doi:10.1006rjmaa.2000.7396, available online at http:rrwww.idealibrary.com on
Sufficiency and Duality in Multiobjective Programming
Ž .
Involving Generalized F, -Convexity
Brahim Aghezzaf and Mohamed Hachimi
Departement de Mathematiques et d’Informatique, Uni´ ´ ¨ersite Hassan II, Faculte des´ ´ Sciences Aın Chock, B.P. 5366 Maarif Casablanca, Morocco¨
E-mail: [email protected] Submitted by K. Mizukami
Received May 2, 2000
Ž .
New classes of generalized F, -convexity are introduced for vector-valued
Ž .
functions. Examples are given to show their relations with F, -pseudoconvex, ŽF,.-quasiconvex, and strictly ŽF,.-pseudoconvex vector-valued functions. The sufficient optimality conditions and duality results are obtained for multiobjective
Ž .
programming involving generalized F, -convex vector-valued functions. 䊚2001 Academic Press
1. INTRODUCTION
In the present paper we consider the following multiobjective nonlinear programming problem,
minimize f xŽ .sŽf1Ž .x , . . . ,fmŽ .x .
ŽMOP. ½subject to xgAsxgXNg xŽ .O0, h xŽ .s0 ,4
where X is an open subset of ⺢n and f: Xª⺢m, g: Xª⺢p, and h:Xª⺢q are differentiable functions at xgA.
Under different assumptions of convexity convexity, generalized convex-Ž ity, generalized -convexity, generalized F-convexity, invexity or general-
. w x w x w x
ized invexity Aghezzaf 1 , Aghezzaf and Hachimi 2 , Weir and Mond 10 ,
w x w x
Egudo 3 , and Gulati and Islam 4 have used proper efficiency or efficiency to establish some duality results.
Ž . w x
The concept of F, -convexity was introduced by Preda 8 as extension
w x w x
of F-convexity 6 and -convexity 9 , and he used the concept to obtain w x
some duality results. In 11 , Xu introduced a mixed type dual for problem
617
0022-247Xr01 $35.00
Copyright䊚2001 by Academic Press All rights of reproduction in any form reserved.
ŽMOP with inequality only, and obtained duality results under generalized. ŽF,.-convexity assumptions.
Ž .
In this paper, we introduce new classes of generalized F, -convexity for vector-valued functions and present more sufficient conditions and
Ž .
duality results for problem MOP .
Notations. Throughout this paper we use the following notations. The
4 4 4
index set Ms 1, 2, . . . ,m, Ps 1, 2, . . . ,p, and Qs 1, 2, . . . ,q. For
Ž . 4
xgA, the index set Es jNg xj s0 and gE denotes the vector for active constraints. If x and yg⺢n, then xOymxiOyi, is1, . . . ,n;
xFymxOy and x/y; x-ymxi-yi, is1, . . . ,n; xy or xty denotes the inner product.
Ž .
For the multiobjective programming problem MOP , the solution is
Ž .
defined in terms of a weak efficient solution in the following sense:
DEFINITION1.1. We say that xgA is an efficient solution for problem ŽMOP if and only if there exists no. xgAsuch that f xŽ .Ff xŽ ..
DEFINITION 1.2. We say that xgA is a weak efficient solution for
Ž . Ž . Ž .
problem MOP if and only if there exists no xgAsuch that f x -f x .
2. PRELIMINARIES w x The following definitions are from Preda 8 :
DEFINITION 2.1. A functional F:X=X=⺢nª⺢ is sublinear if for any x,xgX,
F x,Ž x;a1qa2.OF x,Ž x;a1.qF x,Ž x;a2. ᭙a1,a2g⺢n, F x,Ž x;␣a.s␣F xŽ ,x;a. ᭙␣g⺢,␣P0,᭙ag⺢n.
Ž . m
Let Fbe a sublinear functional and the function fs f1, . . . ,fm :Xª⺢
Ž . m Ž .
be differentiable at xgX and s 1, . . . ,m g⺢ , and d⭈,⭈ :X= Xª⺢.
Ž .
