( F, ρ )-Convexity in Nonsmooth Vector Optimization over Cones

Transcription

1 Appied Maheaics, 25, 6, 7-9 Pubished Oie Jauary 25 i SciRes hp://wwwscirporg/joura/a hp://dxdoiorg/4236/a2562 Higher-Order Miiizers ad Geeraized ( F, ρ )-Covexiy i Nosooh Vecor Opiizaio over Coes S K Sueja, Suia Shara, Mai Kapoor 2* Depare of Maheaics, Mirada House, Uiversiy of Dehi, Dehi, Idia 2 Depare of Maheaics, Moia Nehru Coege, Uiversiy of Dehi, Dehi, Idia Eai: * aiapoor@gaico Received Noveber 24; revised 29 Noveber 24; acceped 6 Deceber 24 Copyrigh 25 by auhors ad Scieific Research Pubishig Ic This wor is icesed uder he Creaive Coos Aribuio Ieraioa Licese (CC BY) hp://creaivecoosorg/iceses/by/4/ Absrac I his paper, we iroduce he cocep of a (wea) iiizer of order for a osooh vecor opiizaio probe over coes Geeraized casses of higher-order coe-osooh (F, ρ)- covex fucios are iroduced ad sufficie opiaiy resus are proved ivovig hese casses Aso, a uified dua is associaed wih he cosidered pria probe, ad wea ad srog duaiy resus are esabished Keywords Nosooh Vecor Opiizaio over Coes, (Wea) Miiizers of Order, Nosooh (F, ρ)-covex Fucio of Order Iroducio I is we ow ha he oio of covexiy pays a ey roe i opiizaio heory [] [2] I he ieraure, various geeraizaios of covexiy have bee cosidered Oe such geeraizaio is ha of a ρ -covex fucio iroduced by Via [3] Haso ad Mod [4] defied he oio of a F-covex fucio As a exeded uificaio of he wo coceps, Preda [5] iroduced he cocep of a ( F, ρ ) -covex fucio Acza gave he oio of a ocay Lipschiz ( F, ρ ) -covex scaar fucio of order [6] ad a differeiabe ( F, ρ ) - covex vecor fucio of order 2 [7] L Croe [8] defied he cocep of a sric oca iiizer of order for a scaar opiizaio probe This cocep pays a fudaea roe i covergece aaysis of ieraive uerica ehods [8] ad i sabiiy * Correspodig auhor How o cie his paper: Sueja, SK, Shara, S ad Kapoor, M (25) Higher-Order Miiizers ad Geeraized (F, ρ)- Covexiy i Nosooh Vecor Opiizaio over Coes Appied Maheaics, 6, 7-9 hp://dxdoiorg/4236/a2562

2 S K Sueja e a resus [9] The defiiio of a sric oca iiizer of order 2 is geeraized o he vecoria case by Acza [7] Recey, Bhaia ad Sahay [] iroduced he cocep of a higher-order sric iiizer wih respec o a oiear fucio for a differeiabe uiobjecive opiizaio probe They proved various sufficie opiaiy ad ixed duaiy resus ivovig geeraized higher-order srogy ivex fucios The ai purpose of his paper is o exed he cocep of a higher-order iiizer o a osooh vecor opiizaio probe over coes The paper is orgaized as foows We begi i Secio 2 by recaig soe ow coceps i he ieraure We he defie he oio of a (wea) iiizer of order for a osooh vecor opiizaio probe over coes Thereafer, we iroduce various ew geeraized casses of coeosooh ( F, ρ ) -covex fucios of higher-order I Secio 3, we sudy severa opiaiy codiios for higher-order iiizers via he iroduced casses of fucios I Secio 4, we associae a uified dua o he cosidered probe ad esabish wea ad srog duaiy resus 2 Preiiaries ad Defiiios Le S R be a oepy ope subse of R Le K R be a cosed covex coe wih oepy ierior ad e ik deoe he ierior of K The dua coe K * of K is defied as The sric posiive dua coe s K { :, for a } K = y R yy y K of K is give by { R { }} s K = y : yy, > for a y K\ A fucio ψ : S R is said o be ocay Lipschiz a a poi u S if for soe >, ψ ( x) ψ ( x) x x x, x wihi a eighbourhood of u A fucio ψ is said o be ocay Lipschiz o S if i is ocay Lipschiz a each poi of S Defiiio 2 [] Le ψ : S R be a ocay Lipschiz fucio, he ψ ( uv ; ) deoes he Care s geeraized direcioa derivaive of ψ a u S i he direcio v ad is defied as ψ ( uv) y u ( y v) ψ ( y) ψ ; = i sup The Care s geeraized gradie of ψ a u is deoed by ψ ( u) ad is defied as : ( ; ), for a ψ u = ξ R ψ uv ξ v v R { } f S R be a vecor vaued fucio give by Le : f = f, f2,, f, fi : S R The f is said o be ocay Lipschiz o S if each f i is ocay Lipschiz o S The geeraized direcioa derivaive of a ocay Lipschiz fucio f : S R a u S i he direcio v is give by ( 2 ) ( ; ) ( ; ), ( ; ),, ( ; ) f uv f uv f uv f uv = The geeraized gradie of f a u is he se f ( u) = f ( u) f ( u), where fi ( u) is he geeraized gradie of f i a u for i =, 2,, Every eee ( A= A A ) f ( u) is a coiuous iear operaor fro R o R ad Au = ( A u,, A u) R for a u S A fucioa F : S S R R is subiear wih respec o he hird variabe if, for a (, ) (i) F( xu, ; A A2) F( xu, ; A) F( xu, ; A2) for a A, A2 R, ad (ii) F( xu, ; αa) = αf( xu, ; A) for a α R xu S S, 8

