type dual has been defined and duality results are established. Let X and Y be topological vector spaces. A set-valued function F from X into Y is a map that associates a unique subset of Y with each point of X. Equivalently, F can be viewed as a function from X into the power set of Y, i.e., F : X ª 2 Y. The domain of F : X ª 2 Y is given by D Ž F . s x g X ¬ F Ž x . / B4 . For E : X, F : E ª 2 Y, denote, F Ž E . s Dx g E F Ž x .. A subset V of Y is said to be a cone if lj g V for every j g V and l P 0. A convex cone is one for which l1 j 1 q l 2 j 3 g V for every j 1 , j 2 g V and l1 , l2 P 0. A pointed cone is one for which V l ŽyV . s 04 , where 0 is the zero element of Y. Let V be a pointed convex cone with int V / B. Then we define three cone orders with respect to V as
j 1 OV j 2
iff j 2 y j 1 g V ,
j 1 FV j 2
iff j 2 y j 1 g V _ 0 4 ,
j 1 -V j 2
iff j 2 y j 1 g int V .
DUALITY FOR SET-VALUED FUNCTIONS
601
The set of all the weak V-minimal points and weak V-maximal points of a set A in Y are defined as w-Min V A s y 0 g A ¬ there exist no y g A for which y -V y 0 4 , w-Max V A s y 0 g A ¬ there eixst no y g A for which y 0 -V y 4 . If y 0 g A is a weak minima of A with respect to cone V then it is denoted by y 0 g w-Min V A. The polar cone VU of V is defined as VU s yU g Y U ¬ ² yU , y : P 0 for all y g V 4 . The following result is due to Wang and Li w14x. LEMMA 1.1.
If V g Y is a pointed con¨ ex cone with int V / B, then
Ži. V q int V ; int V Žii. ² yU , y : ) 0 for any yU g VU _ 04 and y g int V. If A, B : R p , a g R, then we define A q B s a q b ¬ a g A, b g B 4
a A s a a ¬ a g A4 and if B : int Rqp then define A B
s arb s Ž a1rb1 , a2rb 2 , . . . , a prbp . ¬ a s Ž a1 , a2 , . . . , a p . g A, b s Ž b1 , b 2 , . . . , bp . g B 4 ,
where Rqp denotes the nonnegati¨ e orthant of R p. DEFINITION 1.1 w13x. Let E ; X be a convex set and F : E ª 2 Y be a set-valued function and V be a pointed convex cone in Y. Then F is said to be V-convex on E if for every x 1 , x 2 g E, t g w0, 1x, tF Ž x 1 . q Ž 1 y t . F Ž x 2 . ; F Ž tx 1 q Ž 1 y t . x 2 . q V .
2. PREINVEX SET-VALUED FUNCTION In this section, we define a new class of set-valued function, called a preinvex set-valued function, as a generalization of a convex set-valued function.
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DEFINITION 2.1. Let E be a subset of X, F : E ª 2 Y and let V be a pointed convex cone in Y. F is said to be V-preinvex on E if there exists a function h defined on X = X and with values in X such that for any x 1 , x 2 g E, t g w0, 1x, tF Ž x 1 . q Ž 1 y t . F Ž x 2 . ; F Ž x 2 q th Ž x 1 , x 2 . . q V . It is implicit in the above definition that for x 1 , x 2 g E and t g w0, 1x, x 2 q th Ž x 1 , x 2 . g E. We call such a set E to be an invex set with respect to h. This definition generalizes the class of V-convex set-valued functions, as in the case where F is a V-convex function on E; then by taking x 1 y x 2 s h Ž x 1 , x 2 . for all x 1 , x 2 g E, F becomes V-preinvex. However, the converse need not be true, that is, a V-preinvex set-valued function need not be V-convex. 2 EXAMPLE 2.1. Let X s R, Y s R 2 , V s Rq , and let F : X ª 2 Y be defined as
F Ž x . s Ž a 1 , a 2 . g Y ¬ a 1 q a 2 P y< x < 4 . Then F is not V-convex on X, as for x 1 s y1 g X, x 2 s 2 g X, t s 5r6, we have tx 1 q Ž 1 y t . x 2 s y Ž 1r2 . F Ž x 1 . s Ž a 1 , a 2 . g Y ¬ a 1 q a 2 P y1 4 , F Ž x 2 . s Ž a 1 , a 2 . g Y ¬ a 1 q a 2 P y2 4 , and tF Ž x 1 . q Ž 1 y t . F Ž x 2 . o F Ž tx 1 q Ž 1 y t . x 2 . q V because, for Žy1, 0. g F Ž x 1 ., Ž0, y2. g F Ž x 2 ., we have t Ž y1, 0 . q Ž 1 y t . Ž 0, y2 . s Ž y5r6, y1r3. g tF Ž x 1 . q Ž 1 y t . F Ž x 2 . but Žy5r6, y1r3. f F Žy1r2. q V. However, F is V-preinvex on X with respect to a function h defined on X = X as
h Ž x1 , x 2 . s x1 y x 2 s x 2 y x1
if either x 1 ) 0, x 2 ) 0 or x 1 - 0, x 2 - 0 if either x 1 ) 0, x 2 - 0 or x 1 - 0, x 2 ) 0.
The following theorem characterizes the generalized Farkas]Minkowski type theorem for preinvex set-valued functions.
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603
THEOREM 2.1. Let E be an in¨ ex subset of X Ž with respect to a function h : X = X ª X .. If the set-¨ alued function F : E ª 2 Y is V-prein¨ ex and G : E ª 2 Z is H-prein¨ ex Ž with respect to same function h ., where V and H are pointed con¨ ex cones in topological ¨ ector spaces Y and Z, respecti¨ ely, then exactly one of the following statements is true: Ži.
there exists x g E such that F Ž x . l Ž yint V . / B G Ž x . l Ž yint H . / B
Žii. x g E,
there exists Ž yU , zU . / Ž0, 0. in VU = HU such that for e¨ ery ² yU , F Ž x . : q ² zU , G Ž x . : P 0.
