Item 987654321/8870

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 161, 367-387 (1991)

On the Optimality Conditions of

Vector-Valued n-Set Functions*

LAI-JIU LIN

Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan, Republic qf China

Submitted by E. Stanley Lee

Received October 17, 1989

In a finite atomless measure space (X, r, n), the optimization problem of vector- valued n-set functions defined on a convex subfamily S of r” = f x x r is investigated. The necessary and sufficient conditions of Pareto optimal solution or proper RP,-solution of optimization problem with differentiable vector valued n-set functions are given. 0 1991 Academic Press, Inc.

1. INTRODUCTION

The general theory for optimizing set functions was first developed by Morris [12]. This type of problem arises in various areas and has many

interesting applications in mathematics, engineering, and statistics, for

example, in fluid flow, electrical insulator design, optimal plasma conline-

ment (see Ref. [ 12]), and Neyman-Pearson lemma of statistics (see

Ref. [3]). There are many results on the optimization problem of set

functions, one can consult Refs. [12, 1, 2, 410, 141. All the previous

results on this type of problem are only confined to set functions of a single set. Corley [3] started to give the concepts of partial derivatives and derivatives of real-valued n-set functions and developed the general theory of n-set functions. In [7], we study the vector valued n-set functions

optimization problem. This paper is a continuous work of [7].

Throughout this paper, we assume that (X, r, p) is a finite atomless

measure space with ,5,(X, r, p) separable. For any n E N, we let [w” be the

n-dimensional Euclidean space. We also let SC P’= TX ... x r be a

subfamily of r” and F: S + Rp, H: S --+ R’, and G: S -+ R” be vector-

valued n-set functions defined on S.

* This research was supported by the National Science Council of the Republic of China,

367

0022-247X/91 $3.00

Copyright ‘i,: 1991 by Academic Press. Inc All rvghts of reproductmn in any form reserved

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368 LAI-JIU LIN

We consider two optimization problems as minimize F(n, , . . . . A,)

(P) subject to (/ii, . . . . ~,)ES, G(/i,, . . . . /i,)SO, H(n,, . . . . /i,)=O,

and

minimize F(n,, . . . . A,)

subject to (A,, . . . . /i”)~r”, G(/i,, . . . . /i,) SO. (Pl) In [7], we define the derivative of vector-valued n-set functions, we establish the necessary and sufficient conditions for the existence of a weak local minimum to problem (Pl ) in terms of the partial derivative of vector- valued n-set functions involved. This paper is a continuous work of [7]. The sufficient conditions of Pareto optimal solution to problem (P) and the necessary conditions of pareto optimal solution of (Pl) with nonconvex differentiable n-set functions are developed. The necessary and sufficient conditions of proper WC -solution to problem (Pl) with convex differentiable vector-valued n-set function are also derived.

2. PRELIMINARIES

We define a pseudometric don T”=rx . . . ~r={(n,,.,.,n,)(n~~r,

i= 1, 2, . . . . n} as

4-W,, . . . . Q,), (A 1, . . . . i [Pu(aiAnj)]2

w

3

i= 1

where (a,, . . . . Q,), (/ii, . . . . A,) E r” and s2, dni denote the symmetric dif- ference for 52, and ni. For f EL,(X, r, p) and Sz~r, the integral jn f dp

will be denoted by (f, xn), where xn denotes the characteristic function of Q.

DEFINITION 2.1. A set function F: r+ R is said to be differentiable at

Sz E r if there exists f E L,(X, r, p) the derivative of F at Q such that FM I= F(Q) + <f, xn - xo > + ,W AA) E&4 A ),

where lim jtw,u~-roW, A)=O.

We define the partial derivatives of n-set functions.

DEFINITION 2.2. Let F: r” + IR and (In,, . . . . Q,) E r”. Then F is said to

have partial derivative with respect to Ai if the set function

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OPTIMALITY OF n-SET FUNCTIONS 369

has derivative hoi at 52,. In this case we define the ith partial derivative of F at WI, . . . . Q,) to be fh ,,..., Rn = b,.

Using the partial derivative of n-set function, we can define the derivative of vector-valued n-set functions.

DEFINITION 2.3 [7]. Let Scf”, F=(F,, . . . . F,): S-+ R”, and

(Q r, . . . . Q,)E S. Then F is said to be differentiable at (Q,, . . . . Q,) if the partials f$, ,,,,, R,, i = 1, 2, . . . . n, of Fj exist for each j= 1, 2, . . . . m and satisfy

F(A 11 . . . . A,) =F(Q,r-,Q,)+

( i= i <fii ,,.... 1 n,~n,-XP,) i=l + W,((Q,, . . . . Q,), (A,, . . . . 4J),

for all (/i, , . . . . A,) E S, where

wF((Q,, . . . . Q”), (A,, ..., A”)) --*. 4(Q, 3 ...3 f&I), (A, > .*., A,))

as 4(Q,, . . . . Q,), (A 1, . . . . A,)) + 0. If F is differentiable at every point (Q,, . . . . Q,) of S, we say that F is differentiable on S.

