函数的C_拟凸性_英文_

函数的C_拟凸性_英文_ C2qua siconvexity of Funct ion s C H E N Y u a n ()Depa rt ment of Mat hematic s , Hengya ng No r mal U niver sit y , Hengya ng H una n 421008 , China ) ( 1such t hat Abstract : The p aper ha s given a sufficient and nece ssa r y co nditio n w hen t here exi st s t? 0 ,0 ) ( )( ) )( ( Π y ? E , f x, f x C , f or x+ 1 - txf t? y -1 2 ? y - C ,0 1 0 2 t hen f i s C2qua sico nvex o n X and so me relating re sult s. Key words : densit y ; C 2qua sico nvex ; st rictl y C 2qua xico nvex () CLC nunber : O221 Document : A Artical IA : 1673 —0313 200706 —0026 —03 t ?[ 0 , 1 ] ,1 Introduction ( ( ) ) ( ) ( ) ( ) f t x + 1 - tx?t f x+ 1 - tf x- C;1 2 1 2 () The qua sico nve xit y a nd t he co nve xit y a re vit al 2f i s st rictl y C 2co nve x o n X , w he n i nt C i s co ncep t s o n t he op ti mizatio n t heo r y. They have ( ) no ne mp t y ,if x, x?X , x?x, t ?0 , 1, i . e .0 < 1 2 1 2 bee n still re sea rched by wo r ke r s o n op e ratio nal re2 ( ( ) ) ( ) ( ) ( ) t < 1 , f t x1 + 1 - tx2 ?t f x1 + 1 - tf x2 -sea rc h [ 1213 ] , mo st of t he se re sult s a re li mit ed to i nt C ; ( ) 3f i s C 2qua sico nve x o n X if t he fi nit e di me n sio nal sp ace . Thi s p ap e r mai nl y y ?E , X, X?1 2 ( ) ( ) f t x+ ( st udie s t he qua sico nve xit y of f unctio n s under t he X , t ?[ 0 , 1 ] , f x, f x?y - C i mp lie s1 1 2 ( ) ) 1 - tx?y - C;i nfi nit e di me n sio nal sp ace , a nd gai n s a n sufficie nt 2 () a nd nece ssa r y co nditio n fo r a nd releva nt o t he r re2 4f i s st rict l y C 2qua sico nve x o n X ,w he n i nt ( sult s a bo ut t he C 2qua sico nve x of f unctio n s. C i s no ne mp t y ,if y ?E , x1 , x2 ?X , x1 ?x2 , t ?0 , ) ( ) ( ) ( ( ) Def in it ion 1 . 1 C i s sai d to be a co ne if tc ?C 1, f x1 , f x2 ?y - C , i mp lie s f t x1 + 1 - t? ) ) x?y - i nt C.fo r ever y c ?C a nd t ?[ 0 , + ?; C i s sai d to be a 2 λE = R , C = R co nve x co ne if C i s a co ne a nd fo r a ny a , b ?C a nd In a p a rticula r ca se w he re , we + λ( λ) ?[ 0 , 1 ] , t he n a + 1 - b ?C; C i s sai d to be a poi get t he defi nitio n of co nve x a nd qua si2co nve x f unc2 nt ed co ne if C i s a co ne a nd C ?- C = { 0} ; C i s sai tio n s i n t he u sual se n se . d to be a clo se d co nve x poi nt ed co ne if C i s a co 2 Some relat ing results of C 2qua siconvexity nve x co ne ,a poi nt e d co ne a nd a clo se d co ne . L et f be a f unctio n f ro m X to E a nd C be a Def in it ion 1 . 2 L et f be a f u nctio n f ro m X to E , X be a no ne mp t y co nve x set i n E , E be a real to2 clo sed co nve x poi nt e d co ne . polo gical vecto r sp ace a nd be Theorem 2 . 