Proof. We will first show that Ži. and Žii. cannot hold simultaneously. Let if possible Ži. and Žii. both be true. Then by Ži., there exists some ˆx g E such that FŽ ˆ x . l Ž yint V . / B GŽ ˆ x . l Ž yint H . / B, i.e.,
ˆy -V 0
for some ˆ y g FŽ ˆ x.
ˆz -H 0
for some ˆ z g GŽ ˆ x. .
Now for any Ž yU , zU . / Ž0, 0. in VU = HU , we have ² yU , ˆ y : q ² zU , ˆ z: - 0 for some ˆ y g FŽ ˆ x. , ˆ z g GŽ ˆ x . , and some ˆ x g E. But this contradicts the assumption that Žii. holds for every x g E. Next, we will show that not Ži. « Žii.. Let Ži. be not true. On the lines of Lin w9x, it is sufficient to show that the set A defined as A s Ž y, z . g Y = Z ¬ there exists x g E such that for some u g F Ž x . and ¨ g G Ž x . , u -V y and ¨ -H z 4 is convex in Y = Z and Ž0, 0. f A.
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Let Ž y 1 , z1 . g A and Ž y 2 , z 2 . g A. Then there exists x 1 g E, x 2 g E such that for some u1 g F Ž x 1 ., ¨ 1 g GŽ x 1 ., u 2 g F Ž x 2 ., ¨ 2 g GŽ x 2 . u1 -V y 1
and
¨ 1 -H z1
u 2 -V y 2
and
¨ 2 -H z 2 .
5
Ž 2.1.
Since u1 g F Ž x 1 ., u 2 g F Ž x 2 ., ¨ 1 g GŽ x 1 ., ¨ 2 g GŽ x 2 . therefore for t g w0, 1x, tu1 q Ž 1 y t . u 2 g tF Ž x 1 . q Ž 1 y t . F Ž x 2 . t¨ 1 q Ž 1 y t . ¨ 2 g tG Ž x 1 . q Ž 1 y t . G Ž x 2 . .
5
Ž 2.2.
By preinvexity of F and G with respect to same h , we have tF Ž x 1 . q Ž 1 y t . F Ž x 2 . ; F Ž x 2 q th Ž x 1 , x 2 . . q V tG Ž x 1 . q Ž 1 y t . G Ž x 2 . ; G Ž x 2 q th Ž x 1 , x 2 . . q H. Hence by virtue of Ž2.2., there exists u g F Ž x 2 q th Ž x 1 , x 2 .. j g V, and ¨ g GŽ x 2 q th Ž x 1 , x 2 .., t g H such that tu1 q Ž 1 y t . u 2 s u q j t¨ 1 q Ž 1 y t . ¨ 2 s ¨ q t , i.e., u s tu1 q Ž 1 y t . u 2 y j OV tu1 q Ž 1 y t . u 2 -V ty 1 q Ž 1 y t . y 2 ¨ s t¨ 1 q Ž 1 y t . ¨ 2 y t OH t¨ 1 q Ž 1 y t . ¨ 2 -H tz1 q Ž 1 y t . z 2
Ž by using Ž 2.1. . . This shows that there exists x s x 2 q th Ž x 1 , x 2 . g E, as E is an invex set with respect to h , such that u -V ty 1 q Ž 1 y t . y 2 ¨ -H tz1 q Ž 1 y t . z 2
for some u g F Ž x ., ¨ g GŽ x .. Hence, t Ž y 1 , z1 . q Ž 1 y t . Ž y 2 , z 2 . g A, COROLLARY 2.2.
t g w 0, 1 x .
If in Theorem 2.1, we assume further that there exists
ˆx g E such that GŽ ˆx . l Žyint H. / B then yU / 0.
The proof follows along similar lines as that of Corollary 3.4 in w9x.
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605
3. LAGRANGIAN DUALITY We are concerned with the following problem. Let R p and R m be ordered by pointed convex cones V and H, respectively, with int V / B, int H / B. p p m Let E0 ; R n, F : E0 ª 2 R , G : E0 ª 2 R , and H : E0 ª 2 R be set-valued maps defined on E0 . Then the problem is W-Min V
FŽ x. GŽ x .
F1 Ž x .
s
G1 Ž x .
,...,
Fp Ž x .
Ž FP.
Gp Ž x .
subject to H Ž x . l Ž yH . / B,
x g E0 .
We assume that for each x g E0 , and for each i s 1, 2, . . . , p, Fi Ž x . ; Rq
and
Gi Ž x . ; int Rq .
Let E s x g E0 ¬ H Ž x . l ŽyH. / B4 be the feasible set of ŽFP.. In this section, all the relations, until otherwise stated, are with respect to the cone order V. DEFINITION 3.1. A point xU g E is said to be a weakly efficient solution Žor an efficient solution. for ŽFP. if there exists yU g F Ž xU . and zU g GŽ xU . such that yU z
ž
U
g W-Min D
xgE
or respectively,
yU z
U
FŽ x. GŽ x .
g Min D
xgE
FŽ x. GŽ x .
/
.