3 S K Sueja e a (i) ad (ii) ogeher ipy F( xu, ;) = () We cosider he foowig osooh vecor opiizaio probe where (NVOP) K-iiize f ( x ) subjec o g( x) Q,,, f = f f :,, p g = g g p : S R, K ad Q are cosed covex coes wih oepy ieriors i R ad R p respecivey We assue ha f i for each i {,, } ad g j for each j {,, p} are ocay Lipschiz o S Le S = { x S : g( x) Q} deoe he se of a feasibe souios of (NVOP) The foowig souio coceps are we ow i he ieraure of vecor opiizaio heory Defiiio 22 A poi x S, is said o be (i) a wea iiizer (weay efficie souio) of (NVOP) if for every x S, S R, f ( x) f ( x) i K; (ii) a iiizer (efficie souio) of (NVOP) if for every x S, f ( x) f ( x) K \ { } Wih he idea of aayzig he covergece ad sabiiy of ieraive uerica ehods, L Croe [8] iroduced he oio of a sric oca iiizer of order As a rece advacee o his pafor, Bhaia ad Sahay [] defied he cocep of a higher-order sric iiizer wih respec o a oiear fucio for a differeiabe uiobjecive opiizaio probe We ow geeraize his cocep ad give he defiiio of a higher-order (wea) iiizer wih respec o a fucio ω for a osooh vecor opiizaio probe over coes Defiiio 23 A poi x S is said o be (i) a wea iiizer of order for (NVOP) wih respec o ω, if here exiss a vecor β ik such ha, for every x S β ω f x f x x, x ik ; (ii) a iiizer of order for (NVOP) wih respec o ω, if here exiss a vecor β ik such ha, for every x S β ω { } f x f x x, x K \ Rear 2 () If f is a scaar vaued fucio, K = R ad ω ( xx, ) = x x, he defiiio of a wea iiizer of order reduces o he defiiio of a sric iiizer of order (see [8] [9] [2] [3]) (2) If K = R, = 2 ad ω ( xx, ) = x x, he defiiio of a (wea) iiizer of order becoes he defiiio of a vecor sric goba (wea) iiizer of order 2 give by Acza [7] (3) If K = R he defiiio of a wea iiizer of order reduces o he defiiio of a sric iiizer of order give by Bhaia ad Sahay [] Rear 22 () Ceary a iiizer of order for (NVOP) wih respec o ω is aso a wea iiizer of order for (NVOP) wih respec o he sae ω (2) A direc ipicaio of he fac ha β ik is ha, a (wea) iiizer of order for (NVOP) wih respec o ω is a (wea) iiizer for (NVOP) (3) Noe ha if x is a (wea) iiizer of order for (NVOP) wih respec o ω, he for a >, i is aso a (wea) iiizer of order for (NVOP) wih respec o he sae ω I he seque, for a vecor fucio f : S R ad (,, A= A A ) f ( u), F( xu, ; A ) deoes he vecor ( F( xu, ; A ),, F( xu, ; A )) We ow defie various casses of osooh ( F, ρ ) -covex fucios of higher-order over coes Defiiio 24 A ocay Lipschiz fucio f : S R is said o be K-osooh ( F, ρ ) -covex of order wih respec o ω a u S o S if here exis a subiear (wih respec o he hird variabe) fucioa 9