Throughout the paper if F=(F,,...,F,): Scf" --) Rp G=(G1,...,G,):

S-, TV” and H= (H,, . . . . H,): S + R’ are differentiable at (Sz,, . . . . a,), we will denote f i’, gV, and h” the ith partial derivatives of Fj, G,, and Hi at (Q r, . . . . Q,) respectively.

For two vectors x = (x1, . . . . xP) and y = (y,, . . . . y,) in p-dimensional Euclidean space Rp, we introduce the following notations

(1) x < y iff xi < yi for all i = 1, 2, . . . . p.

(2) x < y iff xi < yi for all i = 1,2, . . . . p and x # y.

(3) x~yiffxi<yiforalli=1,2 ,..., p.

The nonnegative orthant and the nonpositive orthant in RP are denoted by

rw; = {xERP;x~o} and Rip = {xElRP;X~O},

respectively, where 0 is the zero vector (0, 0, . . . . 0) in RP. We also denote (x, y) =Cip,, x,y, as the inner product of x=(x,, . . . . xP) and y= (Y 1, ..*, yP) in Rp. For a set E c Rp, the set of all interior points of E will be denoted by int E and the set of closure of E in Rp will be denoted by i?.

DEFINITION 2.4. A set E c IRP is said to be [w P,-convex if E + R ", is a

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370 LAI-JIU LIN

DEFINITION 2.5. A point x* is a lower efficient point of E c RP if x* E E

and there is no XE E such that x <x*. We denote the set of all lower efficient points of E by g(E).

LEMMA 2.6 [ 143. Suppose that for a point x* E E E Rp, there exists a $~int lR$ such that (ji, x*) < (,L, x) for xgE. Then x*~g(E).

DEFINITION 2.7. A point x* E Rp is said to be a properly efficient point

ofEcRPifx*E_e(E)andE+RP,-x*nRP={O}.

DEFINITION 2.8. Given a p-dimensional vector-valued function f =

(fi, . . . . f,): X+ Rp, L.eLI(X, I’, p), i= 1, . . . . p, we say that f separates

Qgf if ((fly xn>, . . . . (f,, xn>) is a properly efficient point of the set

y= {((fi, x/i>, ...> <fp, x/l>); ‘4 ET).

It follows from [12, Proposition 3.2 and Lemma 3.33, for any (a, /i, A) E TX TX [0, 11, there exist sequences {Q,) and {A,> in r such that

Xn. + RXn,n and _{x/l, -5} _{(1 - 4xn\n} ₍₁₎

implies

XR.“n.“(Rnn+

AX.4

+ (1 -n)x,

where IV* stands for the w*-convergence. The sequence { V,(n) = 52, u A, u (Q n A)> satisfying (1) and (2) is called the Morris sequence associated with (Q, A, A).

DEFINITION 2.9. A subfamily S of r” is convex if given (Sz,, . . . . Sz,),

V r, . . . . A,)E S, and 1~ [0, 11, there exists a Morris sequence {V:(n)} in r associated with (Qi, ni, 1) for each i= 1, . . . . n such that ( VW. 11 ..*, V:(A)) E S for all k E N, where N is the set of natural numbers.

DEFINITION 2.10. A set function F = (F, , . . . . F,): S + Rp is called convex

on a convex subfamily S of r” if for each (Q,, . . . . Sz,) and (A,, . . . . A,) ES, il E [O, l] there exists a Morris sequence { Vf(l)} in r associated with (52,, ni, 2) for each i= 1, . . . . n such that (V:(A), . . . . V:(~))E S for all k~ N and

lim

k-m F( V:(A), . . . . V;(l))

= (k5m F,( V&l), . . . . Vi(A)), . . . . ki$m F,( V:(l), . . . . V;(A))) S 1F(A,, . . . . A,)+ (1 -A)F(Q,, . . . . 52,).

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OPTIMALITY OF n-SET FUNCTIONS 371 LEMMA 2.11 [15, Lemma 2.41. Let E be a IRT-convex set. Then yO~ E satisfies

[E+RP,-y,]nRP={O} ijjf there exists a vector p E int R “, such that

<I4 Yo) d (I4 Y> for any y E E.

LEMMA 2.12 (Liapunov [13]). Let fI, . . . . fPE L,(X, f, p), then the set {((f, , x,, ), . . . . (f,, x,, )), A E r} is convex and compact.

3. MAIN RESULTS

Throughout this paper, we will denote A = {(A r, . . . . A,) E P’, G(A I, . . . . A,)sO}, A’= {(A, ,..., A,)E~“, G(A, ,..., A,)<O}, and a=

w 1 ,..., A,)ES, G(Ar ,..., A,)sO, H(.4, ,..., A,)=O}. The following

lemma follows immediately from the definition of convex subfamily and properties of Morris sequence.

LEMMA 3.1. Let (52,) . . . . 12,) E F’ and G : r” + R” be convex, then for

each 6 > 0 the set

= {(A,, . . . . A,) E f”; d((A,, . . . . A,), (Q,, . . . . Q,,)) < 6, GM,, . . . . A,) < 0)

is a convex subfamily of r”.

