Keywords--Multiobjective optimization, Preinvexity, Optimality conditions. ..... Since s ⢠int Q, there exist a ball N with center at zero, such that s + N C Q. ..... We will call Xo ⢠F a global weak efficient solution for problem (P1) if it is a weak efficient.
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ELSEVIER
computers & mathematics
with applications Computers and Mathematics with Applications 48 (2004) 885-895 www.elsevier.com/locate/camwa
P r e i n v e x Functions and W e a k Efficient Solutions for S o m e Vectorial Optimization Problem in B a n a c h Spaces L.
BATISTA DOS SANTOS* IMECC-UNICAMP CP 6065, 13081-970, Campinas-SP, Brazil lucelina@ime, unicamp, br
R. OSUNA-G6MEZ
t*
Departamento de Estadfstica e Investigacidn Operativa Facultad de Matem£ticas Universidad de Sevilla, Sevilla, 41012, Spain rafaela©us,
es
M. A. ROJAS-MEDAR
#*
IMECC-UNICAMP CP 6065, 13081-970, Campinas-SP, Brazil marko©ime, unicamp, br
A. RUFIJ~N-LIZANA* D e p a r t a m e n t o de Estad~stica e Investigaci6n O p e r a t i v a F a c u l t a d de M a t e m £ t i c a s Universidad de Sevilla, Sevilla, 41012, S p a i n ruf ian%us, e s
(Received November 2002; revised and accepted May PO03) Abstract--In this work~ we introduce the notion of preinvex function for functions between Bar nach spaces. By using these functions, we obtain necessary and sufficient conditions of optimality for vectorial problems with restrictions of inequalities. Moreover, we will show that this class of problems has the property that each local optimal solution is in fact global. © 2004 Elsevier Ltd. All rights reserved.
K e y w o r d s - - M u l t i o b j e c t i v e optimization, Preinvexity, Optimality conditions.
*This author is supported by CNPq-Brasil. t This author is partially supported by MCYT-Spain, Grant MTM2004-01433. :~ The authors are partially supported by MCYT-Spain Grant BFM2003-06579. # This author is partially supported by CNPq-Brazil Grant 301354/03-0, and FAPESP-Brazil Grant 01/07557-3. The authors would like to thank very much the anonymous referees whose remarks helped to improve the first version of this paper in some important points. 0898-1221/04/$ - see front matter @ 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2003.05.013
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L. BATISTA DOS SANTOS
et al.
1. I N T R O D U C T I O N AND FORMULATION OF THE PROBLEM Throughout this paper, E, F, and G will be real Banach spaces. We shall consider the following problem of optimization: Minimize f(x), subject to - g(~) e K,
(P)
xEScE, where f and g are mappings from E into F and G, respectively, and where S and K are two subsets of E and G. We assume that the spaces F and G are ordered by cones Q c F, K c G and that these cones are closed, convex, and with nonempty interior. We denote by jr = {x E S : - g ( x ) E K } the feasible set of (P). We can consider the following partial order in F: y, z E F ,
y-_ o, .., ;- w* o f (x2 + )vr/(Xl, x2)) _< .~w* o f (Xl) + (1 -- ,~) OJ* 0 f (.T2) ,
Vw* E Q*, Vw* E Q*,
where the first equivalence follows from the definition of __F, the second from Lemma 2.4 and the third from the linearity of w*. | DEFINITION 2.5. Let f : ~ C E ~ F. We say that f is directionally differentiable at xo in the direction d, denote f ' ( x o , d), ff the foIlowing limit ex/sts lira f (xo + Ad) - f (x0) ~--*0+ The following property of the directionaily differentiable preinvex functions will be extensively used in the rest of the paper. LEMMA 2.6. Let f : ~ C E --* F be a preinvex function on S C f~, directionMly differentiable. Then, (w* o f)t (x, r/(z,y)) 0,
(16)
Vy e S, Vw* e Q*. PROOF. First, we show the implication ( 3 ) . We assume that • is a weak efficient solution and that (16) is not true. In this case, there exist y E S and w* E Q*, such that
(~* o f)' (~, ~ (~, y)) < 0.
(17)
Since S is open and S: E S, we have that ~ + A~](~,y) E S, for A > 0 sufficiently small. From (17), we obtain lira w* o f (S: + AT/(S:,y)) - w * of(S:) < 0, X--+O+
and therefore, for A > 0 sufficient small, we get
~* ( f (x + A~ (x, Y)) - f (S:)) < 0. Since w* E Q*, w* # 0, we have f (S: + A~ (S:, y)) -4F f (~), with ~ + ),~(~, y) E S. This is a contradiction with the fact that • is a weak efficient solution. Now, we prove the reverse implication ( ~ ) . To do this, we assume that condition (16) is true and that S: is not a weak efficient solution. In this case, there exists y E S, such that f(y) " 0,
vucT,
(23)
and
(#*, g (~)) = O.
(24)
(,* o 9)' (~, ~ (~, x0)) < ,* o g(~o) - ,* o 9 (~) = ,* o ~(~o) < 0
(25)
But,
(where the first inequality is consequence of Lemma 2.6, the equality is obtained (24) and the last inequality from g(xo) -