4 S K Sueja e a F : S S R R ad a vecor = (,,, ) R such ha, for each A f ( u) ρ ρ ρ ρ 2 (, ; ) ρω(, ) f x f u F xu A xu K ad a x S If he above reaio hods for every u S F, ρ -covex of order wih respec o ω o S Rear 23 () If f is a scaar vaued fucio ad K = R, he above defiiio reduces o he defiiio of a (ocay Lipschiz) ( F, ρ ) -covex fucio of order wih respec o ω give by Acza [6] (2) If f is a differeiabe fucio, K = R, = 2 ad ω ( xx, ) = x x he defiiio of a K-osooh ( F, ρ ) -covex fucio of order wih respec o ω becoes he defiiio of a vecor ( F, ρ ) -covex fucio of order 2 give i [7] = A xu, for soe fucio η : S S R ad = 2 F, - covexiy of order wih respec o reduces o ρ ( ηθ, ) -ivexiy, where ω( xx, = θ xx,, iroduced (3) If K = R, F ( xu, ; A) η ω, K-osooh ( ) ρ ) by Naha ad Mohapara [4] (4) If f is a differeiabe fucio, K = R ad (, ; ) F xua = aη ( xu, ), a R, for soe fucio η : S S R, he above defiiio becoes he defiiio of a higher-order srogy ivex fucio give by Bhaia ad Sahay [] Defiiio 25 A ocay Lipschiz fucio f : S R is said o be K-osooh ( F, ρ ) -pseudocovex ype I of order wih respec o ω a u S o S if here exis a subiear (wih respec o he hird variabe) fucioa F : S S R R ad a vecor A f u ad a x S, Equivaey, he f is said o be K-osooh ρ R such ha, for each ρω F xu, ; A i K f x f u xu, ik ρω f x f u xu, i K F xu, ; A ik If f is K-osooh ( F, ρ ) -pseudocovex ype I of order wih respec o ω a every u S o be K-osooh ( F, ρ ) -pseudocovex ype I of order wih respec o ω o S Ceary, if f is K-osooh ( F, ρ ) -covex of order wih respec o ω, he f is K-osooh (, ) he f is said F ρ - pseudocovex ype I of order wih respec o he sae ω, however he coverse ay o be rue as show by he foowig exape 2 Exape 2 Cosider he foowig osooh fucio f : S R, S = ( 2, 2) R, f ( x) = ( f( x), f2( x) ) K= xy, : x, y x ad { } f ( x) = 2 x, x< x x, x 6 =, 2 3 Defie F : S S R R as Here f ( ) = [ 2, ] ad f 2 Le : S S R The, a u = ω be give by 2 2 f 2 ( x) (, ; ) F xua = a x u ω xu, = x u, 3 3 x x x< 2, = x 3, x = ad (, ) ρ = ρω f x f u xu, ik x> F xu, ; A ik, for every x S ad A f Hece, f is K-osooh ( F, ρ ) -pseudocovex ype I of order 3 wih respec o ω a u o S However, for A =, 2 x = ad (, ; ) ρω(, ) f x f u F xu A xu K,