Proof: Suppose (A,, . . . . A,), (fir, . . . . 8,) E B6((SZ1, . . . . a,)) and A E [0, 11. Then (A,, . . . . A,), (Sz,, . . . . hn)~I’“,

(aI, . . . . Q,))c6,

G(A,: . . . . A,)<O, G(&,, . . . . fi,)<O, 4(/l,, . . . . A,), d((fiil, . . . . a,), (a,, . . . . Q,))<4 and for each i = 1, 2, ,,., n, there exists a Morris sequence {V:(A)} in r associated with (d,, Ain) such that (V:(A), ,.., V:(A)) E I”’ for all k E IV. Since

lim

k-a dU v’r(4, . . . . W)), (52,, . . . . Q,)) = lim k-t~ i llx~,i,-x~tll~l}1’2

1 i= I =

( $, Ilki, + (1 - A)xfi, - x&i)1’2

=

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372 LAI-JIU LIN

GA i IlLi-X&l

₍

_>

w +(1-l) i IIXa,-X&l

( >

l/2

i=l i=l

=A i [p(niAQi)]2

₍

_>

112 + (1-A) f [p(BiAsZi)]’

i= 1 (

112

i=l >

= Ad((Al, . . . . A,), (Q,, . . . . Q,)) + (1 - 2) w%, .-., aA (J-21, . . . . Q,))

<16+(1-1)6=6.

Hence there exists a natural number M, such that

4(W), . . . . W)), (Q,, *-.,

Q,)) < 6

forall k>M,, (3) since G is convex,

iii-i-i G( V$l), . . . . V-;(i)) 5 AG(A,, . . . . A,) + (1 -1) G(Si,, . . . . si,) < 0.

k-w

Therefore, there exists a natural number M, such that

G( V;(A), . . . . V;(A)) < 0. ₍₄₎

Let M=max{M,, M,}, then from (3) and (4), we see that if k>M, 4(W), ee.3 ma), (Ql, .*-, Q,)) < 6 and G( V’;(A), . . . . V;(l)) < 0. Thus (l’!(A), . . . . I’;(A))E&((& . . . . a,)) for ka M. This shows that b((Q, 9 **., 0,)) is a convex subfamily of r”.

COROLLARY 3.2. Let (Q,, . . . . 52,) E F’ and G: F’ + IL!” be a convex set function, then the set A’ is a convex subfamily of r”.

ProoJ It is easy to see that A’= lJ,“= 1 B,((Q,, . . . . Q,)) where &n((Q,, .--9 Q”))

= {(A,,..., A,)Er”;d((A ,,..., A,),(Q, ,..., Q,))<mandG(A, ,..., A,)<O) and the corollary follows immediately from Lemma 3.1.

For any SC r, we denote 3 the w*-closure of xs = (x,,; A ES} in L,(X, r, ,u), then r= {f~ &,(A’, r, p); 0 <J< l} Cl, Corollary 3.61. For f~ F, we denote N(f) the family of all w*-neighborhood off in i=‘. Since i= is w*-compact and L,(X. r, p) is separable, i= is metrizable [l]. Therefore r x . . . x 7= (r)” is also metrizable.

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OPTIMALITY OF n-SET FUNCTIONS 373 LEMMA 3.3 [7]. Let F= (F,, . . . . F,): Y’+ Rp be differentiable and conuex on f”, then for all (/iI, . . . . A,), (Q,, . . . . Q2,) E r”

F(A 1, . . . . 4)

2 F(sZ,, . . . . Qrl)+ i (fi’&i-Xn,) Y...? .c, <fipJ/l-XQ,)

( i= I )

A set function F: S + iRp is said to be w*-continuous at (a,, .,., Sz,) E S, if for any sequence {(Szf, . . . . C$)> in S and for each i = 1, . . . . n, xnk --% x0,

as k -+ co implies F(Q,, . . . . 52,) = lim,, o. F(@, . . . . QE). F is said to be w*-continuous on S, if F is w*-continuous at each point (52,) . . . . 52,) E S.

LEMMA 3.4. Let S be a convex subfamily of r” and F: S -+ Rp be a w*-continuous and convex set function. Then the set F(S) is RP,-convex.

Proof The proof of this lemma is similar to Lemma 3.1 of [2]

LEMMA 3.5. Let F: r” + Rp be w*-continuous and G: r” --* W” be convex. Suppose that there exists (b,, . . . . 6,) E r” such that G@, , . . . . fi,) -C 0. Then F(A) = F(A’) and F(A) is RP,-convex.

Proof Since (si,, . . . . 8,) < 0, A’ is not empty. Let (Q,, . . . . Sz,) E A’ and

(A r, . . . . /i,) E A, for each i = 1, . . . . n and each positive integer m, let { Vk,, } be the Morris sequence in r such that

By the convexity of G,

lim

G(( V;,,, . . . . v;,,))$-+((f& ,..., Q,))+ G((A, ,..., A,))<O.

k-a:

Thus there exists a natural number M such that

G((v;,,, ..a, v;,,)) < 0 for k>M.

This shows that ( V!,,k, . . . . Vz,,) E A’ for k 2 M. Hence

(X b$, ***9 Xv+Xx= {(Xe,, ..‘, I!&), (B,, . . . . &J-q crx ... x r We note that (xn,, . . . . xnJ is a cluster point of {(xv; p, . . . . XC”, ,), m, k E IV}.