DEFINITION 2.2. The function fi is said to be f,i -convex at x, if for all xgX we have
F x,Ž x;ⵜf xiŽ ..qid2Žx,x.Of xiŽ .yf xiŽ ..
m Ž .
The vector-valued function f:Xª⺢ is F, -convex at x if each of its
Ž .
components fi is F,i -convex at x.
Ž .
DEFINITION 2.3. The function fi is F,i -quasiconvex at x, if for all xgX we have
f xiŽ .Of xiŽ .«F xŽ ,x;ⵜf xiŽ ..O yid2Žx,x..
Ž .
We say that f is F, -quasiconvex at x if each of its components fi is ŽF,i.-quasiconvex at x.
Ž .
DEFINITION 2.4. The function fi is F,i -pseudoconvex at x, if for all xgX we have
f xiŽ .-f xiŽ .«F xŽ ,x;ⵜf xiŽ ..-yid2Žx,x..
Ž .
The function f is F, -pseudoconvex at x if each of its components fi is ŽF,i.-pseudoconvex at x.
The following convention will be used. If f is an m-dimensional vector-
Ž Ž ..
valued function, then F x,x;ⵜf x denotes the vector of components
Ž Ž .. Ž Ž ..
F x,x;ⵜf x1 , . . . ,F x,x;ⵜfm x .
Ž . Ž .
We now define weak strictly F, -pseudoconvex, strong F, -pseudo-
w x Ž . Ž .
convex 5 , weak F, -quasiconvex, and sub-strictly F, -pseudoconvex functions.
Ž .
DEFINITION 2.5. The function f is said to be weak strictly F, -pseu- doconvex at x, if for all xgX we have
f xŽ .Ff xŽ .«F x,Ž x;ⵜf xŽ ..-yd2Žx,x..
Ž .
The class of weak strictly F, -pseudoconvex functions does not con-
Ž .
tain the class of F, -convex functions, but does contain the class of
Ž .
strictly F, -pseudoconvex functions.
Ž .
DEFINITION 2.6. The function f is said to be strong F, -pseudocon- vex at x, if for all xgX we have
f xŽ .Ff xŽ .«F x,Ž x;ⵜf xŽ ..F yd2Žx,x..
Ž .
The class of strong F, -pseudoconvex functions does not contain the Ž .
class of F -pseudoconvex functions, but does contain the class of ŽF,.-convex and weak strictly ŽF,.-pseudoconvex functions.
Ž . Ž .
It is clear that both strong F, -pseudoconvexity and F, -pseudo-
Ž .
convexity imply F, -convexity. The following examples show that, in
Ž .
general, no relationship holds between strong F, -pseudoconvexity and ŽF,.-pseudoconvexity.
Ž . 2 Ž . Ž 2
EXAMPLE 2.1. Define a function f: X s⺢ ª⺢ by f x s x y
3 2. 3 Ž . Ž .
2x,x yx and F:⺢ ª⺢ by F x,x;a sa⭈ xyx , and let s0. f
Ž . Ž .
is strong F, -pseudoconvex at xs0 but is not F, -pseudoconvex at
3 2
Ž . Ž .
xs0, because the function f x1 sx yx is not F, -pseudoconvex at xs0.
Ž . 2 Ž . Ž Ž EXAMPLE 2.2. Define a function f:X s⺢ ª⺢ by f x s x xy
2 3
. Ž .. Ž . Ž .
2 ,x xy3 and F:⺢ ª⺢by F x,x;a sa⭈ xyx , and let s0. f
Ž . Ž .
is F, -pseudoconvex at xs0 but is not strong F, -pseudoconvex at
t t
Ž . Ž . Ž .Ž . Ž . Ž .
xs0, because f 2 Ff 0 butⵜf 0 2y0 s 8,y6 g 0, 0 .
Ž .
Remark 2.1. For the scalar functions the class of weak strictly F, -
Ž .
pseudoconvex functions, the class of strong F, -pseudoconvex functions,
Ž .
and the class of F, -pseudoconvex functions coincide.
Ž .