5 so ha f is o K-osooh (, ) S K Sueja e a F ρ -covex of order 3 a u o S Defiiio 26 A ocay Lipschiz fucio f : S R is said o be K-osooh ( F, ρ ) -pseudocovex ype II of order wih respec o ω a u S o S if here exis a subiear (wih respec o he hird variabe) fucioa F : S S R R ad a vecor A f u ad a x S, Equivaey, ρ R such ha, for each ρω F xu, ; A xu, ik f x f u ik ρω f x f u i K F xu, ; A xu, ik If he above reaio hods for every u S, he f is said o be K-osooh ( F, ρ ) -pseudocovex ype II of order wih respec o ω o S We ow give a exape o show ha a K -osooh ( F, ρ ) -pseudocovex ype II fucio of order wih respec o ω ay fai o be a K -osooh ( F, ρ ) -covex fucio of order wih respec o ω 2 Exape 22 Cosider he foowig osooh fucio f : S R, S = (, 2) R, f ( x) = ( f( x), f2( x) ) K= xy, : x, y x ad { } f ( x) = x, x x, x > 2 Here f ( ) = [ 2, ] ad f 2 Le F : S S The, a u =, for every, x S, f ( x) =, 4, ; e x u F xua = a e R R be give by x 4, x = ( x 2 ), x > ω ( xu, ) = x u ad (, ) 6 ρ = ρω f x f u ik x F xu, ; A xu, ik, ad A f ( ) Therefore, f is K-osooh (, ) However, for x = 54 ad A = ( 2, α ), [ ] Thus, f is o K-osooh (, ) F ρ -pseudocovex ype II of order wih respec o ω a u o S 2 α2 4,, (, ; ) ρω(, ) f x f u F xu A xu K F ρ -covex of ay order wih respec o ω a u o S Defiiio 27 A ocay Lipschiz fucio f : S R is said o be K-osooh ( F, ρ ) -quasicovex ype I of order wih respec o ω a u S o S if here exis a subiear (wih respec o he hird variabe) fucioa F : S S R R ad a vecor ρ R such ha, for each A f ( u) ad a x S, i (, ; ) ρω(, ) f x f u K F xu A xu K If he above reaio hods a every u S, he f is said o be K-osooh ( F, ρ ) -quasicovex ype I of order wih respec o ω o S Defiiio 28 A ocay Lipschiz fucio f : S R is said o be K-osooh ( F, ρ ) -quasicovex ype II of order wih respec o ω a u S o S if here exis a subiear (wih respec o he hird variabe) fucioa F : S S R R ad a vecor A f u ad a x S, If f is K-osooh (, ) ρ R such ha, for each ρω(, ) i (, ; ) f x f u xu K F xu A K F ρ -quasicovex ype II of order wih respec o ω a every u S, he f is said

6 S K Sueja e a F ρ -quasicovex ype II of order wih respec o ω o S Rear 24 Whe f is a differeiabe fucio, K = R ad (, ; ) F xua = aη ( xu, ), a R for soe fucio η : S S R, Defiiio ae he for of he correspodig defiiios give by Bhaia ad Sahay [] o be K-osooh (, ) 3 Opiaiy I his secio, we obai various osooh Friz Joh ype ad Karush-Kuh-Tucer (KKT) ype ecessary ad sufficie opiaiy codiios for a feasibe souio o be a (wea) iiizer of order for (NVOP) O he ies of Crave [5] we defie Saer-ype coe cosrai quaificaio as foows: Defiiio 3 The probe (NVOP) is said o saisfy Saer-ype coe cosrai quaificaio a x if, for a B g( x), here exiss a vecor ξ R such ha Bξ iq Rear 3 The foowig icusio reaio is worh oicig For (,, λ = λ λ ) R ad (,, p µ = µ µ p ) R, p ( λ f µ g)( x) = λ i fi µ jg j ( x) i= j= Thus, p λ f x µ g x i i j j i= j= p λ i fi( x) µ j g j ( x) i= j= ( f ( x) λ g( x) µ ) = ( f g)( x) f ( x) g( x) λ µ λ µ (2) Sice a wea iiizer of order for (NVOP) is a wea iiizer for (NVOP), he foowig osooh Friz Joh ype ecessary opiaiy codiios ca be easiy obaied fro Crave [5] Theore 3 If a vecor x S is a wea iiizer of order wih respec o ω for (NVOP) wih S = R, he here exis Lagrage uipiers λ K ad µ Q o boh zero, such ha ( λ f µ g )( x ) µ g( x) = The ecessary osooh KKT ype opiaiy codiios for (NVOP) ca be give i he foowig for Theore 32 If a vecor x S is a wea iiizer of order wih respec o ω for (NVOP) wih S = R ad if Saer-ype coe cosrai quaificaio hods a x, he here exis Lagrage uipiers λ K \ ad µ Q, such ha { } ( λ f µ g )( x ) (3) µ g( x) = (4) Proof Assue ha x S is a wea iiizer of order wih respec o ω for (NVOP), he by Theore 3 here exis λ K ad µ Q, o boh zero, such ha (3) ad (4) hod If possibe, suppose λ = The, µ ad (3) reduces o So here exiss B g( x) such ha ( g)( x) g( x) µ µ B µ = (5) 2