Since i= is metrizable in the w*-topology and l’x ‘. . . x r is metrizable, there exists a subsequence {(V:, . . . . V;)} of { ( V!,,k, . . . . Vi,,)} such that x5 asl-tm Iv* + x,,, for each i= 1, . . . . n. Since F is w*-continuous, we have

lim

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374 LAI-JIU LIN

This shows that F(A,, . . . . A,,) EF(A’). Therefore F(A) = F(A’) and the lemma follows immediately from Lemma 3.4 and Corollary 3.2.

We say (Ql, . . . . SZ,)EA (resp. (Q,, . . . . 0,) E A) is a Pareto optimal solution to problem (P) (resp. (Pl )) if

JIQ, 7 ..*, 52,) E g(a) (resp. F(Q2,, . . . . 9,) E g(A)).

DEFINITION 3.6. (a,, ..,, 52,) E A is said to be a proper IWP,-solution of

(Pl)ifF(A)+RP,--F(Q,,...,IR,)nIW~={O}.

LEMMA 3.7. Suppose that S is a nonempty subfamily of r”, F=

VI, . . . . Fr): S+ Rp, and F(S) is RP, -convex. Then (Q,, . . . . Q,)E S is a proper RP,-solution of (Pl) if and only if (Q,, . . . . Sz,) is optimalfor (Pl(2))

for some ,I E int R:, where

min i &FJA,, . . . . A,)

i=l _W(A))

subject to (A,, . . . . A,) E S.

Proof The proof of this lemma is the same as Theorem 3.1 of [2].

THEOREM 3.8. Let F= (F,, . . . . F,): r” + Rp be w*-continuous and

convex and G: r”+R” be convex. Suppose that there exists

@ 1, . . . . fi,)~r” such that G(fi,, . . . . fi,) < 0. Then (Q,, . . . . 52,) E A is a proper RP,-solution if and only zf (Q,, . . . . a,) is optimal for (MPl(A)) for

some 1= (A,, . . . . 1,) E int R “, where

min i &Fi(A,, . . . . A,)

i=l _WPl(l))

subject to (A,, . . . . A,) E A.

Proof: This theorem follows immediately from Lemmas 3.5 and 3.7.

DEFINITION 3.9. A point (52,) . . . . 0,) E r” is said to be a local minimum

to problem (Pl) if there exists 6 > 0 such that F(Q,, . . . . 52,) 5 F(A,, . . . . A,)

for all (A, ,..., A,)E~“, G(A, ,,,., A,)50 satisfying d[(A, ,..., A,,),

(0 19 . . . . QJI < 6.

LEMMA 3.10 [3, Corollary 3.91. In problem (Pl) if F: P+ Iw' and

G= (G,, . . . . G,): r” + R” are differentiable at (O,, . . . . a,,) EY. Suppose

that (Q,, . . . . (fi

Q,) is a local minimum to problem (Pl ) and that there exists , , . . . . 6,) E r” such that

G#, , . . . . Qn)+ 2 <gs I...., sj,Jrj,-xXn,)4 j= 1, . . . . m. (5)

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Then there exist A,, . . . . A,, such that

(

fi+ f Ajg’j,Xn,-Xo, 3O ,=l > (6) .for all Ai E Z, i = 1, . . . . n. /ljGj(Ol, . . . . s2,) = 0 2 1, . . . . I, 2 0 Gj(Q,, . . . . s2,) < 0, j = 1, . . . . m,

where & ,,..., on is the ith partial derivative of G, at (6,, . . . . a,,).

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THEOREM 3.11 (Necessary and Sufftcient Conditions for Constrained

Local Minimum). In problem (Pl), if F= (FI, . . . . F,): Z” + Rp and

G= (G,, . . . . G,): Z” -+ 58”’ are convex on Z” and differentiable at

(Q 1, . . . . Q,)E Z”, suppose that (Q,, . . . . S2,) is a proper RP,-solution of

problem (Pl). Suppose further that there exist (a,, . . . . d,) and

(B, , . . . . B,) E Z” such that

G(Q,,

. . . . QJ+ i <di ,,...,

ri.3

Xri,-Xn,)?

. ..Y

$, <g$

,.__)

si,m,-Xn,)

i=l >

-co

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and

G(B 1, . . . . B,) < 0. (11)

Then there exist A= (A,, . . . . I.,) E int WC, u = (u,, . . . . uL,) E RT such that

( f, Ajfq+ j=l f ujgg”, a,,-~~,) 20, i= 1, . . . . m, n,u, (12)

PjGj(Qt > ...y Qn) = 0, j = 1, . . . . m (13)

Gj(Ql, ..-T Q,)<O, j = 1, . . . . m. ₍₁₄₎

Conversely, if there exist A= (A,, . . . . A,) E int IJ!:, u = (ul, . . . . y,) E RT, and

(B I, .a*, B,)Er” such that (ll), (12), (13), and (14) hold, then (Q ,,.,., Q,)

is a proper IF! P,-solution of problem (Pl ).