On using the parametric approach, we consider the following optimization problem W-Min F1 Ž x . y l1G1 Ž x . , . . . , Fp Ž x . y l p Gp Ž x . subject to x g E,
Ž P. l
l s Ž l1 , . . . , l p . g Rqp . The following lemma for set-valued functions can easily be proved on the lines of the corresponding result proved for real valued functions Žin terms of cones. by Chandra et al. w3x. LEMMA 3.1. Let xU g E. Then xU is weakly efficient for ŽFP. with y rzU g F Ž xU .rGŽ xU . as a weakly efficient ¨ alue of ŽFP. if and only if xU is weakly efficient for ŽP.lU where lU s yU rzU and 0 g F Ž xU . y lU GŽ xU . as a weakly efficient ¨ alue of ŽP.lU . U
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BHATIA AND MEHRA
Let L s BqŽ R m , R p . be the set of all bounded continuous linear functions s : R m ª R p such that sŽH. ; V Ži.e., s is nonnegative with respect to cones V and H.. The weak dual problem associated with ŽFP. is defined as W-Max D W-Min sgL
F
ž
G
q
sŽ H . G
/Ž
E. .
Letting
c Ž s . s W-Min
ž
F
q
G
s W-Min D
xgE
ž
sŽ H .
F G
G q
/Ž
E.
sŽ H . G
/Ž . x
the dual problem can be rewritten as W-Max D c Ž s . ,
Ž DFP.
sgL
i.e., we have to determine the weak maximal elements of Ds g L c Ž s . with respect to the cone V. THEOREM 3.2 ŽWeak Duality Theorem.. Suppose x 0 is feasible for ŽFP. and s0 is feasible for ŽDFP.. Then for any y 0 g F Ž x 0 .. z 0 g GŽ x 0 ., and ¨ 0 g c Ž s0 ., y0 l ¨0. z0 Proof. Suppose, to the contrary, that there exists some y 0 g F Ž x 0 ., z 0 g GŽ x 0 ., and ¨ 0 g c Ž s0 . for which y0 z0
- ¨0
i.e., ¨0 y
y0 z0
g int V .
Ž 3.1.
Since, ¨ 0 g c Ž s0 . s W-Min Dx g E Ž FrG q sŽ H .rG.Ž x ., hence for any x 1 g E with y 1 g F Ž x 1 ., z1 g GŽ x 1 ., and w 1 g H Ž x 1 ., we have y1 z1
q
s0 Ž w 1 . z1
l ¨0.
Ž 3.2.
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607
But as x 0 g E, so in particular, for y 0 g F Ž x 0 ., z 0 g GŽ x 0 ., and w 0 g H Ž x 0 . l ŽyH. Žthis set is non-empty because of the feasibility of x 0 for ŽFP.., we have from Ž3.2., y0 z0
q
s0 Ž w 0 . z0
l ¨0.
Ž 3.3.
Now, w 0 g H Ž x 0 . l Ž yH . « w 0 g yH which implies s0 Ž w 0 . O 0
Ž since s0 g L . .
Also z 0 ) 0 and V is a convex cone hence s0 Ž w 0 . z0
O 0,
i.e., y
s0 Ž w 0 .
g V.
z0
Ž 3.4.
From Ž3.1. and Ž3.4., we get ¨0 y
ž
y0 z0
q
s0 Ž w 0 .
/
z0
g int V
on account of Lemma 1.1. Hence, y0 z0
q
s0 Ž w 0 . z0
- ¨0
which contradicts Ž3.3.. The result follows. THEOREM 3.3 ŽStrong Duality Theorem.. Suppose that V and H are pointed con¨ ex cones in R p and R m , respecti¨ ely, with int V / B, int H / B. Let E0 be an in¨ ex subset of R n with respect to function h : R n = R n ª R n. Let F, yG be V-prein¨ ex and H be H-prein¨ ex on E0 with respect to same h and further assume that there exist ˆ x g E0 such that H Ž ˆ x . l Žyint H. / B. Ž . If x 0 is a weakly efficient solution for FP then there exists y 0 g F Ž x 0 ., z 0 g GŽ x 0 . such that y0 z0
g W-Max D c Ž s . . sgL
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BHATIA AND MEHRA
Proof. Since x 0 is a weakly efficient solution for ŽFP. hence there exists y 0 g F Ž x 0 ., z 0 g GŽ x 0 . such that y0 z0
g W-Min D
xgE
F G
Ž x. .
Therefore by Lemma 3.1, x 0 is a weakly efficient solution for ŽP.l 0 where l0 s y 0rz 0 with 0 g F Ž x 0 . y l0 GŽ x 0 . as a weakly efficient value of ŽP.l 0 . This shows that the system F Ž x . y l 0 G Ž x . l Ž yint V . / B H Ž x . l Ž yint H . / B
5
Ž 3.5.
has no solution. Since F, yG are V-preinvex with respect to h and l0 P 0 hence F y l0 G is V-preinvex with respect to h on E0 . Therefore by Theorem 2.1 and Corollary 2.2, there exists Ž uU , ¨ U . g VU = HU such that for all x g E0 ² uU , F Ž x . y l0 G Ž x . : q ² ¨ U , H Ž x . : P 0 U
U
Ž u , ¨ . / Ž 0, 0 . ,
U
u / 0.
Ž 3.6. Ž 3.7.
Let x s x 0 in Ž3.6.. We get ² uU , F Ž x 0 . y l 0 G Ž x 0 . : q ² ¨ U , H Ž x 0 . : P 0.
Ž 3.8.
Since y 0 g F Ž x 0 . and z 0 g GŽ x 0 ., hence, we have ² uU , y 0 y l 0 z 0 : q ² ¨ U , H Ž x 0 . : P 0 giving ²¨ U , H Ž x0 . : P 0
ž
by using l0 s
y0 z0
/
.
Ž 3.9.
Also x 0 is feasible for ŽFP. therefore H Ž x 0 . l Ž yH. / B. Choose w 0 g H Ž x 0 . l ŽyH.; then ²¨ U , w0 : P 0
Ž by using Ž 3.9. . .
Ž 3.10.
Moreover w 0 g yH and ¨ U g HU , therefore ² ¨ U , w 0 : O 0.
Ž 3.11.
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609
Combining Ž3.10. and Ž3.11., we get ² ¨ U , w 0 : s 0.
Ž 3.12.