7 Now, sice Saer-ype coe cosrai quaificaio hods a x, we have for a B g( x) ξ R such ha Bξ iq Sice µ Q \{ }, we ge B S K Sueja e a, here exiss a vecor µ ξ < I paricuar, µ Bξ < O he corary (5) ipies ξ B µ = This coradicio jusifies λ Now, we give sufficie opiaiy codiios for a feasibe souio o be a higher-order (wea) iiizer for (NVOP) Theore 33 Le x be a feasibe souio for (NVOP) ad suppose here exis vecors λ K, λ > ad µ Q, µ such ha Furher, assue ha f is K-osooh (, ) Q-osooh (, ) ( f ( x) λ g( x) µ ) (6) µ g( x) = (7) F ρ -covex of order wih respec o ω a x o S ad g is F σ -covex of order wih respec o he sae ω a x o S If ρ ik ad σ Q, he x is a wea iiizer of order wih respec o ω for (NVOP) Proof Assue o he corary ha x is o a wea iiizer of order wih respec o ω for (NVOP) The, for ay β ik, here exiss a vecor ˆx S such ha, As β ω f xˆ f x xˆ, x ik ρ ik, he above reaio hods i paricuar for β = ρ, so ha we have ρω As (6) hods, here exis A f ( x) ad B g( x) f xˆ f x xˆ, x ik (8) such ha A λ B µ = (9) Sice f is K-osooh ( F, ρ ) -covex of order wih respec o ω a x o S, we have Addig (8) ad (), we ge As λ K \{ }, we obai Aso, sice g is Q-osooh (, ) have However, ˆx S, µ Q ad (7) ogeher give Addig () ad (2), we ge which ipies ha ( ˆ) ( ˆ, ; ) ρω( ˆ, ) f x f x F xxa xx K () F xxa ˆ, ; ik λ F xxa ˆ, ; < () F σ covex of order wih respec o ω a x o S ad ( ˆ) ( ˆ ) ( ˆ ) µ g x g x F xxb, ; σ ω xx, ( ˆ ) ( xx ˆ ) µ Q, we µ F xxb, ; σ ω, (2) ( ˆ ) ( ˆ ) ( ˆ ) λ F xxa, ; µ F xxb, ; µσ ω xx, <, λ ( ˆ, ; ) ( ˆ, ; ) ( ˆ if xxai µ jf xxbj µσ ω xx, ) < i j 3

8 S K Sueja e a Usig subieariy of F uder he assupio λ > ad µ, we obai which o usig (9) ad (), gives ( ˆ λ µ ) µσ ω( ˆ ) F xx, ; A B xx, <, ( xx) µσ ω ˆ, < This is ipossibe as µ Q ad σ Q, so ha µσ, ad or is a o-egaive fucio Hece x is a wea iiizer of order wih respec o ω for (NVOP) Theore 34 Suppose here exiss a feasibe souio x for (NVOP) ad vecors λ K, λ > ad µ Q, µ such ha (6) ad (7) hod Moreover, assue ha f is K-osooh ( F, ρ ) -pseudocovex ype I of order wih respec o ω a x o S ad g is Q -osooh ( F, σ ) -quasicovex ype I of order wih respec o he sae ω a x o S If ρ ik ad σ Q, he x is a wea iiizer of order wih respec o ω for (NVOP) Proof: Le if possibe, x be o a wea iiizer of order wih respec o ω for (NVOP) The, for ay β ik, here exiss ˆx S such ha, Sice β ω f xˆ f x xˆ, x ik ρ ik aig, i paricuar, β = ρ i he above reaio, we obai ρω As (6) hods, here exis A f ( x) ad B g( x) Sice f is K-osooh (, ) As λ K \{ } Now, ˆx S, we have f xˆ f x xˆ, x ik (3) such ha (9) hods F ρ -pseudocovex ype I of order wih respec o ω a x o S, (3) ipies F xxa ˆ, ; ik eas g( xˆ ) Q, so ha g( x) { g( x) g( x) } If µ, he (5) ipies ( ˆ) Sice g is Q-osooh (, ) so ha λ F xxa ˆ, ; < (4) µ ˆ This aog wih (7) gives µ ˆ (5) g x g x iq F σ -quasicovex ype I of order wih respec o ω a x o S, herefore ( ˆ, ; ) σ ω( ˆ, ) F xxb xx Q, ( ˆ ) ( xx ˆ ) µ F xxb, ; σ ω, (6) If µ =, he aso (6) hods Now, proceedig as i Theore 33, we ge a coradicio Hece, x is a wea iiizer of order wih respec o ω for (NVOP) s Theore 35 Assue ha a he codiios of Theore 33 (Theore 34) hod wih λ K, λ > The x is a iiizer of order wih respec o ω for (NVOP) Proof: Le if possibe, x be o a iiizer of order wih respec o ω for (NVOP), he for ay β ik here exiss ˆx S such ha ( ˆ) β ω( ˆ, ) \{ } f x f x x x K (7) 4