Proof Since (a,, . . . . Q,) is a proper R P,-solution of problem (Pl ), it

follows from Theorem 3.8 that there exists A= (A,, . . . . I,,) E int W: such that (4 W,, . . . . 4) 2 (A FW,, . . . . a,)>

for all (A,, . . . . AJEA. Then by Lemma 3.10, there exists p= (pr, . . . . F,)E lRy such that (12), (13), (14) are true.

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376 LAI-JIU LIN

Conversely, if there exist I = (I,, . . . . A,) lint rW$, p = (p,, . . . . &,) E WQ

and (B,, . . . . B,)E~” such that (ll), (12), (13), and (14) hold. Since F and

G are differentiable and convex on r”, it follows from Lemma 3.3 that for all (A I, . . . . A,) E r” Ft’(n , , . . . . 4) - FWJ,, . . . . Q,) 2 ,cl <filJn,-Xn,L .**9 i (f’,xn.-xd) (. (15) i=l G(A 1, . . . . A) - G(Q, , . . . . f&z) L ( is Wh,-Xn,) 9***3 i mXn,-X*,)) (16) i= 1

Since IEint KIT, pEE[Wm+, it follows from (15), 16), and (12) that <A FM,, ..a, 4) -W-2,, A.., 52,)) + <pL, GUI, . . . . A,)- G(Q,, . . . . f&z))

2 A ig, (fi’dn,-XXn,) Y...Y ic, <f”&,-x*,))) ((

+ PL, i: WXn,-h,) 7*..3 ( ( i= 1

i$ ( gim, Xn, - XQ,)))

= i ( i Ajfj+ f /ijLig’, Xn,-xfi;) 20.

i=l j=l j=l

As p E Rm,, /+GJQ,, . . . . Q,) = 0, j= 1, . . . . m, we have (4 F(A,, . . . . A,,)-f’@2,, . . . . Q,)>

2 (A., F(A,, . . . . A,)-J-W,, . . . . Q,,)) + <pL, W,, . . . . 4)-G(Q,, . . . . Q,)) 2 0.

For any (/II, . . . . A,) E A, (Q,, . . . . a,) is a proper RP,-solution follows immediately from Theorem 3.8.

Remark 3.12. In Theorem 3.11, if the condition

(

5 Ajf”+ 2 jijggij,Xn,-Xa, 20

j=l j= 1 >

for all i = 1, . . . . n and all ,4 i E r is replaced by

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As a consequence of Theorem 3.11, we have the following two theorems.

THEOREM 3.13. In problem (PI ), let F= (F1, . . . . F,): r” -+ 53 p, and

G = (G,, . . . . G,): r” -+ ET” be convex on r” and differentiable at (a,, ..,, Q,) E r”. Suppose that (52,, .,., Q,) is a proper RP,-solution of problem (Pl). Suppose further that there exist (fi, , . . . . d,) and (B 1, . . . . B,) E r” such that

G(Q,>-,a,)+

( i= i: <g: ,,.... I h~xn.-x&-,i~, Cd’: ,..., ti,m,-xn,) 1 -co

and G(B,, . . . . B,) < 0, then there exists p = (pL1, . . . . pL,) E 0;s: such that for each i= 1, . . . . n,

(

f"+ f /ijg' ) . ..) fi" + f pjgo separates Qi, ₍₁₇₎

j= 1 j=l

PjGj(Qn, ...y Qn)=o, j= 1, . . . . m (18)

Gj(Q,, ...y Q,) < 0, j = 1, . . . . m, (19)

where di,..., d” denotes ith partial derivative of G, at (B,, ..,, 5,).

ProojI It follows from Theorem 3.11, there exist 1= (A,, . . . . 1,) E int[WP,, p=(pi,...,p,)~ EW”, such that (12), (13), and (14) are true. Without loss of generality, we may assume that I,!‘=, Aj= 1. In view of (12), we have for each i = 1,2, . . . . n,

(4 ( (f"? Xn, - Xn, >, . ..Y <fiP, XA,-XC?,>)> + f Pjg', XA,-XC?, a0

j=l >

for all n+r. (20)

Since C:=, Ai = 1, it follows from (20) that (~,((fjl,Xn,-Xn,),..., (fi”,Xn,-Xn,>)

Therefore for each i = 1, . . . . n and (/ii, . . . . /i,) E r”,

( (( 2, f i‘ + f pjgg”, XA, 3 ...? f ip + jJ Pj g”, X.4, j= 1 > ( j=l >)i 2 IL, ( Cl fi’+ t I*ig",XQ, )...) fi”+ g /Jjg’,XQ, . (21) j=l ! ( .j= I >)>

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378 LAI-JIU LIN

Then by Lemma 2.6 and (21),

where

Since Yi is convex by Liapunov’s Lemma (Lemma 2.12), we get by (21) and Lemma 2.11 that

((

f j1 + ~ ,Uj gi’, Xn, ) . ..) f jp + ~ CLj g”, Xn;

j=l > ( j=l >I

is a properly efficient point of Yi for each i= 1, . . . . n. This shows that for each i= 1, . . . . n

(

fil+ 5 Sgv,...,f”+ f Ajg”

> separates Qi

j=l j=l

and the proof of the theorem is completed.