Since uU g VU , uU / 0, choose j g V such that ² uU , j : s 1. Define s0 : R m ª R p as s0 Ž w . s ² ¨ U , w :j . Then clearly s0 g L. Further, s0 Ž w 0 . s 0
² uU , s0 Ž w . : s ² ¨ U , w : .
and
Ž 3.13.
We will show that y 0rz 0 g c Ž s0 .. To the contrary, let if possible, y 0rz 0 f c Ž s 0 .. This implies that there exists ˆ x g E such that for some ˆ y g FŽ ˆ x ., ˆz g GŽ ˆx ., and w ˆ g H Ž ˆx ., s0 Ž w y0 ˆy ˆ. q z0 ˆz ˆz giving y0 z0
y
ž ˆˆ y
z
q
s0 Ž w ˆ.
ˆz
/
g int V ,
hence,
¦
uU ,
y0 z0
y
ž ˆˆ y
z
q
s0 Ž w ˆ.
ˆz
/;
as uU g VU , uU / 0.
)0
Therefore, we have ² uU , ˆ y y l0 ˆ z : q ² uU ? s0 Ž w ˆ.: - 0
ž
using the fact that l 0 s
y0 z0
/
which in view of Ž3.13. gives ² uU , ˆ y y l0 ˆ z : q ²¨ U , w ˆ : - 0. But this contradicts Ž3.8.. Hence y 0rz 0 g c Ž s0 .. Next, we have to show that y0 g W-Max D c Ž s . . z0 sgL Assume to the contrary. Then there exists s1 g L such that for some
¨ 1 g c Ž s1 ., we have
y0 z0
- ¨1 « ¨1 y
y0 z0
g int V .
Ž 3.14.
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Now, x 0 is feasible for ŽFP. hence H Ž x 0 . l ŽyH. / B. Therefore for w 0 g H Ž x 0 . l ŽyH., we have s1Ž w 0 . O 0. Since z 0 ) 0 and H is a convex cone, therefore s1 Ž w 0 . z0
O0
i.e., y
s1 Ž w 0 . z0
g V.
Ž 3.15.
From Ž3.14. and Ž3.15., we get ¨1 y
ž
y0 z0
q
s1 Ž w 0 . z0
/
g int V
on account of Lemma 1.1. Hence, we have y0 z0
q
s1 Ž w 0 .
- ¨1.
z0
Ž 3.16.
But Ž3.16. is a contradiction to the fact that y0 z0
q
s1 Ž w 0 . z0
g
D
xgE
F G
q
s1 Ž H . G
Ž x.
and ¨ 1 g c Ž s1 . s W-Min D
xgE
ž
F G
q
s1 Ž H . G
/Ž .
x .
Therefore s0 is a weakly efficient solution for ŽDFP. with y 0rz 0 g c Ž s0 . as a weakly efficient value of ŽDFP.. THEOREM 3.4. Suppose the assumptions of the pre¨ ious theorem Ž Strong Duality Theorem. are satisfied. Let x 0 be a weakly efficient solution for ŽFP. and ¨ 0 g W-Max Ds g L c Ž s .. Then for e¨ ery y 0 g F Ž x 0 ., z 0 g GŽ x 0 ., y 0rz 0 l ¨ 0 . Moreo¨ er there exists ˆ y 0 g F Ž x 0 . and ˆ z 0 g GŽ x 0 . for which ¨ 0 l ˆ y 0rzˆ0 . Proof. Since x 0 is a weakly efficient solution for ŽFP. hence it is feasible for ŽFP. and ¨ 0 g W-Max Ds g L c Ž s .. Therefore there exists s0 g L such that ¨ 0 g c Ž s0 ..
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Hence by Theorem 3.2 ŽWeak Duality Theorem. for every y 0 g F Ž x 0 . and z 0 g GŽ x 0 ., y0 z0
l ¨0.
On using Theorem 3.3 ŽStrong Duality Theorem., we get an existence of ˆy 0 g F Ž x 0 ., ˆz 0 g GŽ x 0 ., and s0 g L such that
ˆy 0 g c Ž s0 . . ˆz 0 But ¨ 0 g W-Max Ds g L c Ž s ., hence ¨ 0 l ˆ y 0rzˆ0 . ACKNOWLEDGMENTS The authors are grateful to Professor R. N. Kaul, University of Delhi, for his constant encouragement in preparing this paper. Thanks are also due to the referee for his valuable suggestions.
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13. T. Tanino and Y. Sawaragi, Conjugate maps and duality in multiobjective optimization, J. Optim. Theory Appl. 13 Ž1980., 473]499. 14. S. Wang and Z. Li, Scalarization and Lagrangian duality in multiobjective optimization, Optimization 26 Ž1992., 315]324. 15. T. Weir, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl. 136 Ž1988., 29]38. 16. T. Weir and V. Jeyakumar, A class of nonconvex functions and mathematical programming, Bull. Austral. Math. Soc. 38 Ž1988., 177]189. 17. W. I. Zangwill, ‘‘Nonlinear Programming: A Unified Approach,’’ Prentice Hall International, Englewood Cliffs, NJ, 1969.
214, 599]612 Ž1997.
