Journal of Mathematical Analysis and Applications 247, 377-383 (2000) doi:10.1006/jmaa.2000.6814, available online at http:/7www.idealibrary.com on ID E J^ L Arcwise Connectedness of Closed Efficient Point Sets E. K. Makarov 1 Institute of Mathematics, National Academy of Sciences of Belarus, Surganova 11, Minsk 220072, Belarus N. N. Rachkovski Belorussian State Pedagogical University, Sovyetskaya, 18, Minsk 220809, Belarus and Wen Song Harbin Normal University, Harbin 150080, People's Republic of China, and Systems Research Institute, Newelska 6, 01-447 Warsaw, Poland Submitted by George Leitmann Received February 16, 1999 1. I N T R O D U C T I O N One of the most important problems of vector optimization is to investigate the structure of efficient point sets. For various applications, the possibility of continuous moving from one optimal solution to any other along optimal alternatives only is of special interest. This possibility is guaranteed if the efficient set is arcwise connected or at least connected. It is well known that in a finite dimensional space the efficient point set of a closed convex cone-compact set with respect to an order defined by a closed convex cone is connected, as well as the efficient solution set for concave objective functions (see [1, 12, 13]). Several authors proved analogous theorems for some subclasses of quasiconcave objective functions (see [3, 7, 9, 14, 15, 19, 20]). Many of the above results were also 1 E-mail: [email protected]. 377 0022-247X/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved. 378 MAKAROV, RACHKOVSKI, AND SONG generalized to the infinite dimensional space settings by Luc [10, 11], Fu [4], Gong [5, 6], and Song [16-18]. However, the results concerning strong topological properties, e.g., contractibility or arcwise connectivity, are rather seldom. Contractibility of the efficient point set and arcwise connectivity of the weak efficient point set are studied in [11] under some stronger assumptions, and contractibility of the efficient solution set for a strongly quasiconcave function (called strictly quasiconcave function in [9, 10]) is studied in [9, 10]. In this paper we prove that the efficient point set Max((2l^O of a compact convex set Q с X in a Hausdorff topological vector space X ordered by a closed convex pointed cone К с X is arcwise connected if the set Мах(<21Ю is closed. 2. PRELIMINARIES Let X be a Hausdorff topological vector space, Q с X, and К с X a closed convex pointed cone. We denote the interior, the closure, and the convex hull of an arbitrary set А с X by int A, cl A, and convЛ, respectively. A point x e Q is said to be an efficient point of Q with respect to К (д: e М а х ( £ > Ю if (Q ~ x) П К = {0}, and it is said to be the least element of Q with respect to К if Q с x + K. Let F be a set-valued mapping from a topological space W to X. By dom F and graph F we denote the domain and graph of F, i.e., d o m F = {w e W\F(w) Ф 0} and graph F = { ( w , x ) e W X X\x e F ( w ) } . Let P be a nonempty subset of X , х1гх2 e Q, and 0 ^ т1 < т 2 < 1. Consider the set-valued map F: [тит2] Q given by F(t) = (((r 2 — 0 / ( т 2 - г,))*, + (U - T i ) / ( r 2 - т 1 ))х 2 + P ) n Q . LEMMA Suppose 2.1. that Let P the set-valued t, - be a map convex cone t - r, *! + zx+P hi "~ Ti Ti h ~ Ti G(/) Then G(t) t = с Fit) and G: [тх,т2] for all t e [т„ т2]. tl E ( r 1 , r 2 ) , Q is given by n g , i f t e [ r zt e \ u t , ] ; F(tJ). 379 CONNECTEDNESS OF EFFICIENT POINT SETS Proof. Since f, e [T,, T2], there exists a e [0,1] such that tx = a r , + (1 - a)r2. Since z1 e. F O j ) , we have z1 e ( a x j + (1 - a)x2 + P) П Q. Hence, for every t e [т^*,], we have t - т t i - t + G(t) ~(ax1 h t1 - t + + (1 - a)x2 + P) + P n Q - a(t Ti) + (1 - a) ' -л:, + P h ~ т, r, - t t - + T, - 7, T, + P T, ~ T, n Q = F ( t ) . For t G [<,, R2], the inclusion G ( 0 С F(t) can be proved similarly. LEMMA 2 . 