The following theorem gives the sufficient conditions for the existence of the proper R “, -solution.

THEOREM 3.14. In problem (Pl) if F and G are differentiable and convex

on r”. Suppose that there exist (B,, . . . . B,) E Z”, A= (A,, . . . . A,) E int RT,

P = (Pl, ..-, P,) E wy such that (ll), (17), (18), and (19) hold, then

VJ 1, ..,, 52,) E r” is a proper IRP, -solution of problem (Pl ).

Proof: By Lemmas 2.11 and 2.12, there exists A = (A,, . . . . A,) E int W$

such that for each i = 1, . . . . n and for all Ai E Z,

( cc

;1, fil+ 2 /l.jjg', Xn, 3 ...T

f”+ f PjLi'3 XAi

j=l > ( j=l >I>

> 1,

( ((

f j1 + f /ijg”, XQ, 9 ee.9 f jp + f Pj 99 Xi?,

j=l > ( j=l >>>

or

j1 + f jljuig”, XA, - XQ, + f Pj L?‘v XA~- XQ, 20 (22)

(13)

OPTIMALITY OF n-SET FUNCTIONS

for all i = 1, . . . . n and ni E r. Without loss of generality, that Cip_, Aj = 1. From (22) and Cp=, 31, = 1, we see that

379

we may assume

for all (A,, . . . . A,) E P. By Theorem 3.11 and Remark 3.12, we complete the proof of the theorem.

DEFINITION 3.15. A set function F: S + [w is called quasiconvex on a convex subfamily S of P if for each (In,, . . . . Sz,), (,4,, . . . . A,,) in S, 2 E [0, 11, there exists a Morris sequence {V;(n)> in r associated with (Q,, /li, 1) for each i = 1, . . . . n such that (V:(n), . . . . V:(n)) E S for all k E N

and _T-

hm

k-a. F( V’;(A), . . . . V:(i)) <max{F(Q,, . . . . Q,), F(A,, . . . . A,)}.

DEFINITION 3.16. A set function F= (F,, . . . . Fp): S+ [wp is called quasiconvex on a convex subfamily S of P, if for each i = 1, . . . . p, F, is quasiconvex on S.

Remark. It is easy to see that if a set function is convex, then it is

quasiconvex, but the converse is not true, in [S], we give an example of a quasiconvex set function which is not convex.

LEMMA 3.17. Let S be a nonempty convex subfamily of r” and F=

F ): S -+ Iwp be differentiable and quasiconvex on S. Zf for any

!2:,‘:::, sz”,,), (A,, . . . . A,)ES with F(A,, . . . . .4,)sF(SZ,, . . . . 9,) then

( ,!, (f i’Y X/l,-Xn,) >...? f i=l <fiPJn,-In,) ) 50.

Proof. Since F is quasiconvex on S, it follows that F, is quasiconvex on

S for each j= 1, . . . . n. Let 1 E (0, l), then there exists a Morris sequence {V;(A)} in r associated with (Q,, ni, I) for each i= 1, . . . . n such that (V;(n), . . . . V;(n)) E S for all k E N and

lim

k-cc Fj( V:(l), . . . . I’$)) < Fj(s2,, . . . . QJ.

Since F is differentiable at ($2,) . . . . Q,,) E S, it follows that Fj(vf/:(l), -.T vE(J-1)

=Fj(Ql, ...y Qn)+ i (f”, x~(A)-xo,)

i=l

(14)

380

where

LAI-JIU LIN

E(( W), . . . . q(m (521, . . . . f-2")) -+ 0

as d((Vf(l), . . . . Vf(l)), (Q,, . . . . Q,)) --) 0.

In theorem 3 of [7], we show that

G-i

k-cc

dW’#), . . . . V:(A)), (Q,, . . . . ~2,))

. E(( W), a.-, V34), (Ql, . . . . 52,)) E o(A).

Hence

lim zy q(n), . ..) v;(n))

k+ao = Fj(Q,, . . . . Q,) + A i cf-‘? X/l, - x0,> + o(A) i=l < Fj(s2,, . . . . a,). That is,

tJ i <fVJn,-xn,>+44a,

for all j= 1, . . . . p. i=l

Dividing both sides of the above inequality by A and letting A --) 0, we have

It follows that

(

if <filJA,-XR,LY

i (fip~xA.-x*+o.

i=l

The following theorem gives sufhcient conditions for existence of a Pareto optimal solution to problem (P) with convex objective function and non-convex constrained functions.

THEOREM 3.18. In problem (P), if S is a convex subfamily of r” and (Q 1, ..a, 52,) ES. Suppose that

(i) F, G, and H are dlyferentiable at (Q,, .,,, a-,,).

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(iii) G,= (G,, , . . . . G,,) and H = (H,, . . . . H,) are quasiconvex on S, where I= {i; Gi(Q,, . . . . a,) =0} = {sl, . . . . sj}.

(iv) There exists u E int RT, v, E If@+, w E Rl, such that

+

(

w, i: (hi’, xn, - xn,>, . . . . i (h”, I,,, - xo,)

(

;= I

i= 1

2 0.

(v)

G(Q,, . . . . 8,) 5 0.