AY975599
Lagrangian Duality for Preinvex Set-Valued Functions Davinder Bhatia Department of Mathematics, S.G.T.B. Khalsa College, Uni¨ ersity of Delhi, Delhi, 110007, India
and Aparna Mehra Department of Mathematics, Acharya Narendra De¨ College, Gobindpuri, Kalkaji, New Delhi, 110019, India Submitted by Koichi Mizukami Received July 31, 1996
In this paper, generalizing the concept of cone convexity, we have defined cone preinvexity for set-valued functions and given an example in support of this generalization. A Farkas]Minkowski type theorem has been proved for these functions. A Lagrangian type dual has been defined for a fractional programming problem involving preinvex set-valued functions and duality results are established. Q 1997 Academic Press
1. INTRODUCTION AND PRELIMINARIES Problems involving set-valued functions have a wide range of applications in economics w8x, nonlinear programming w17x, differential inclusion w4x, and many more. Tanino and Sawaragi w12x and Corley w5x developed the duality theories for multiobjective programming problems which implicitly involve set-valued functions. Later, Corley w6x considered maximization of concave set-valued functions, in possibly infinite dimensions, and established an existence result and developed Lagrangian duality theory for a programming problem involving such functions. More recently, Lin w9x generalized the Moreau]Rockafeller type theorem and the Farkas]Minkowski type theorem for convex set-valued functions and obtained necessary and sufficient optimality conditions for the existence of a Geoffrion 599 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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efficient solution for a set-valued problem. He further defined Mond]Weir and Wolfe type duals and established duality results. The notion of preinvexity for scalar-valued functions was introduced into literature by Weir and Jeyakumar w16x and Weir w15x by relaxing the convexity assumption on the domain set of the functions. The advantage with preinvex functions is that their nonnegative linear combination is also preinvex. Several contributions have been made in the past Žsee w2, 3, 7, 10, 11x and the references therein . in developing the duality theory for nonlinear fractional programming problems by using a parameterization technique given by Bector w1x. Motivated by these ideas, in the present paper, we have extended the class of cone-convex set-valued functions to the class of cone-preinvex set-valued functions. A fractional programming problem involving set-valued functions has been considered. A weakly efficient solution of this problem has been related to the weakly efficient solution of a multiobjective set-valued programming problem obtained by using a parameterization technique. Also a Lagrangian type dual has been defined and duality results are established. Let X and Y be topological vector spaces. A set-valued function F from X into Y is a map that associates a unique subset of Y with each point of X. Equivalently, F can be viewed as a function from X into the power set of Y, i.e., F : X ª 2 Y. The domain of F : X ª 2 Y is given by D Ž F . s x g X ¬ F Ž x . / B4 . For E : X, F : E ª 2 Y, denote, F Ž E . s Dx g E F Ž x .. A subset V of Y is said to be a cone if lj g V for every j g V and l P 0. A convex cone is one for which l1 j 1 q l 2 j 3 g V for every j 1 , j 2 g V and l1 , l2 P 0. A pointed cone is one for which V l ŽyV . s 04 , where 0 is the zero element of Y. Let V be a pointed convex cone with int V / B. Then we define three cone orders with respect to V as
j 1 OV j 2
iff j 2 y j 1 g V ,
j 1 FV j 2
iff j 2 y j 1 g V _ 0 4 ,
j 1 -V j 2
iff j 2 y j 1 g int V .
DUALITY FOR SET-VALUED FUNCTIONS
601
The set of all the weak V-minimal points and weak V-maximal points of a set A in Y are defined as w-Min V A s y 0 g A ¬ there exist no y g A for which y -V y 0 4 , w-Max V A s y 0 g A ¬ there eixst no y g A for which y 0 -V y 4 . If y 0 g A is a weak minima of A with respect to cone V then it is denoted by y 0 g w-Min V A. The polar cone VU of V is defined as VU s yU g Y U ¬ ² yU , y : P 0 for all y g V 4 . The following result is due to Wang and Li w14x. LEMMA 1.1.
If V g Y is a pointed con¨ ex cone with int V / B, then
Ži. V q int V ; int V Žii. ² yU , y : ) 0 for any yU g VU _ 04 and y g int V. If A, B : R p , a g R, then we define A q B s a q b ¬ a g A, b g B 4
a A s a a ¬ a g A4 and if B : int Rqp then define A B
s arb s Ž a1rb1 , a2rb 2 , . . . , a prbp . ¬ a s Ž a1 , a2 , . . . , a p . g A, b s Ž b1 , b 2 , . . . , bp . g B 4 ,
where Rqp denotes the nonnegati¨ e orthant of R p. DEFINITION 1.1 w13x. Let E ; X be a convex set and F : E ª 2 Y be a set-valued function and V be a pointed convex cone in Y. Then F is said to be V-convex on E if for every x 1 , x 2 g E, t g w0, 1x, tF Ž x 1 . q Ž 1 y t . F Ž x 2 . ; F Ž tx 1 q Ž 1 y t . x 2 . q V .
2. PREINVEX SET-VALUED FUNCTION In this section, we define a new class of set-valued function, called a preinvex set-valued function, as a generalization of a convex set-valued function.
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DEFINITION 2.1. Let E be a subset of X, F : E ª 2 Y and let V be a pointed convex cone in Y. F is said to be V-preinvex on E if there exists a function h defined on X = X and with values in X such that for any x 1 , x 2 g E, t g w0, 1x, tF Ž x 1 . q Ž 1 y t . F Ž x 2 . ; F Ž x 2 q th Ž x 1 , x 2 . . q V . It is implicit in the above definition that for x 1 , x 2 g E and t g w0, 1x, x 2 q th Ž x 1 , x 2 . g E. We call such a set E to be an invex set with respect to h. This definition generalizes the class of V-convex set-valued functions, as in the case where F is a V-convex function on E; then by taking x 1 y x 2 s h Ž x 1 , x 2 . for all x 1 , x 2 g E, F becomes V-preinvex. However, the converse need not be true, that is, a V-preinvex set-valued function need not be V-convex. 2 EXAMPLE 2.1. Let X s R, Y s R 2 , V s Rq , and let F : X ª 2 Y be defined as
F Ž x . s Ž a 1 , a 2 . g Y ¬ a 1 q a 2 P y< x < 4 . Then F is not V-convex on X, as for x 1 s y1 g X, x 2 s 2 g X, t s 5r6, we have tx 1 q Ž 1 y t . x 2 s y Ž 1r2 . F Ž x 1 . s Ž a 1 , a 2 . g Y ¬ a 1 q a 2 P y1 4 , F Ž x 2 . s Ž a 1 , a 2 . g Y ¬ a 1 q a 2 P y2 4 , and tF Ž x 1 . q Ž 1 y t . F Ž x 2 . o F Ž tx 1 q Ž 1 y t . x 2 . q V because, for Žy1, 0. g F Ž x 1 ., Ž0, y2. g F Ž x 2 ., we have t Ž y1, 0 . q Ž 1 y t . Ž 0, y2 . s Ž y5r6, y1r3. g tF Ž x 1 . q Ž 1 y t . F Ž x 2 . but Žy5r6, y1r3. f F Žy1r2. q V. However, F is V-preinvex on X with respect to a function h defined on X = X as
h Ž x1 , x 2 . s x1 y x 2 s x 2 y x1
if either x 1 ) 0, x 2 ) 0 or x 1 - 0, x 2 - 0 if either x 1 ) 0, x 2 - 0 or x 1 - 0, x 2 ) 0.