2 . closed If P and Q are closed sets, then F is closed, | i.e., graph F is a set. Define a continuous mapping p: [TJ, T 2 ] X X -* О? X X by - (((T2 - t ) / ( r 2 - тj))дсJ + ((t - Xj)/(T 2 - r , ) ) ^ ) ) . Then graph F = р~\[тх, T 2 ] X P) П ([т„ т 2 ] X (2). Since the sets [т„ т 2 ] X P and [т1,т2] X Q are closed, graph F is closed. | Proof. p(t,y) = U,y LEMMA 2 . 3 . and xk Then e (Зл the sequence Let Qx Э Q2 Э . . . FEE A sequence /^e least element {xk)k e N of Qk converges <~){Qk: к <e N) Ф 0 w i t h respect to with respect to the unique of compact subsets to K, for each least element of X , к e.N. of Q0 •= K. Proof. Since X is Hausdorff and each Qk is compact, {Qk}k e N is a decreasing sequence of nonempty closed sets in compact space Q v By [2, Proposition 1-11.4], we have Q0 Ф 0 and by [2, Proposition 1-11.5], U*}* 6 N has an adherent point л:0 e Q0. Suppose that there exists z e Q0 such that z <£ x0 + K. Since Jt0 + К is closed, there exists a neighbourhood U of zero in X such that ( z - U) П (*„ + K) = 0 . Hence, z £ x0 + U + K. Take к e N such that xk e * 0 + U. Then xk + К d x 0 + U + К and, therefore, z<£xk + K. On the other hand, we have z e Q0 с Qk с xk + K. By this contradiction, we assert that <20 <zx0 + K. Hence, x0 is the least element of Q0. Since К is pointed, the set Q0 has the unique least element x 0 . Therefore, is the unique adherent point of { x k ) k ^ N . By [2, Proposition 1-11.5], we have x k -* дс0. | 3. MAIN R E S U L T Recall that a set A of a topological space is said to be arcwise connected if for every two points x , y e A there exists a continuous function ф: [0,1 ] - > A such that ф(0) = ф( 1) = у. 380 MAKAROV, RACHKOVSKI, AND SONG THEOREM 3 . 1 . compact Let К <zX be a closed convex pointed cone and Q and convex. If the set М а х ( ( 2 1 Ю cXbe is closed, then it is arcwise connected. Proof. Take any xv x2 e Мах(<2|Ю. The set-valued map F0 : [0,1] Q given by F 0 ( / ) = ((1 - t)xx + tx2 + К) П Q has compact nonempty values for each t e [0,1]. By [8, Theorem 6.3 and Lemma 6.2], we have 0 Ф M a x ( F 0 ( f ) | / O с Мах(<21Ю for all t e [0,1]. For each txm = т/2, m = 0,1,2, we choose an arbitrary zlm e M a x ( F 0 ( O | / O . Since F 0 (/^) = {*,} and F0(t\) = {x2}, we have z\ = xx and z\ = x2. Now we can define the set-valued map Ft :[0, \] Q by setting FXU) = ((1 - 2t)z{0 + 2tz\ + K) П Q for t e [ 0 , 1 / 2 ] and F^t) = ((2 - 2t)z\ + (21 - 1 )z\ + K) n Q for t e [1/2,1]. By Lemma 2.1, we have F 0 ( f ) э F,(/) for all / e [0,1]. It can be easily seen that F,(f) is compact and nonempty for each t e [0,1], and 0 Ф MaxCFjWItf) с Мах((21Ю. Let us now iterate this process. Suppose that for some e N we have already defined the set-valued mappings Fi: [0,1] Q, i = 0 , 1 , . . . , к by putting f m = № for t e + i - ' K + 2 V - CK+1 + k] n Q + where t'm = 2~'m, m = 0 , 1 , . . . , 2 ' . We also suppose that F0(t) z> F j ( 0 э ••• D F t ( ( ) for all r e [0,1] and F^J = {z'J с M a x ( Q | / 0 for m = 0 , 1 , . . . , 2', i = 0 , 1 , . . . , k. It can be easily seen that Ft(t) is compact and nonempty for each ( £ [ 0 , 1 ] , i = 0,...,k, and 0 Ф Max(F)(OI^) с Мах(б|Л:). Thus, for each tk + 1 = 2~k~1m, m e {0, l,...,2k+1), we can 1 choose an arbitrary e Max(F A (r* + Ol^O с Мах((2|/0. Note that +1 k k zf„ = z , n = 0,1,..., 2 , by assumption. Now we can define the set-valued mapping Fk+l :[0,1] Q by putting W ) [2k+\tkm\\-t)zkm^+2k+\t-t^)zkm\\+K]nQ = forte[C\C+\], where m = 0 , 1 , . . . , 2 * + 1 — 1. Since 2 * + 1 ( C i ~ '> = 0 * V i " 0 Л С Л by Lemma 2.1, we obtain Fk(t) з Fk+l(t) for all t e [0,1]. It can be easily seen that Fk+l(t) is compact and nonempty for each t e [0,1] and 0 Ф Max(Fk+{t)\K) с M a x ( Q | / 0 . Obviously, Fk + l(t^+l) = {z* + 1 } с M a x ( e | / 0 for m = 0 , 1 , . . . , 2 * + Define the set-valued mapping F : [0,1] ~> Q by F ( 0 = N{F T (():TEN). CONNECTEDNESS OF EFFICIENT POINT SETS 381 Note that for every t E [0,1] and к G N, Fk(t) is compact and nonempty. Moreover, we have Fk(t) D F h 1 ( / ) . Thus, we can assert [2, Proposition 1-11.4] that Fit) = n(Fk(t):k G M) Ф 0 for all t G [0,1] and, therefore, dom F = [0,1]. Let T = U { 2 ~ k m : m = 0 , 1 , . . .,2k, к G N} с [0,1]. Obviously, cl T = [0,1]. If t e T, then t = t* for some m = 0 , 1 , . . . , 2*, к G N and, hence, F0(t) D F , 0 ) D - D F t ( f ) = F t + 1 ( / ) = - = F ( 0 = {z k J С Мах(б1Ю, i.e., ( f £ , z £ ) G graph F . Now suppose that ( E [ 0 , 1 ] \ Г . Then for each к G N we can find m{k) E { 0 , 1 , . . . , 2 * - 1} such that t G [t^k),tkm{k)+xl Clearly, tkm(k) - t and tm(k)+\ / as A: -> + oo. By the definition of Fk, we have Fk(t) = ( 2 * 0 * ( t ) + 1 - f ) z * w + 2*(f - J* ( *,)2r* ( * )+ 1 + К) n Q. Let fik = 2 * a * w + 1 - 0 e [0,1] and * t = + (1 - /3k)zk(k)+1. Then F , ( / ) = (л^ + Ю П (3. Since JC^ is the least element of Fk(t) with respect to К for each к G N, by Lemma 2.3, we obtain x k -> x 0 G n f F ^ O ) : i e N) = F ( f ) # 0 and F ( 0 + t By compactness, there exist some subnets {/3 t(M) } e M of {fi k } k e N c o n " verging to p 0 G [0,1], of ( z * w } t e N converging to о e g , and (^mdtuM+ileM o f { z £ ( * ) + 1 } t e N converging to w e Q. Let us denote «м : = A ( M ) , == and == z « ( # M ) ) + 1 , м e M. Hence, * t ( / x ) = « Л + (1 - «M)wM converges to x 0 = (3Qv + (1 - /30)w G g . By Lemma 2.2, for every к G N, graph F t is closed. Hence, graph F = П (graph Fk: к <e N] is closed. Since O * ^ » , v j , „ e graph F for any ц е М , we have (/, и) G graph F and (f, w) G graph F. Hence, f , w G x 0 + AT. Since x0 = /30v + (1 — /30)w and is pointed, we assert that X 0 G [v,w). Note that {v,w} G с1Мах(<21Ю = Мах(<2|Ю. Therefore, {ЛГ0} = Q П (x0 + K) D Q П F ( 0 = F(t). Thus, F is a single-valued (point-valued) map. By Lemma 2.2, graph F is closed. Since graph F с [0,1] X Q and Q is compact, we assert that graph F is compact as a closed subset of a compact set. Thus, F is continuous. The proof is completed. | The assumption of convexity on Q in Theorem 3.1 is essential. EXAMPLE 3.1. Suppose that X = IR2, Q = {U,Y) G [R2U = 0,Y G [0,1]} U {(x,y) G R2|JC G [0,1], у = 0} and К = U2+ == {(*,Y) G [R 2 |* > 0, у > 0]. Clearly, Q is compact and not convex and Max(g|AT) = {(1,0), (0,1)} is closed, but not arcwise connected. Let now I be a locally convex space. Then we can use the following sufficient conditions for the set Max(<2|/0 to be closed given by Fu [4]. Let A be a subset of X. A point JC е Л is said to be an F-point if for every open set U with x G (U О A) — K, there exists a neighborhood W of 382 MAKAROV, RACHKOVSKI, AND SONG x such that WnA с (UnA) - K. The set A is said to be an F-set if every point of A is an F-point. THEOREM 3.2 [4]. Let Q be a subset of a locally convex space X and К be a closed convex cone such that there exists a closed convex cone S with К \ {0} с int S and 0 £ int S. If Q is a compact F-set, then Max(<2|/0 is closed. EXAMPLE 3.2. L e t X = IR3, Q = conv({(.*, y, z) E (R3|;T2 + у 2 < 1, z = 0} U {(1,0,1)}), К = {(*, у, z) е R 3 |;t = у = 0, z ;> 0}. Clearly, Мах(<2|К) is arcwise connected, but not closed. 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