(vi) H(SZ,, . . . . sZ,)=O.

Then (52 1, . . . . 62,) is a Pareto optimal solution to problem (P),

Proof. Suppose that (a,, . . . . 0,) is not a Pareto optimal solution to

problem (P). Then there exists (A,, . . . . A,)E r” such that F(A 1, . . . . A,) - WI,, . . . . Q,) < 0,

W I, . . . . AJso,

H(A,, . . . . A,) = 0.

Hence

G&f,, . . . . 4) 5 G,(Q1, . . . . Q,,) = 0, WA 1, . . . . A,) = H(Q,, . . . . Q,) = 0.

By the convexity of F and quasiconvexity of G, and H, Lemmas 3.3 and 3.18, we have

jl, <.I-“~

Xn,

- X62,>>

. ..Y

ic, (f"wn*,))

5 F(A 1, . . . A,) -f-W,, . . . . Q,) < 0, (23)

(24)

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382 LAI-JIU LIN

Since u > 0, u 2 0, w 2 0, it follows from (23), (24), (25) that we have

This inequality contradicts hypothesis (iv). Hence (52,) . . . . 0,) is a Pareto optimal solution to problem (P).

DEFINITION 3.19. Let S be a nonempty subfamily of r” and let

F= (F,, . . . . F,): S + Rp be differentiable on S. The set function F is said

to be pseudoconvex on S if for each (Q,, ..,, 52,) and (/ii, . . . . A,) in S, with

( j$l We X/l, - xn,>3 ee.9 ig, uip, X/i, - XQ,)) L 0 we have

HA, , . . . . 4 4 F(Q,, . . . . 0,).

Remark. It follows from Lemma 3.3 that if F is a convex set function,

then it is pseudoconvex, but the converse is not true. In [8], we give an example to show that a pseudoconvex set function is not convex.

THEOREM 3.20. In problem (P), suppose that

(i) F, GI, and H are differentiable at (52,) . . . . Sz,) E S c P’, where

I= (i; G,(SZ,, . . . . 0,) = 0} = (So, . . . . sj}.

(ii) There exist ueint !J!c, VEIWC, and WEIR; such that

( (. u, ,g, W’,Xn,-X*,)3 .-*9 f <P?x”i-Xni))) i=l

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(iii) G(SZ,, . . . . Q,) 5 0. (iv) H(S2,, . . . . 52,) = 0.

(v) Cf’=, u,F, + Cic, viGi + c>= 1 wjHj is pseudoconuex on S.

Then (Sz,, . . . . s2,) is a Pareto optimal solution to problem (P).

Proof. Assume that (52,) . . . . 52,) is not a Pareto optimal solution

to problem (P), then there exists (A,, . . . . A,) E A, G(n , , . . . . /1,) s 0, H(/f 1, . . . . A,) = 0 such that

FM ,r . . . . A,) d F(Q,, . . . . Q,). By (i), (iv), we see that

G,(Al, . . . . A,,) 6 0 = G&2,, . . . . i-2,) and

H(A 1, . . . . A,) = H&2,, . ..) s2,) = 0. Since u E int R “, , u E Ri, , it follows that

<u, FM,, . . . . 4)) + (0, GM,, . . . . 4)) + (w, WA,, . . . . 4)

< <u, F(Q,, . . . . Q,)) + (0, G,(fJ2,, . . . . Q,J> + (w, H(Q,, . . . . Q,)>. By assumption, Xi”= r u,F, + Cic, uiGi + C>=, wjHj is pseudoconvex (52 1, . . . . Q,), we have at

+ w, ( i <hi’,

xn,- ~a,),

. . . . n

i=l ic, wh,-h,) ’ ~0 (27) 1,

for all (A,, . . . . LI,)E~. But (27) contradicts hypothesis (ii). Hence (Q r, . . . . a,) is a Pareto optimal solution to problem (P).

THEOREM 3.21. In problem (P), suppose that S is a convex subfamily of

r”, w I, .**, Q,)ES, and

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384 LAI-JIU LIN

(ii) There exist uEint IL!:, v~[Wm+, WER’, such that

P

(a) 1 u,F, ispseudoconvex on S,

i= 1 (b) c viGi is quasiconvex on S, iel (c) i wiHi is quasiconvex on S, i=l (d) ( (

UP ig, uil~Xn,-Xo,)~ **.> i, wp~xA,-xQi)))

+ v, i (g”, Xn,-Xni), ..* u i= I

7 ;c, (g”, XKX,.,))

+ w, ( (

i (hi’, xn, -xn,>, . . . . i (hi’, xni- xni> 20

i= 1 i= I >>

for all (A,, . . . . A,)EA.

(iii) (v, G(Q,, . . . . Q,)) =O. (iv) G(Q,, . . . . 52,) 5 0.

(v) H(Q,, . . . . J-2,)=0.

Then (Q,, . . . . 52,) is a Pareto optimal solution to problem (P).

Proof: Since (v, G(B,, . . . . Q,)) =O, G(Q1, . . . . Q,)sO, ~20, it follows

that

viGi(12,, . . . . 0,) = 0 for all i.