The following theorem characterizes the generalized Farkas]Minkowski type theorem for preinvex set-valued functions.
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603
THEOREM 2.1. Let E be an in¨ ex subset of X Ž with respect to a function h : X = X ª X .. If the set-¨ alued function F : E ª 2 Y is V-prein¨ ex and G : E ª 2 Z is H-prein¨ ex Ž with respect to same function h ., where V and H are pointed con¨ ex cones in topological ¨ ector spaces Y and Z, respecti¨ ely, then exactly one of the following statements is true: Ži.
there exists x g E such that F Ž x . l Ž yint V . / B G Ž x . l Ž yint H . / B
Žii. x g E,
there exists Ž yU , zU . / Ž0, 0. in VU = HU such that for e¨ ery ² yU , F Ž x . : q ² zU , G Ž x . : P 0.
Proof. We will first show that Ži. and Žii. cannot hold simultaneously. Let if possible Ži. and Žii. both be true. Then by Ži., there exists some ˆx g E such that FŽ ˆ x . l Ž yint V . / B GŽ ˆ x . l Ž yint H . / B, i.e.,
ˆy -V 0
for some ˆ y g FŽ ˆ x.
ˆz -H 0
for some ˆ z g GŽ ˆ x. .
Now for any Ž yU , zU . / Ž0, 0. in VU = HU , we have ² yU , ˆ y : q ² zU , ˆ z: - 0 for some ˆ y g FŽ ˆ x. , ˆ z g GŽ ˆ x . , and some ˆ x g E. But this contradicts the assumption that Žii. holds for every x g E. Next, we will show that not Ži. « Žii.. Let Ži. be not true. On the lines of Lin w9x, it is sufficient to show that the set A defined as A s Ž y, z . g Y = Z ¬ there exists x g E such that for some u g F Ž x . and ¨ g G Ž x . , u -V y and ¨ -H z 4 is convex in Y = Z and Ž0, 0. f A.
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Let Ž y 1 , z1 . g A and Ž y 2 , z 2 . g A. Then there exists x 1 g E, x 2 g E such that for some u1 g F Ž x 1 ., ¨ 1 g GŽ x 1 ., u 2 g F Ž x 2 ., ¨ 2 g GŽ x 2 . u1 -V y 1
and
¨ 1 -H z1
u 2 -V y 2
and
¨ 2 -H z 2 .
5
Ž 2.1.
Since u1 g F Ž x 1 ., u 2 g F Ž x 2 ., ¨ 1 g GŽ x 1 ., ¨ 2 g GŽ x 2 . therefore for t g w0, 1x, tu1 q Ž 1 y t . u 2 g tF Ž x 1 . q Ž 1 y t . F Ž x 2 . t¨ 1 q Ž 1 y t . ¨ 2 g tG Ž x 1 . q Ž 1 y t . G Ž x 2 . .
5
Ž 2.2.
By preinvexity of F and G with respect to same h , we have tF Ž x 1 . q Ž 1 y t . F Ž x 2 . ; F Ž x 2 q th Ž x 1 , x 2 . . q V tG Ž x 1 . q Ž 1 y t . G Ž x 2 . ; G Ž x 2 q th Ž x 1 , x 2 . . q H. Hence by virtue of Ž2.2., there exists u g F Ž x 2 q th Ž x 1 , x 2 .. j g V, and ¨ g GŽ x 2 q th Ž x 1 , x 2 .., t g H such that tu1 q Ž 1 y t . u 2 s u q j t¨ 1 q Ž 1 y t . ¨ 2 s ¨ q t , i.e., u s tu1 q Ž 1 y t . u 2 y j OV tu1 q Ž 1 y t . u 2 -V ty 1 q Ž 1 y t . y 2 ¨ s t¨ 1 q Ž 1 y t . ¨ 2 y t OH t¨ 1 q Ž 1 y t . ¨ 2 -H tz1 q Ž 1 y t . z 2
Ž by using Ž 2.1. . . This shows that there exists x s x 2 q th Ž x 1 , x 2 . g E, as E is an invex set with respect to h , such that u -V ty 1 q Ž 1 y t . y 2 ¨ -H tz1 q Ž 1 y t . z 2
for some u g F Ž x ., ¨ g GŽ x .. Hence, t Ž y 1 , z1 . q Ž 1 y t . Ž y 2 , z 2 . g A, COROLLARY 2.2.
t g w 0, 1 x .
If in Theorem 2.1, we assume further that there exists
ˆx g E such that GŽ ˆx . l Žyint H. / B then yU / 0.
The proof follows along similar lines as that of Corollary 3.4 in w9x.
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605
3. LAGRANGIAN DUALITY We are concerned with the following problem. Let R p and R m be ordered by pointed convex cones V and H, respectively, with int V / B, int H / B. p p m Let E0 ; R n, F : E0 ª 2 R , G : E0 ª 2 R , and H : E0 ª 2 R be set-valued maps defined on E0 . Then the problem is W-Min V
FŽ x. GŽ x .