1 If f i s a f unctio n f ro m X to E a clo sed co nve x C ( ) ( ( poi nt ed co ne . We sa y t hat a nd t he re e xi st s t0 ?0 , 1,such t hat f t0 x1 + 1 () ) ) 1f i s C 2co nve x o n X if fo r x, x?X a nd- tx?y - C , t he n t he set1 2 0 2 Received item :2007 —09 —10 ()Foundation item :Suppo rted by teaching fo undatio n of Hengya ng No r mal U niver sit y J Y0737 () Biogra phy :Chen Yuan 1968 —,male ,bo r n in L eiyang Hunan Pro vince ,Dep art ment of Mat hematics , Hengya ng No r mal U2 niver sit y ,Ma ster of Science , Majo ring i n no nli nea r a nalysi s a nd game t heo r y. 27 :函数的 C 2 拟凸性 陈源 2007 年第 6 期 ( ) ( ( ) ) ( ) set , fo r Π x0 ?X , f x1 ?y -M = { t ?[ 0 , 1 ] : f t x1 + 1 - tx2 ?C , we ha ve f x?y ( ) ( ) y - C , f o r al l x , y ?X}x, - C , x ?U x, U xi s a nei gh bo r boo d of 0 0 0 t he n f i s C 2qua sico nve x o n X iff t here e xi st s t?i s de n se i n [ 0 ,1 ] . 0 Proof . Suppo se M i s no t de n se i n [ 0 , 1 ] . ( ) 0 , 1such t hat ( ) ε( ) ) ( The n t here e xi st s t′?0 , 1\ M a nd a 2nei gh bo r2 0 x1+ 1 - t0 x2 ? y - C , f o r Π y ? E ,f t ( ε)of t,′ s. t . ( ) ( ) hoo d U t,′ f x1 , f x2 ? y - C ( ε)( )Φ U t,′ 1 ( ) ?M = Proof . ?The nece ssa r y co nditio n i s o b2 Defi ne vio u s ( )}′ t > t2 ) ( = in f { t ?M |t 1 ?The sufficie nt co nditio n i s t e stifie d a s t < t}′ () follo wi ng : t= su p { t ?M |3 2 () Suppo se f i sn ’t C 2qua sico nve x o n X , t he nFro m 1,we ha ve t1 ?t2 , t he refo re ( ) 0 ?t< t?1t he re e xi st s x1 , x2 ?X ,t ?0 , 1,such t hat 2 1 ) ( ) ( Si nce t, 1 - t?0 , 1, t he n we ca n f et ch() )( 0 0 p, pf t x txy - C , f o r Π y ? E , + 1 - 1 2 1 2 ?M , p?t, p?t, s. t .1 1 2 2 ( ) ( ) ( )f x, f x? y - C4 1 2 ) ( ) ( pma x{ t, 1 - t}tL et x = t x+ 1 - t x.tp- < - 2 0 0 1 2 1 2 1 L et Beca u st X i s a no ne mp t y op r n co nve x set , ) ( t = tp+ 1 - tp ( 0 1 0 2 f ro m t heo re m 1 we k no w t he re i s t?M , t? t n ?n n The n fo r ever y x, x?X , we ha ve( )1 2 x - tn x 1 ( )n ) ?, a nd x2 =?X . It i s o bvio u s t hat( )1 - t n ) )( px( ) ( + 1 - + 1 - 1 2 t x tx= tpx 1 2 0 1 1 ( ))( n n ( ) x?x, t he n x = 1 - tx+ tx .2 2 n 2 n 1 ) )( ) ( ( + 1 - tpx+ 1 - p2 x2 0 2 1 ( n)( ) ( ) ?U x,Beca u se x?X , f x?y - C , x 2 2 2 2 He nce i s a nei gh bo r hoo d of x, he nce )( 2 ) ( ( ) U x(( ) ( f t x + 1 - tx= f tpx+ 1 - px2 1 2 0 1 1 1 2 ( )( )n n ) ) ( ( ) ( ) ( ) ) )f x( f t x + 1 - tx ? y - C1 ( ( ) tpx+ 1 -px2 + 1 - ? y - C , f x=n 1 n n 0 2 1 2 2 () Fro m 4, we k no w t hat it bri ng s to a co nt ra2 Beca u se p, p?M , t he n1 2 ( ( dict . ) ) f p1 x1 + 1 -px? y -1 2 C , If X i s a no ne mp t y op e n co n2 )( )( Corollary 2 . 