Therefore

c v,G,(!~~, . . . . Q,) = 0.

isI

For any (/i 1, . . . . A,,) E S with G(/i 1, . . . . A,) 5 0, we have

1 viGi(-4,, ---y A,) < C viGi(Q,, **v Q,).

isI iel

Since ~jp,viGi is quasiconvex on S, it follows that

( VI7 .g, <b+‘~Xn,-Xa,)9 *..3 (. $, (E?J? X/i, - XQ,))) G 0 (28) for (AI, . . . . &)~a. As vj=O for each Jo (1, . . . . m}\Z=(t,, . . . . tr), we have

(19)

for all (A,, . . . . A,) E A. Similarly ZlJ,I wjHj(A19 ...3 An) =

CJ= 1 wjHj(Q19 ...9 O,)=O for all (Al, . . . . A,)E A. As cl=1 wiH, is

quasiconvex on S, we have

( w, i (h”,xn,-xn,),..., ( i=l i=l i W’,x,,,-xn,) >> GO (30) for all (A,, . . . . A,)E~. By (ii)(d), (28) (29), and (30) we have

for all (A r, . . . . A,) E a. Since Cf= r uiFi is assumed to be pseudoconvex on S, we have

(u, F(A,, -.., A,)> a (4 F(Q,, ..., QJ)

for all (A,, . . . . A,)E~. For uEintK!P,, it follows from Lemma 2.6 that (Q , , . . . . Q,) is a Pareto optimal solution to problem (P).

LEMMA 3.22 [S]. In problem (Pl), (Q,, . . ..a.,) is a Pareto optimal

solution tf and only if (Q,, . . . . Q,) minimize each F, on the constraint set

c,= ((4, . ..> A,) E r”, Fi(A 1, . . . . A-1 G Fi(Ql, ...) Qn),

if jandG(A,, . . . . A,)<O}. (31)

The following theorem establishes necessary conditions for a Pareto optimal solution of problem (Pl ) when the set functions are differentiable.

THEOREM 3.23. Let the set functions F= (F,, . . . . F,): Z” + Rp and G = (G ,, . . . . G,): Z” + R” be differentiable on Z”. Suppose that (64,) . . . . 52,) is a

Pareto optimal solution of (PI) and for each s= 1, . . . . p there exist

(tis,) . ..) @,) E Z” such that G(Q2,, . . . . QrJ+

( i=l i <g& ,,._, o”,Js:-X52,) ,...? ,cl c&j ,..., ri;Jb:-Xa,) > -co

and for each j = 1, . . . . p, j # s

(

jl, (f” q,...,&“T xc2: - Xn, > > < 09

then there exist v = (a,, . . . . vp) E int Rc, c,“= , v, = 1, A = (A,, . . . . A,) E rWy

such that

i (5 vjfb, ,_._, n.+ 2 12igb, ,..,, nn3XA,-X*,)20 (32)

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386 LAI-JIU LIN

for all (A,, . . . . A,) E r”

f v,G,(sZ 1) . . . . .a,) = 0

j=l

Gj(Q, 3 .**) 52,) 2 0, j= 1, . . . . m,

where f ji, ,,..., A.3 g! ,,..., ,,, are the ith partial derivatives of Fj and Gj at

(A I, . . . . A,), respectively.

Proof: Since (a,, . . . . Q,) is a Pareto optimal solution of (Pl ), it follows

from Lemma 3.23 that (a,, . . . . 52,) minimizes each Fj on the constraint set Cj of (31). Then by Lemma 3.9 for each i = 1, . . . . n, j = 1, . . . . p, there exist

Blj, -*9 Pmj, Y lj3 .*.9 Yj- 1, j9 Yj+ 1, j9 .-3 ypj such that

( f& ,.._, a,+ k=l IE BIG&j ,,,,.__, *.+ k=l 2 Y&f;, ,..., R,~XA,-X0; > >O (33) k#j

for all Ai E I’

kgl BkjG/c(Q1) ...p Qn) ~0,

Gk(G1, . . . . 52,) < 0, k = 1, . . . . m.

Letting j= 1, . . . . p in (33) and then summing up, we obtain

((l+~~Y,j)f~,,...,~.+ **. +(l+~~:Y,)f~,,...,~”

+ i f bkj<8$~,...,fZn~ XA,-~Xni) 2’

j=l k=l >

for all A,ET.

Letting

Ps = l+ E Ysj,

A =c/=lbkj j=l ‘j=&.p k Cj”= 1 Pj ’ i#s

then c,?= 1 vi= 1, v = (vl, . . . . v,)~int Rc, A = (A,, . . . . 2,)~ R’J and for all

i=l , .-*, n

j$, vjfb, ,..., i2n+j~,n,Pi2, ,__,, O.~XAi-XCJ, 2o

> fora AiEr.

5 ljGj(12 ,,...,Qn)= ? i A-

j=l j=, i=I~P-l~jGj(n,,...,a,)

=& jt ,t BjiGj(Q1, .-y Qn)=O*

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Hence

f ( i vjfbl,...,*,+ i Ljgi,,....f2,~

XA,-*Cd.) 2 O

i=l j= 1 j=l

for all (A,, . . . . /i,) E P. We complete the proof of the theorem.

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