DEFINITION 2.7. The function f is said to be weak F, -quasiconvex at x, if for all xgX we have
f xŽ .Ff xŽ .«F x,Ž x;ⵜf xŽ ..O yd2Žx,x..
Ž . Ž .
Every F, -quasiconvex function is weak F, -quasiconvex. However,
Ž . Ž .
there exist functions which are weak F, -quasiconvex but not F, - quasiconvex.
Ž . 2 Ž . Ž Ž
EXAMPLE 2.3. Define a function f:X s⺢ ª⺢ by f x s x xy
2 2 3
. Ž .. Ž . Ž .
2 ,x xy2 and F:⺢ ª⺢by F x,x;a sa⭈ xyx , and let s0.
Ž . Ž .
f is weak F, -quasiconvex at xs0 but is not F, -quasiconvex at
Ž . Ž .2 Ž .
xs0, because the function f x1 sx xy2 is not F, -quasiconvex at xs0.
Ž .
DEFINITION2.8. The function f is said to be sub-strictly F, -pseudo- convex at x, if for all xgX we have
f xŽ .Of xŽ .«F x,Ž x;ⵜf xŽ ..F yd2Žx,x..
Ž . Ž .
Every strictly F, -pseudoconvex function is sub-strictly F, -pseudo- convex. However, the converse is not necessarily true.
Ž . 2 Ž . Ž 2Ž
EXAMPLE2.4. Define a function f: X s⺢ ª⺢ by f x s x xy
2 2 3
. Ž . . Ž . Ž .
2 ,yx xy2 and F:⺢ ª⺢byF x,x;a sa⭈ xyx , and let s0.
Ž . Ž .
f is weak F, -quasiconvex at xs0 but is not F, -quasiconvex at
2 2
Ž . Ž . Ž .
xs0, because the function f x1 sx xy2 is not strictly F, -pseu- doconvex at xs0.
Ž .
Remark 2.2. For the scalar functions the class of sub-strictly F, -
Ž .
pseudoconvex functions and class of strictly F, -pseudoconvex functions coincide.
3. SUFFICIENT OPTIMALITY CONDITIONS
In this section, we obtain sufficient conditions for a feasible x to be
Ž .
efficient for MOP in the form of the following theorems
Ž . THEOREM 3.1. Suppose that there exists a feasible solution x for MOP
m p q
and¨ectors ug⺢ ,¨g⺢ , and wg⺢ such that Ž .a uⵜf xŽ .q¨ⵜg xŽ .qwⵜh xŽ .s0,
Ž .b ¨g xŽ .s0,
Ž .c u)0,¨P0, wP0.
Ž 1. Ž 2.
If f is strong F, -pseudocon¨ex, gE is F, -quasicon¨ex, and h is
3 1 2 3
ŽF, .-quasicon¨ex at x with u q¨E qw P0, then x is an efficient
Ž .
solution for MOP .
Ž .
Proof. Suppose that x is not an efficient solution for MOP . Then,
Ž . Ž . Ž . Ž . Ž . Ž .
there exists an xgA such that f x Ff x , gE x OgE x , h x sh x . From the hypothesis on f, gE, and h, we have
1 2
F x,Ž x;ⵜf xŽ .. F y d Žx,x., Ž .1
2 2
F xŽ ,x;ⵜgEŽ .x .O y d Žx,x., Ž .2
1 3
F x,Ž x;ⵜh xŽ .. O y d Žx,x.. Ž .3
Ž . Ž . Ž .
Multiplying 1 , 2 , and 3 with u,¨E, andw, respectively, we get
1 2
uF x,Ž x;ⵜf xŽ ..-yu d Žx,x.,
2 2
¨EF x,Ž x;ⵜgEŽ .x .O y¨E d Žx,x.,
1 3
wF x,Ž x;ⵜh xŽ ..O yw d Žx,x.. By the sublinearity of F, we summarize to get
F x,Ž x;uⵜf xŽ .q¨ⵜg xŽ .qwⵜh xŽ ..
OuF x,Ž x;ⵜf xŽ ..q¨EF x,Ž x;ⵜgEŽ .x .qwF x,Ž x;ⵜh xŽ ..
