Some Properties of Semi-E-Convex Function and Semi-E-Convex Programming*

Size: px

Start display at page:

Download "Some Properties of Semi-E-Convex Function and Semi-E-Convex Programming*"

Loren Wells
6 years ago
Views:

1 The Eighh Ieraioal Symposium o Operaios esearch ad Is Applicaios (ISOA 9) Zhagjiajie Chia Sepember Copyrigh 29 OSC & APOC pp Some Properies of Semi-E-Covex Fucio ad Semi-E-Covex Programmig* Xiusu Che College of Mahemaics ad saisics Chogqig Techology ad Busiess Uiversiy Chogqig 467 People s epublic of Chia Absrac I ef 1Yag shows ha some of he resuls obaied i ef 2 o E-covex programmig are icorrec i ef 3Che iroduce semi-e-covex fucios o correc he mai resuls i ef2 i ef 4Duca ad Lupsa show ha he resuls obaied i ef 2 cocerig he characerizaio of a E-covex fucio f i erms of is E-epigraph are icorrec Ad give some characerizaios of E-covex fucios usig wo oio of epigraph ( epi E ( f ) ad epi ( f ) ) I his oe some ew properies of semi-e-covex fucios are discussed ad ew characerizaios of semi-e-covex fucios usig a ew epigraph of epi E ( f ) ad slack 2-covex se are proposed more ew resuls of semi- E-covex programmig are give Keywords E-covex ses; semi-e-covex fucios; epigraphs; semi-e-covex programmig; slack 2-covex ses 1 Iroducio The covexiy of fucios is impora i he discussio of opimizaio ad variaio iequaliies o weak he covexiy of fucios araced more aeio of researchers see [1-8] Youess iroduced he coceps of E- covex se ad E-covex fucio i ef 2 Che iroduced he defiiios of semi-e-covex fucio ad quasi-semi- E-covex fucio i ef 3 For coveiece we recall hese defiiios Ad give oher relaed coceps which is required i he laer discussios Defiiio 11 [2] Le E : be a fucio A subse M is said o be E-covex if ( 1 M for all x y M ad all [1] Defiiio 12 [2] Le M be a oempy subse of ad le E : be a fucio A fucio f : M is said o be E-covex o M if M is E-covex ad * The work is suppored by he Projec of Chogqig Muicipal Educaio Commissio( NOKJ9732) ad Naural Sciece Foudaio Projec of CQ CSTC29BB xousuche1@siacom

2 34 The 8h Ieraioal Symposium o Operaios esearch ad Is Applicaios f (( 1 ( 1 f ( ) f ( for all x y M ad all [1] Defiiio 13 [3] A fucio f : M is said o be semi- E-covex o a se M iff here is a mappig E : such ha M is a E-covex se ad f (( 1 ( 1 f ( f ( for each x y M ad all [1] Defiiio 14 [3] A fucio f : M is said o be quasi-semi- E-covex o a se M iff here is a mappig E : such ha M is a E-covex se ad f (( 1 maxf ( for each x y M ad [1] Defiiio 15 A fucio f : M is said o be sricly quasi-semi E-covex o a se M iff here is a mappig E : such ha M is a E-covex se ad for each x y M wih f ( we have f (( 1 maxf ( (1) Defiiio 16 A fucio f : M is said o be srogly quasi-semi E-covex o a se M iff here is a mappig E : such ha M is a E-covex se ad for each x y M wih x y we have f (( 1 maxf ( (1) Defiiio 17 [3] The fucio f : M is said o be pseudo-semi- E-covex o E-covex se M if here exiss a sricly posiive fucio b : such ha f ( f ( (1 ( 1) x for all x y M ad [1] I ef 2 he coceps of E-covex ses ad E-covex fucios are give is properies are proposed ad he relaed resuls are used i he sudy of E-covex programmig Uforuaely some of he resuls obaied i ef 2 are icorrec Ideed i ef 1 ad ef 3 Yag ad Che shows ha some of he resuls obaied i ef 2 o E-covex programmig are icorrec respecively bu does o prove ha he resul which makes he coecio bewee a E-covex fucio ad is E-epigraph is icorrec I ef 4 Duca ad Lupsa show ha he resul obaied i ef 2 o he characerizaio of a E-covex fucio f i erms of is E-epigraph(E e(f )) is o rue Ad give some characerizaios of E-covex fucio usig oio of epi E ( f ) ad epi ( f ) I his oe we discuss more ew properies abou semi-e-covex fucios ad give also some characerizaios of semi-e-covex fucio usig a ew oio of epigraph (ie epi E ( f ) ) which is firs iroduced i he oe We propose some ew resuls of semi-e-covex programmig