9 S K Sueja e a Proceedig o siiar ies as i proof of Theore 33 (Theore34) ad usig (7) we have As λ K s, we ge ( ˆ, ; ) K\ { } F xxa λ F xxa ˆ, ; < This eads o a coradicio as i Theore 33 (Theore 34) Hece, x is a iiizer of order wih respec o ω for (NVOP) 4 Uified Duaiy O he ies of Cabii ad Carosi [6], we associae wih our pria probe (NVOP), he foowig uified dua probe (NVUD) (NVUD) K-axiize f ( y) ( δ ) µ g( y) λ, (8) subjec o ( λ f µ g)( y) δµ g( y), (9) where y S, ik λ K \, µ Q δ, is a - paraeer Noe ha Wofe dua ad Mod-Weir dua ca be obaied fro (NVUD) o aig δ = ad δ = respecivey Defiiio 4 Give he probe (NVOP) ad give a vecor i K, we defie he foowig Lagrage fucio:, { } ad { } L ( x, λµ, ) = f ( x) µ g( x), x S, λ K, µ Q λ y λµ be feasibe for (NVUD) If f is F ρ -covex of order wih respec o ω a y o F σ -covex of order wih respec o he sae ω a y o S, wih λ >, µ ad Theore 4 (Wea Duaiy) Le x be feasibe for (NVOP) ad (,, ) K-osooh (, ) S ad g is Q-osooh (, ) he, Proof: Assue o he corary ha λρ µσ, (2) f ( y) ( δ ) µ g( y) f ( x) ik λ f ( y) ( δ ) µ g( y) f ( x) ik (2) λ Sice ( y, λµ, ) is feasibe for (NVUD), herefore by (2), here exis A f ( y) ad B g( y) such ha A λ B µ = (22) Sice f is K-osooh ( F, ρ ) -covex of order wih respec o ω a y o S, we have Addig (2) ad (23), we obai λ ρω f x f y F xya, ; xy, K (23) ( δ ) µ ρω g y F xya, ; xy, ik 5