1 2 3 2
-yŽu q¨ qw .d Žx,x.O0,
Ž . Ž .
which contradicts a because F x,x; 0 s0.
Ž .
In the next theorem, we replace the strong F, -pseudoconvexity of f
Ž .
by the weak strictly F, -pseudoconvexity.
Ž .
THEOREM 3.2. Suppose that there exists a feasible solution x for MOP
m p q
and¨ectors ug⺢ ,¨g⺢ , and wg⺢ such that Ž .a uⵜf xŽ .q¨ⵜg xŽ .qwⵜh xŽ .s0,
Ž .b ¨g xŽ .s0,
Ž .c uG0,¨P0, wP0.
Ž 1. Ž 2.
If f is weak strictly F, -pseudocon¨ex, gE is F, -quasicon¨ex, and h is
3 1 2 3
ŽF, .-quasicon¨ex at x with u q¨E qw P0, then x is an efficient
Ž .
solution for MOP .
Ž .
Proof. Assume that x is not an efficient solution for MOP . Then, there exists an xgA such that
f xŽ .yf xŽ .F0, gEŽ .x ygEŽ .x O0, h xŽ .yh xŽ .s0. Ž .4 The assumptions on f, gE, and h together with 4 givesŽ .
1 2
F x,Ž x;ⵜf xŽ .-y d Žx,x.,
2 2
F xŽ ,x;ⵜgEŽ .x .O y d Žx,x.,
1 3
F x,Ž x;ⵜh xŽ .. O y d Žx,x., and now the proof is similar to that of Theorem 3.1.
Ž .
In our final sufficiency result below, we invoke the weak F, -quasi-
Ž .
convexity of f and the sub-strictly F, -pseudoconvexity of gE.
Ž .
THEOREM 3.3. Suppose that there exists a feasible solution x for MOP
m p q
and¨ectors ug⺢ ,¨g⺢ , and wg⺢ such that Ž .a uⵜf xŽ .q¨ⵜg xŽ .qwⵜh xŽ .s0,
Ž .b ¨g xŽ .s0,
Ž .c Žu,¨.P0,¨E)0, wP0.
Ž 1. Ž 2.
If f is weak F, -quasicon¨ex, gE is sub-strictly F, -pseudocon¨ex,and
3 1 2 3
Ž .
h is F, -quasicon¨ex at x with u q¨E qw P0, then x is an
Ž .
efficient solution for MOP .
Ž .
Proof. If x is not an efficient solution of MOP , then there exists an xgA such that
f xŽ .yf xŽ .F0, gEŽ .x ygEŽ .x O0, h xŽ .yh xŽ .s0. Ž .5
Ž 1. Ž 2.
From the weak F, -quasiconvexity of f, the sub-strictly F, -pseudo-
Ž 3. Ž .
convexity of gE, and the F, -quasiconvexity of h, from 5 we obtain
1 2
F x,Ž x;ⵜf xŽ .O y d Žx,x.,
2 2
F xŽ ,x;ⵜgEŽ .x .F y d Žx,x.,
1 3
F x,Ž x;ⵜh xŽ .. O y d Žx,x., and now the proof is similar to that of Theorem 3.1.
We can also prove Theorem 3.3 by replacing the assumption that gE is
Ž 2.
sub-strictly F, -pseudoconvex with the assumption that gE is strictly ŽF,2.-pseudoconvex and relaxing the condition ¨E)0 to the condition
¨EG0 only.
4. MIXED TYPE DUALITY
Let J1 be a subset of P and J2sPrJ1, let K1 be a subset of Q and K2sQrK1, and let e be the vector of⺢m whose components are all ones.
Ž .
We consider the following mixed type dual for MOP ,
¡maximize f yŽ .q¨J1gJ1Ž .y eqw hK1 K1Ž .y e,
s.t. uⵜf yŽ .q¨ⵜg yŽ .qwⵜh yŽ .s0,
¨J2gJ2Ž .y P0,
~
ŽXMOP.
w hK2 K2Ž .y s0,
¨P0,
¢ uP0,u et s1.