F1 Ž x .
s
G1 Ž x .
,...,
Fp Ž x .
Ž FP.
Gp Ž x .
subject to H Ž x . l Ž yH . / B,
x g E0 .
We assume that for each x g E0 , and for each i s 1, 2, . . . , p, Fi Ž x . ; Rq
and
Gi Ž x . ; int Rq .
Let E s x g E0 ¬ H Ž x . l ŽyH. / B4 be the feasible set of ŽFP.. In this section, all the relations, until otherwise stated, are with respect to the cone order V. DEFINITION 3.1. A point xU g E is said to be a weakly efficient solution Žor an efficient solution. for ŽFP. if there exists yU g F Ž xU . and zU g GŽ xU . such that yU z
ž
U
g W-Min D
xgE
or respectively,
yU z
U
FŽ x. GŽ x .
g Min D
xgE
FŽ x. GŽ x .
/
.
On using the parametric approach, we consider the following optimization problem W-Min F1 Ž x . y l1G1 Ž x . , . . . , Fp Ž x . y l p Gp Ž x . subject to x g E,
Ž P. l
l s Ž l1 , . . . , l p . g Rqp . The following lemma for set-valued functions can easily be proved on the lines of the corresponding result proved for real valued functions Žin terms of cones. by Chandra et al. w3x. LEMMA 3.1. Let xU g E. Then xU is weakly efficient for ŽFP. with y rzU g F Ž xU .rGŽ xU . as a weakly efficient ¨ alue of ŽFP. if and only if xU is weakly efficient for ŽP.lU where lU s yU rzU and 0 g F Ž xU . y lU GŽ xU . as a weakly efficient ¨ alue of ŽP.lU . U
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BHATIA AND MEHRA
Let L s BqŽ R m , R p . be the set of all bounded continuous linear functions s : R m ª R p such that sŽH. ; V Ži.e., s is nonnegative with respect to cones V and H.. The weak dual problem associated with ŽFP. is defined as W-Max D W-Min sgL
F
ž
G
q
sŽ H . G
/Ž
E. .
Letting
c Ž s . s W-Min
ž
F
q
G
s W-Min D
xgE
ž
sŽ H .
F G
G q
/Ž
E.
sŽ H . G
/Ž . x
the dual problem can be rewritten as W-Max D c Ž s . ,
Ž DFP.
sgL
i.e., we have to determine the weak maximal elements of Ds g L c Ž s . with respect to the cone V. THEOREM 3.2 ŽWeak Duality Theorem.. Suppose x 0 is feasible for ŽFP. and s0 is feasible for ŽDFP.. Then for any y 0 g F Ž x 0 .. z 0 g GŽ x 0 ., and ¨ 0 g c Ž s0 ., y0 l ¨0. z0 Proof. Suppose, to the contrary, that there exists some y 0 g F Ž x 0 ., z 0 g GŽ x 0 ., and ¨ 0 g c Ž s0 . for which y0 z0
- ¨0
i.e., ¨0 y
y0 z0
g int V .
Ž 3.1.
Since, ¨ 0 g c Ž s0 . s W-Min Dx g E Ž FrG q sŽ H .rG.Ž x ., hence for any x 1 g E with y 1 g F Ž x 1 ., z1 g GŽ x 1 ., and w 1 g H Ž x 1 ., we have y1 z1
q
s0 Ž w 1 . z1
l ¨0.
Ž 3.2.
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607
But as x 0 g E, so in particular, for y 0 g F Ž x 0 ., z 0 g GŽ x 0 ., and w 0 g H Ž x 0 . l ŽyH. Žthis set is non-empty because of the feasibility of x 0 for ŽFP.., we have from Ž3.2., y0 z0
q
s0 Ž w 0 . z0
l ¨0.
Ž 3.3.
Now, w 0 g H Ž x 0 . l Ž yH . « w 0 g yH which implies s0 Ž w 0 . O 0
Ž since s0 g L . .
Also z 0 ) 0 and V is a convex cone hence s0 Ž w 0 . z0
O 0,
i.e., y
s0 Ž w 0 .
g V.
z0
Ž 3.4.
From Ž3.1. and Ž3.4., we get ¨0 y
ž
y0 z0
q
s0 Ž w 0 .
/
z0
g int V
on account of Lemma 1.1. Hence, y0 z0
q
s0 Ž w 0 . z0
- ¨0
which contradicts Ž3.3.. The result follows. THEOREM 3.3 ŽStrong Duality Theorem.. Suppose that V and H are pointed con¨ ex cones in R p and R m , respecti¨ ely, with int V / B, int H / B. Let E0 be an in¨ ex subset of R n with respect to function h : R n = R n ª R n. Let F, yG be V-prein¨ ex and H be H-prein¨ ex on E0 with respect to same h and further assume that there exist ˆ x g E0 such that H Ž ˆ x . l Žyint H. / B. Ž . If x 0 is a weakly efficient solution for FP then there exists y 0 g F Ž x 0 ., z 0 g GŽ x 0 . such that y0 z0
g W-Max D c Ž s . . sgL
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BHATIA AND MEHRA
Proof. Since x 0 is a weakly efficient solution for ŽFP. hence there exists y 0 g F Ž x 0 ., z 0 g GŽ x 0 . such that y0 z0
g W-Min D
xgE
F G
Ž x. .
Therefore by Lemma 3.1, x 0 is a weakly efficient solution for ŽP.l 0 where l0 s y 0rz 0 with 0 g F Ž x 0 . y l0 GŽ x 0 . as a weakly efficient value of ŽP.l 0 . This shows that the system F Ž x . y l 0 G Ž x . l Ž yint V . / B H Ž x . l Ž yint H . / B
5
Ž 3.5.
has no solution. Since F, yG are V-preinvex with respect to h and l0 P 0 hence F y l0 G is V-preinvex with respect to h on E0 . Therefore by Theorem 2.1 and Corollary 2.2, there exists Ž uU , ¨ U . g VU = HU such that for all x g E0 ² uU , F Ž x . y l0 G Ž x . : q ² ¨ U , H Ž x . : P 0 U
U
Ž u , ¨ . / Ž 0, 0 . ,
U
u / 0.