3 Some Properies of Semi-E-Covex Fucio ad Semi-E-Covex Programmig 35 2 Some Properies of semi-e-covex Fucios I his secio some relaios bewee differe oios abou semi-e-covex fucios ad properies of semi-e-covex fucios are give Ad he characerizaios of semi-e-covex fucios i erms of a ew oio of epigraph epi E ( f ) is obaied If M is a oempy subse of ad E : M M ad f : M are wo fucios we cosider he followig hree ses: E e( f ) = {( x M f ( ) a} epi ( f ) = {( x M f ( a} epi E ( f ) = {( z M ) f ( z) a} epi E ( f ) = {( Ex M ) f ( a} The four ses E e( f ) epi(f ) epi E ( f ) ad epi E ( f ) are o equal Obviously we have epi E ( f ) epi ( f ) The followig heorem proposes relaios amog differe oios abou semi-e-covex fucios Theorem 21 A srogly quasi-semi- E-covex fucio o a se M is also a sricly quasi-semi- E-covex fucio o he se M Theorem 22 Le M be a oempy subse of ad le f : M ad E fucio o M he epi( f ) E e( f ) Theorem 23 Le M be a oempy subse of : be wo fucios If M is a E-covex se ad f is a semi-e-covex ad le f : M ad E : be wo fucios If epi( f ) E e( f ) ad f is E-covex fucio o M The f is a semi-e-covex fucio o M Proof For x y M ad [1] ( x f ( )( y f( epi( f )hus ( x f ( )( y f ( E e( f ) which implies ha f ( E f ( as f is E-covex fucio o M we have f (( 1 ( 1 f ( ) f ( ( 1 f ( Theorem 24 Le M be a oempy subse of ad le f : M ad E : be wo fucios If epi( f ) E e( f ) ad f is covex o E (M ) The f is a semi-e-covex fucio o M Proof Be similar o he proof of Theorem 23 Theorem 25 Le M be a oempy subse of ad le f : M ad E : be wo fucios If ) epi( f E e( f ) ad epi E ( f ) is covex

4 36 The 8h Ieraioal Symposium o Operaios esearch ad Is Applicaios he f is a semi-e-covex fucio o M Proof For x y M ad [1] ( x f ( )( y f ( epi ( f ) From he codiio epi( f ) E e( f ) we have ( x f ( )( y f ( E e( f ) hus f ( E f ( This implies ha ( Ex f ( )( Ey f ( epi E ( f ) As epi E ( f ) is covex we have (( 1 Ex Ey(1 f ( f ( ) epi E ( f ) The we have f (( 1 ( 1 f ( ) f ( ( 1 f ( Defiiio 21 Le E : be a mappig M be a oempy E-covex subse of A fucio f : M is said o a logarihmic E -covex o E-covex se M if l( f ( ) is E -covex o E-covex se M ie 1 f (( 1 f ( E x y M [1] Defiiio 22 Le E : be a mappig M be a oempy E-covex subse of A fucio f : M is said o a logarihmic semi- E -covex o E-covex se M if l( f ( ) is semi- E -covex o E-covex se M ie 1 f (( 1 f ( x y M [1] Theorem 26 Le M be a oempy subse of for a mappig E : ad a fucio f : he f be a logarihmic E -covex o E-covex se M f be E -covex o M f be quasi- E -covex o M (ie f (( 1 maxf ( E ) Proof For [1] ad x y M we have 1 f (( 1 f ( E ( 1 f ( ) f ( maxf ( E Theorem 27 Le M be a oempy subse of E : be fucio ad le f : be logarihmic semi- E -covex o E-covex se M f be semi- E -covex o M f be quasi-semi- E -covex o M Proof For [1] ad x y M we have 1 f (( 1 f ( ( 1 f (

