Existence of solutions for vector optimization

Existence of solutions for vector optimization

Appl. Math. Lett. Vol. 9, No. 6, pp. 19-22, 1996 Copyright(~)1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/96 $15...

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Appl. Math. Lett. Vol. 9, No. 6, pp. 19-22, 1996 Copyright(~)1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/96 $15.00 + 0.00

Pergamon

PII: S0893-9659(96)00088-2

E x i s t e n c e of Solutions for Vector O p t i m i z a t i o n K. R. KAZMI Department of Mathematics, Jamia Millia Islamia New Delhi 110025, India (Received April 1995; revised and accepted April 1996)

A b s t r a c t - - I n this paper, we prove the existence of a weak minimum for constrained vector optimization problem by making use of vector variational-like inequality and preinvex functions. K e y w o r d s - - - V e c t o r optimization problem, Vector variational-like inequality problem, Preinvex function, Weak minimum, KKM-Fan theorem.

1. I N T R O D U C T I O N We denote the n o r m on R m by I[" II. ( Rm, R ~ ) is an ordered Hilbert space with an ordering in R m defined by the convex cone R m +, Vy, x E R m,

y < x v* x - y E R'~.

If int R ~ denotes the interior of R ~ , then a weak ordering in •m is also defined by Vy, x E R "~,

y ~ x ¢~ x -

y ~ int R ~ .

A subset K c R n is said to have "rpconnectedness" p r o p e r t y if, for each x, y E K , A E [0, 1], there exists a vector ~?(x, y) E R '~, such t h a t y + A~?(x, y) E K. Consider a vector optimization problem w- min f ( x ) subject to x E K,

(1.1)

where K c R '~ is n o n e m p t y "~?-connected" set, f : R n ~ R m is a vector-valued function, and w - m i n denotes weak minimum. We note t h a t the problem (1.1) has a weak m i n i m u m at x = x0 E K if and only if f ( x ) - f (xo) ~ - int R ~ ,

Vx e K,

see, for example, [1,2]. , R m is called RT-invex, with respect ot a function ~? : Following [3], a function f : K K × K ---* R m, if, for each x, y E K f ( x ) - f ( y ) -- ( i f ( y ) , r/(x, y)) e R ~ ,

(1.2)

where f~(y) denotes the Fr~chet derivative of f at y. If we take R and R+ in place of R m and R ~ , respectively, then f is called invex. Invex functions were first considered by H a n s o n [4], Typeset by .A,A4S-TEX 19

20

K.R. KAZMI

who showed that if, instead of the usual convexity conditions, the objective function and each of the constraints of a nonlinear program axe all invex for the same r/(x, y) then the sufficiency of the Kuhn-Tucker conditions [5], and weak [6] duality still holds. Morever, Craven and Glover [7] showed that the class of real-valued invex functions is equivalent to the class of functions whose stationary points are global minima. Following Ben-Israel and Mond [8] and Hanson and Mond [9], consider a function f : K - - o R m having the property that there exists a function r/: K x K ---* R m such that, for each x, y • K and A • [0, 1], y + Ar/(x, y) • K and

)~f(x) q- (1

-

~k)f(y)

- f

(y q- ~r/(x, y)) • R~.

(1.3)

It is observed that if f is Fr~chet differentiable and satisfies (1.3) that f also satisfies (1.2). This can be seen by rewriting (1.3) as - f(y)) - [f(y +

y)) - S(x)] • R T ,

and then divided by A > 0 and taking the limit as A ~

0+ gives

f(x) - f(y) - (f'(y),u(x,y)) •R~. In view of this observation, the functions satisfying (1.3) will be called R~-preinvex. It is also noted that the set K should have the "r/-connectedness" property. Note also that if r/(x, y) a(x, y)(x - y) where0 < a(x, y) < 1, then K should be star-shaped [10]. If we take R and R+ in place of R m and R~', respectively, and if f satisfies (1.3), then f is called preinvex. In many papers, see, for example, [11-14] dealing with the existence of optimal solutions for vector optimization, some kind of compactness in the value space of the objective functions is assumed. This is often difficult to verify in applications. In this paper, a sufficient condition is given for the existence of optimal solutions for problem (1.1) by making use of vector variational-like inequality and preinvex functions.

2. E X I S T E N C E

OF SOLUTIONS

First we establish the equivalence relation between the vector optimization problem (1.1) and the vector variational-like inequality problem of finding x0 E K such that

(f'(xo), rl(x, xo)) ~ - int R~,

Vx E K.

(2.1)

We need the following lemma [15]. LEMMA 2.1. Let (X, P ) be an ordered topological vector space with a closed, pointed and convex cone P with int P ~ 0. Then, Vx, y E X, we have (i) (ii) (iii) (iv)

y y y y

-

x x x x

E E E E

int P and y ~ int P imply x ~ int P; P, and y ~ int P imply x ¢ int P; - int P and y ~ - int P imply x q~ - int P; and - P and y q~ - int P imply x ~ - int P.