Ž 3.6. Ž 3.7.
Let x s x 0 in Ž3.6.. We get ² uU , F Ž x 0 . y l 0 G Ž x 0 . : q ² ¨ U , H Ž x 0 . : P 0.
Ž 3.8.
Since y 0 g F Ž x 0 . and z 0 g GŽ x 0 ., hence, we have ² uU , y 0 y l 0 z 0 : q ² ¨ U , H Ž x 0 . : P 0 giving ²¨ U , H Ž x0 . : P 0
ž
by using l0 s
y0 z0
/
.
Ž 3.9.
Also x 0 is feasible for ŽFP. therefore H Ž x 0 . l Ž yH. / B. Choose w 0 g H Ž x 0 . l ŽyH.; then ²¨ U , w0 : P 0
Ž by using Ž 3.9. . .
Ž 3.10.
Moreover w 0 g yH and ¨ U g HU , therefore ² ¨ U , w 0 : O 0.
Ž 3.11.
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609
Combining Ž3.10. and Ž3.11., we get ² ¨ U , w 0 : s 0.
Ž 3.12.
Since uU g VU , uU / 0, choose j g V such that ² uU , j : s 1. Define s0 : R m ª R p as s0 Ž w . s ² ¨ U , w :j . Then clearly s0 g L. Further, s0 Ž w 0 . s 0
² uU , s0 Ž w . : s ² ¨ U , w : .
and
Ž 3.13.
We will show that y 0rz 0 g c Ž s0 .. To the contrary, let if possible, y 0rz 0 f c Ž s 0 .. This implies that there exists ˆ x g E such that for some ˆ y g FŽ ˆ x ., ˆz g GŽ ˆx ., and w ˆ g H Ž ˆx ., s0 Ž w y0 ˆy ˆ. q z0 ˆz ˆz giving y0 z0
y
ž ˆˆ y
z
q
s0 Ž w ˆ.
ˆz
/
g int V ,
hence,
¦
uU ,
y0 z0
y
ž ˆˆ y
z
q
s0 Ž w ˆ.
ˆz
/;
as uU g VU , uU / 0.
)0
Therefore, we have ² uU , ˆ y y l0 ˆ z : q ² uU ? s0 Ž w ˆ.: - 0
ž
using the fact that l 0 s
y0 z0
/
which in view of Ž3.13. gives ² uU , ˆ y y l0 ˆ z : q ²¨ U , w ˆ : - 0. But this contradicts Ž3.8.. Hence y 0rz 0 g c Ž s0 .. Next, we have to show that y0 g W-Max D c Ž s . . z0 sgL Assume to the contrary. Then there exists s1 g L such that for some
¨ 1 g c Ž s1 ., we have
y0 z0
- ¨1 « ¨1 y
y0 z0
g int V .
Ž 3.14.
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Now, x 0 is feasible for ŽFP. hence H Ž x 0 . l ŽyH. / B. Therefore for w 0 g H Ž x 0 . l ŽyH., we have s1Ž w 0 . O 0. Since z 0 ) 0 and H is a convex cone, therefore s1 Ž w 0 . z0
O0
i.e., y
s1 Ž w 0 . z0
g V.
Ž 3.15.
From Ž3.14. and Ž3.15., we get ¨1 y
ž
y0 z0
q
s1 Ž w 0 . z0
/
g int V
on account of Lemma 1.1. Hence, we have y0 z0
q
s1 Ž w 0 .
- ¨1.
z0
Ž 3.16.
But Ž3.16. is a contradiction to the fact that y0 z0
q
s1 Ž w 0 . z0
g
D
xgE
F G
q
s1 Ž H . G
Ž x.
and ¨ 1 g c Ž s1 . s W-Min D
xgE
ž
F G
q
s1 Ž H . G
/Ž .
x .
Therefore s0 is a weakly efficient solution for ŽDFP. with y 0rz 0 g c Ž s0 . as a weakly efficient value of ŽDFP.. THEOREM 3.4. Suppose the assumptions of the pre¨ ious theorem Ž Strong Duality Theorem. are satisfied. Let x 0 be a weakly efficient solution for ŽFP. and ¨ 0 g W-Max Ds g L c Ž s .. Then for e¨ ery y 0 g F Ž x 0 ., z 0 g GŽ x 0 ., y 0rz 0 l ¨ 0 . Moreo¨ er there exists ˆ y 0 g F Ž x 0 . and ˆ z 0 g GŽ x 0 . for which ¨ 0 l ˆ y 0rzˆ0 . Proof. Since x 0 is a weakly efficient solution for ŽFP. hence it is feasible for ŽFP. and ¨ 0 g W-Max Ds g L c Ž s .. Therefore there exists s0 g L such that ¨ 0 g c Ž s0 ..
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Hence by Theorem 3.2 ŽWeak Duality Theorem. for every y 0 g F Ž x 0 . and z 0 g GŽ x 0 ., y0 z0
l ¨0.
On using Theorem 3.3 ŽStrong Duality Theorem., we get an existence of ˆy 0 g F Ž x 0 ., ˆz 0 g GŽ x 0 ., and s0 g L such that
ˆy 0 g c Ž s0 . . ˆz 0 But ¨ 0 g W-Max Ds g L c Ž s ., hence ¨ 0 l ˆ y 0rzˆ0 . ACKNOWLEDGMENTS The authors are grateful to Professor R. N. Kaul, University of Delhi, for his constant encouragement in preparing this paper. Thanks are also due to the referee for his valuable suggestions.
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