5 Some Properies of Semi-E-Covex Fucio ad Semi-E-Covex Programmig 37 maxf ( Defiiio 23 [7] See ef 3 or ef 4 Le A ad B be wo subses of We say ha A is slack 2-covex wih respec o B if for every x y A B ad every [1] wih he propery ha ( 1 x y B we have ( 1 x y A The followig heorem gives a sufficie codiio for f o be a semi-e-covex fucio usig he se epi E ( f ) Theorem 28 Le M be a oempy subse of ad le f : M ad E M : be wo fucios If M is a E-covex se ) is a covex se ad f ( E f ( x M epi E ( f ) is a slack 2-covex se wih respec o M ) he f is a semi-e-covex fucio o M Proof Le x y M ad [1] The ( Ex f ( ) ( Ey f ( ( M ) ) epi E ( f ) Sice E (M ) is a covex se we have ( 1 Ex Ey M ) ; hece (( 1 ) Ex Ey(1 f ( f ( ( M ) ) E Sice epi ( f ) is a slack 2-covex se wih respec o ( M) ) he (( 1 ) Ex Ey(1 f ( f ( epi E ( f ) I follows ha here exis z M such ha (1 Ex ) ( ) Ey ( ) Ez ( ) ad f ( z) (1 f ( f ( Hece f (( 1 f ( z) (1 f ( f ( The f f is semi-e-covex fucio o M 3 Some esuls of Semi-E-Covex Programmig Le us cosider he followig programmig problem: (P) Mi f ( s x M x : g i ( i 12 m where f : ad g i : i 12 m are fucio o we have he followig resuls Theorem 31 If x M is a fixed poi of he mappig E : (ie x Ex ) ad x is a local miimum of he problem (P) o a E-covex se M ad f : is semi-e-covex o he se M he x is global miimum of problem (P) o M Proof Le x M be a oglobal miimum of he problem (P) o M he here is y M such ha f f ( x ) sice fucio f : is (

6 38 The 8h Ieraioal Symposium o Operaios esearch ad Is Applicaios semi-e-covex o he se M ad x M is a fixed poi of he mappig E i implies ha f (( 1 x f ((1 x ) ( 1 f ( x ) f ( x ) For ay small (1) which coradics he local opimaliy of x for problem (P) Hece x is global miimum of problem (P) o M Theorem 32 If x M is a fixed poi of he mappig E : (ie x Ex )ad x is a local miimum of he problem (P) o a E-covex se M ad f : is pseudo-semi-e-covex o he se M he x is global miimum of problem (P) o M Proof Le x M be a oglobal miimum of he problem (P) o M he here is y M such ha f ( x ) sice fucio f : is s pseudo-emi-e-covex o he se M ad x M is a fixed poi of he mappig E i implies ha f (1 x ) f ( (1 )) ( x f ( x ) ( 1) y x ) f ( x ) For ay small (1) which coradics he local opimaliy of x for problem (P) Hece x is global miimum of problem (P) o M Theorem 33 Assume fucio f : is a srogly quasi-semi- E-covex o a se M he he global opimal soluios of problem (P) is uique Proof Le x 1 x 2 M be wo differe global opimal soluios of problem (P) he f ( x1 ) f ( x2 ) Sice M is E -covex ad f is srogly quasi-semi-e-covex he f (( 1 x1 ) x2 )) maxf ( x1) f ( x2 ) = f ( x 1 ) (1) This coradics he opimaliy of x1 for problem (P) The he global opimal soluio of he problem (P) is uique Theorem 34 Le M be a oempy subse of E : be fucio ad le f : be pseudo-semi- E -covex o E-covex se M u M be fixed poi of map E (ie u Eu ) ad f ( E Ev Eu v M (1) The u is a miimum of fucio f o M Proof Le u M be a o miimum of fucio f o M he here is

7 Some Properies of Semi-E-Covex Fucio ad Semi-E-Covex Programmig 39 v M such ha f ( f ( sice f is pseudo-semi-e-covex o E-covex se M we have f ( ( )) f ( (1 ) f ( ( 1) v f ( ) ( 1) v For all (1) ad f ( ( )) f ( ) ( 1) v Leig we have f ( v Which coradics he codiio (1)Hece u is a miimum of fucio f o M efereces [1] YANG XM O E-covex ses E-covex fucios ad E-covex programmig Joural of Opimizaio Theory ad Applicaios Vol 19 pp [2] YOUNESS EA E-covex ses E-covex fucios ad E-covex programmig Joural of Opimizaio Theory ad Applicaios Vol 12 pp [3] CHENXS Some properies of semi-e-covex fucios Joural of Mahemaical Aalysis ad Applicaios Vol 275pp [4] DUCAD I ad LUPSA LO he E-epigraph of a E-covex fucio Joural of Opimizaio Theory ad ApplicaiosVol129 No 2 pp [5] NOOMA Fuzzy preivex fucios Fuzzy Ses ad Sysems 64 (1994) [6] ABOU-TAI IASULAIMAN WT Iequaliies via covex fucios Iera J Mah Mah Sci Vol [7] LUPSA L Slack covexiy wih respec o a give se Iiera Semiar o Fucioal Equaios Approximaio ad Covexiy Babes-Bolyai Uiversiy Publishig House Cluj-Napoca omaia pp (i omai

FIXED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE

FIXED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE Mohia & Samaa, Vol. 1, No. II, December, 016, pp 34-49. ORIGINAL RESEARCH ARTICLE OPEN ACCESS FIED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE 1 Mohia S. *, Samaa T. K. 1 Deparme of Mahemaics, Sudhir Memorial