Note that we get a Mond᎐Weir dual for J1s⭋ and K1s⭋ and a
Ž .
Wolfe dual for J2s⭋and K2s⭋ in XMOP , respectively.
Ž . Ž .
We shall prove various duality results for MOP and XMOP under
Ž .
weak assumptions of F, -convexity.
Ž .
THEOREM4.1 Weak Duality . Assume that for all feasible x for problem ŽMOP. and all feasible y,Ž u,¨,w for problem. ŽXMOP ,.
Ž .a ¨J gJŽ .⭈ qw hK KŽ .⭈ is F,Ž ␣.-quasicon¨ex at y, and assume that
2 2 2 2
one of the following conditions holds:
Ž .b u)0, and fŽ .⭈ q¨J gJŽ .⭈ eqw hK KŽ .⭈ e is strong F,Ž .-pseudo-
1 1 1 1
con¨ex at y,with ␣quP0.
Ž .c u)0, and ufŽ .⭈ q¨J gJŽ .⭈ qw hK KŽ .⭈ is F,Ž .-pseudocon¨ex at
1 1 1 1
y,with ␣qP0.
Then the following cannot hold:
f xŽ .Ff yŽ .q¨J1gJ1Ž .y eqw hK1 K1Ž .y e. Ž .6
Ž . Ž .
Proof. Let x be feasible for MOP and let y,u,¨,w be feasible for ŽXMOP . Then, we have.
¨J2gJ2Ž .x qw hK2 K2Ž .x O¨J2gJ2Ž .y qw hK2 K2Ž .y . Ž .7
Ž . Ž .
From 7 and the hypothesis a , we obtain
F xŽ ,y;¨J2ⵜgJ2Ž .y qwK2ⵜhK2Ž .y .O y␣d2Žx,y.. Ž .8
Ž .
By the feasibility of y,u,¨,w and the sublinearity of F, we have F x,Ž y;uⵜf yŽ .q¨J1ⵜgJ1Ž .y qwK1ⵜhK1Ž .y .
qF xŽ ,y;¨J2ⵜgJ2Ž .y qwK2ⵜhK2Ž .y .
PF x,Ž y;uⵜf yŽ .q¨ⵜg yŽ .qwⵜh yŽ ..s0. Ž .9
Ž . Ž .
Relation 9 together with 8 yields
F x,Ž y;uⵜf yŽ .q¨J1ⵜgJ1Ž .y qwK1ⵜhK1Ž .y .P␣d2Žx,y.. Ž10. On the other hand, suppose contrary to the result that 6 holds. SinceŽ . x is
Ž . Ž .
feasible of MOP and¨P0, 6 implies
f xŽ .q¨J1gJ1Ž .x eqw hK1 K1Ž .x e
Ff yŽ .q¨J1gJ1Ž .y eqw hK1 K1Ž .y e. Ž11. Ž .
Multiplying 11 with u, we get
uf xŽ .q¨J1gJ1Ž .x qw hK1 K1Ž .x -uf yŽ .q¨J1gJ1Ž .y qw hK1 K1Ž .y . Ž12.
Ž . Ž .
By hypothesis b and 11 , we have
F xŽ ,y;ⵜf yŽ .q¨J1ⵜgJ1Ž .y eqwK1ⵜhK1Ž .y e.F yd2Žx,y.. Ž13. Ž .
Multiplying 13 with u, we obtain
F x,Ž y;uⵜf yŽ .q¨J1ⵜgJ1Ž .y qwK1ⵜhK1Ž .y .
-yud2Žx,y.O␣d2Žx,y., Ž14. Ž .
which contradicts again 10 .
Ž . Ž .
When hypothesis c holds, 12 implies
F x,Ž y;uⵜf yŽ .q¨J1ⵜgJ1Ž .y qwK1ⵜhK1Ž .y .
-yd2Žx,y.O␣d2Žx,y., Ž15. Ž .
which contradicts 10 .
w x It may be noted that Theorem 4.1 contains Theorem 2.1 of Xu 11 . In
Ž .
fact, let f be a vector function, if for all i, fi is both F,i-quasiconvex