Let W = R m \ ( - int R~). THEOREM 2.1. Let the set K be satisfy the ~l-connectedness property, and let the function f be Nr~-preinvex and Frdchet differentiable. Then the vector optimization problem (1.1) and the vector variational-like inequality problem (2.1) have the same solution set. PROOF. Let x0 be a w e a k minimum of problem (1.1). I f x E K and 0 < a <_ 1, then x 0 + arl(x, Xo) • K since K have the r/-connectedness property. Hence, a -1 f / ( x 0 +

(z, z0)) - I (x0)] • w ,

• (0,1].

Vector Optimization

21

Since W is closed and f is Fr6chet differentiable, it follows that

( f ' ( x o ) , ~ ( x , xo)) ¢ - i n t R ~+ . Conversely, let x0 E K satisfy (2.1) Since f is R~-preinvex and K has the ~-connectedness property, then it follows that

f ( x ) - f (XO) - (f' (Xo), 71(x, xo)) E R~. Hence,

f ( x ) - f (xo) ~ - int R ~

(by (iv) of Lemma 2.1).

This completes the proof. Now we are able to prove the main result of this paper, but before stating this, we quote the following theorem (KKM-Fan Theorem [16]) which play an important role in the proof of our main result. THEOREM 2.2. Let E be a subset of the topological vector space X . For each x E E, let a closed set F ( x ) in X be given such that F ( x ) is compact for at least one x E E. If the convex hull of every finite subset {xl, x 2 , . . . , xn} of E is contained in the corresponding union U~=lF(xi) , then

nxesF(x) ~ 0. THEOREM 2.3. Let K be a nonempty compact convex set in ]~'~ and satisfy the ~?-connectedness property; let f : K ~ R m be ~dchet differentiable and R~-preinvex; let the set

(y E K : ( f ' ( x ) , ~ ( y , x ) ) E - i n t R ~ } be convex for each fixed x E K; let ~ be continuous and satisfy ~l(x, x) = 0 for each x E K. Then the vector optimization problem (1.1) has a weak minimum x0. PROOF. By Theorem 2.1, it is sufficient to show that the vector variational-like inequality problem (2.1) has a solution x0. For y E K , define

F(y) = ( x e K : (f'(x),~?(y,x)) ~ - i n t R ~ } . Let ( x l , x 2 , . . . , xm} C K . Claim that the convex hull of every finite subset ( x l , x 2 , . . . , xm} is contained in the corresponding union U'~=lF(xi ), i.e., cony {xl, x 2 , . . . , xm} c U~=lF(X~ ). Indeed let t~i > m OQ = 1. Suppose that x = ~i=lm aiXi ~ U ,n -- 0, 1 < i < m, with ~-~i=l i = I F ( X i ), then

(f'(x), ~/(x,, x)) E - int R~,

Vi.

Since, for each fixed x E K , the set

G(x) = {y G K : (f'(x),~?(y, x)) e - i n t R ~ } is convex. Hence, by convex property of G(x), we have

f~

aix~

, ~/

aix~,

(~x~

E - int R ~ ,

\i=l

which is a contradiction and our claim is then verified. Next, we claim that for each y E K , F(y) is closed. Indeed, let y E K and let a sequence ( x k } c f ( y ) satisfy IIx~ - ~ll ~ 0.

22

K . R . KAZMI

Since f ' ( . ) is continuous on K, {f'(xk)} ----0 if(x) uniformly on K . Then

t[(f' (xk),~(y, xk)) -- (f'(x),~(y,x))[[ [[(f' (Xk),~(y, xD) -- (f'(x),~(y, xk))H + II( f t ( x ) , ~1(Y, x k ) ) -- ( f ' ( x ) , ~(y, x))II

<

Ilf' (Xk)

- f ' ( x ) [ [ 1]~ (y, x/c)II +

Ilf'(x)ll

I1~ (y, xk) -- 7](y, x)l ]

,0

as n , c¢, since ~/is continuous on K. Since W is closed and (f'(xk), ~(y, Xk)l • W, Vk,

• w, t h a t is

(f'(x), y(y, x)) ¢ - int R~, and our claim is verified. Also F(y) is nonempty since y • F ( y ) , for all y • K . Since F(y) is closed, and hence, compact for each y • K . Thus, by Theorem 2.2, it follows that there exists at least one point xo • NveKF(y), that is x0 • K such that

( f (x0), 77(y, x0)) ¢ - int R T,

Yy • K.

This completes the proof. It is remarked t h a t the results of this paper generalize some results of Chen and Craven [17].

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