1 a nd f px+ 1 - C fo r x2 Π y ?E , 2 1 p2 ?y - ( ) ve x set , fo r Π x0 ?X , f) ( ) x?y - C , we ha ve( f x, f x?y -0 C 1 2 ( ) ( ) ( ) f x?y - c , x ?U x0 , U x0 i s a nei gh bo r hoo d While t?M , t herefo re0 ( ( ) ( ) ( of iff ( x0 , t he n f i s st rict l y C 2qua sico nve x o n Xf tpx+ 1 - px+ 1 - tpx0 1 1 1 2 0 2 1 ( ) ( t he re e xi st s t0 ?0 , 1suc h t hat ) ) ) 1 - + px? y - C2 2 ) ( Π y ? E , ( ) )( The n C fo r + 1 - )f t x( 2 ?y -f txtx? y - i nt C , f o u Π y ? E ,xt 1 + 1 - 0 1 0 2 ( ) ( ) ( ) ( ) f x, f x?y -x, f xf C , ? y - C.1 2 1 2 If X i s a no ne mp t y op e n co n2 Corollary 2 . 2 na mel y t ?M , we have t < t1 ( ) ( ( ) ( ) - )< ?If t > t, ′ f ro m t -p2 = t0 p1ve x set ,fo r Π x0 ?X , f x0 ?y - C , we ha ve f xp2 ( ) ( ) t- t?y - C , x ?U x, U xi s a nei gh bo r hoo d of x,1 2 0 0 0 () B ut t ?M , we ha ve t ?t1 fo r m 2t he n f i s C 2co nve x o n X iff fo r x, x?X , t here1 2 ( )It bri ng s to a co nt ra dict . t?0 , 1suc h t hat e xi st s 0 ) ( )( f x- C t) ) ) + 1 - ( ( ( 0 2 ( ) f tx+ 1 - tx?tf x?If t < t,′ we have t > tfo r m0 1 0 2 0 1 2 If X i s a no ne mp t y op e n co n2 ( ) ( ) Corollary 2 . 3 p- t = 1 - tp- p< t1 0 1 2 1 t- 2 i s o noe mp t y , fo r C ve x set , w he n i nt ?X ,Π x0() B ut t ?M , we ha ve t ?tfo r m 3. 2 ( ) ) ( ) ( f x?y -C , we ha ve f x?y -C , x ?U x,0 0 It al so bri ngs to a co nt ra dict . ( ) x, t he n i s a nei gh bo r hoo d of f i s st rict l y U x0 M i s de n se i n [ 0 ,1 ] . 0 The refo re , t he set C 2co nve x o n X iff fo r ?x, t here x, x?X , x2 Theorem 21 2 If X i s a no ne mp t y op e n co nve x 1 2 1 ? 1994-2013 China Academic Journal Electronic Publishing House. All rights reserved. 28 衡阳师范学院学报 2007 年第 28 卷 () f unctio ns [J ] . O r Tra nsactio ns ,1999 ,13 1:48251 . ( ) e xi st s t0 ?0 , 1,such t hat [ 7 ] Yo une ss E A . E2co nvex set s , E2co nvex f unctio ns , and ) ( )( ) ) ) + 1 - tf x( ( ( - int C f tx+ 1 - tx?tf x0 2 0 1 0 2 0 1 e2co nvex Pro gra mming [ J ] . J o ur nal of Op timizatio n and () Applicatio n ,1999 ,102 2:4392450 . [ 8 ] Mukherjee R N , Reedy L V . Semico ntinuit y and qua si2 参考文献 :co nvex f unctio ns [ J ] . J . Op tim. t heo r y Appl , 1997 , 94 [ 1 ] Ning Ga ng. Crit eria of g2qua sico nvex f unctio n [J ] . Mat h2 () 3:7152726 . () ematics in Practice a nd Theo ry ,2006 ,36 1:2242226 . [ 9 ] Ya ng X M ,L iu S Y. 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On E2co nvex set s , E2co nvex f unctio ns ,a nd E2 5382540 . co nvex Pro gramming [J ] . J o ur nal of Op timizatio n t heo r y [ 13 ] Beckenbach E f . Co nvex f unctio ns [ J ] . Bull . A mer . () a nd Applicatio n ,2001 ,109 3:6992703 . Mat h . Soc ,1948 ,54 :4392460 . [ 6 ] Xinmin Ya ng. Q ua sico nvext y of upper semi2co ntinuo us 函数的 C 2 拟凸性 陈源 () 衡阳师范学院 数学系 , 湖南 衡阳 421008 ( ) 摘要 : 给出了在存在 t? 0 ,1满足0 ) )) ( )( ( ( x+ 1 - txΠ y ? E , f x, f x f t C , f or C , 0 1 0 2 ? y -1 2 ? y - 的条件下函数 f 具有 C2拟凸性的充分必要条件和一些相应的结果 。 关键词 : 稠密性 ; C2拟凸性 ; 严格 C2拟凸性 ? 1994-2013 China Academic Journal Electronic Publishing House. All rights reserved.