10 S K Sueja e a As λ K \{ }, we ge Aso, sice g is Q-osooh (, ) have or, Addig (24) ad (25), we ge ( δ ) µ g( y) λf( xya) λρω( xy), ;, > (24) F σ -covex of order wih respec o ω a y o S ad µ Q, we µ g x g y F xyb, ; σ ω xy, (25) > (, ; ) (, ; ) ( ) (, ) µ δµ λ µ λ ρ µ σ ω g x g y F xya F xyb xy p i i j j i= j= > ( ) µ g x δµ g y λ F xya, ; µ F xyb, ; λ ρ µ σ ω xy, Usig subieariy of F uder he assupio ha λ > ad µ, ogeher wih (22), () ad (2), we obai g( x) δµ g y µ < δµ < As x S, gx Q ad µ Q, so ha µ g( x) ad we have g( y) This coradics he feasibiiy of ( y, λµ, ), hece he resu Theore 42 (Wea Duaiy) Le x be feasibe for (NVOP) ad (,, ) y λµ be feasibe for (NVUD) wih λ > ad µ Suppose he foowig codiios hod: (i) If δ=, ρ K, L (, λµ, ) is K-osooh ( F, ρ ) -pseudocovex ype II of order wih respec o ω a y o S, ad (ii) If δ=, λρ µσ, f is K-osooh ( F, ρ ) -pseudocovex ype II of order wih respec o ω a y o S ad g is Q-osooh ( F, σ ) -quasicovex ype I of order wih respec o ω a y o S The, we have f ( y) ( δ ) µ g( y) f ( x) ik λ Proof: Case (i): Le δ = ad o he corary assue ha, Sice x is feasibe for (NVOP) ad Addig (26) ad (27), we ge Tha is, As L (, λµ, ) is K-osooh (, ) = ( ) L C C,,,, C y λµ f ( y) g( y) f ( x) ik λ µ (26) µ, herefore Q g x λ µ µ g x Furher, ik so ha K (27) { } µ µ { } i f y g y f x g x K λ λ ( λµ ) ( λµ ) L x,, L y,, ik F ρ -pseudocovex ype II of order wih respec o ω, we have for a 6

11 S K Sueja e a Sice, λ K \{ }, we ge ρω F xyc, ; xy, ik λρω( xy) λf xyc, ;, <, or so ha i= Now, sice ( y, λµ, ) is feasibe for (NVUD), Therefore, here exiss Cˆ ( y, λµ, ) which is a coradicio, as λ K { } λρω( xy) λif xyc, ; i, <, ( λ ) λρω F xy, ; C xy, < (28) λ = λ ( λ f µ g)( y) λ f µ g ( y) = λifi λ iiµ g y i= λ i= = λ µ i= λ i λ i fi µ g y λ i= = L i i fi g y ( y ), λµ, λ L such ha λ Cˆ = Subsiuig i (28) ad he usig (), we ge ( xy) λρω, <, \, ρ K ad or is a o-egaive fucio Case (ii): Le δ =, he we have o prove ha Le if possibe, f ( y) f ( x) ik f ( y) f ( x) ik Sice f is K-osooh ( F, ρ ) -pseudocovex ype II of order wih respec o ω a y o S, we have As λ K \{ }, we ge { ρω } F xya, ; xy, ik λρω( xy) λf xya, ;, < (29) Sice x is feasibe for (NVOP) ad ( y, λµ, ) is feasibe for (NVUD), we have If g x g y Q µ, (3) ipies i { g( x) g( y) } µ (3) 7

12 S K Sueja e a As g is Q-osooh ( F, σ ) -quasicovex ype I of order wih respec o ω a y o S, we ge * Sice Q, we have If µ =, he aso (3) hods Sice (,, ) (22) hods Addig (29) ad (3), we ge or { (, ; ) σ ω(, ) } F xyb xy Q ( xy) µ F xyb, ; µσ ω, (3) y λµ is feasibe for (NVUD), by Rear 3, here exis A f ( y) ad B g( y) λ F xya, ; µ F xyb, ; λρ µσ ω xy, <, p λ F xya, ; µ F xyb, ; λρ µσ ω xy, < i i j j i= j= such ha Usig subieariy of F wih he fac ha λ > ad µ ad he usig (22) ad (), we obai ( λρ µσ) ω( xy), < This coradics he assupio ha λρ µσ, hece he resu Theore 43 (Srog Duaiy) Le x be a wea iiizer of order wih respec o ω for (NVOP) wih S = R, a which Saer-ype coe cosrai quaificaio hods The here exis λ K \{ }, µ Q such ha ( x, λ, µ ) is feasibe for (NVUD) Furher, if he codiios of Wea Duaiy Theore 4 (Theore 42) hod for a feasibe x for (NVOP) ad a feasibe ( y, λµ, ) for (NVUD), he x is a wea axiizer of order wih respec o ω for (NVUD) Proof: As x is a wea iiizer of order wih respec o ω for (NVOP), by Theore 32 here exis λ K \, µ Q such ha { } ( λ f µ g )( x ) Sice δ {,}, Equaios (32) ad (33) ca be wrie as ( λ f µ g )( x ), (32) µ g( x) = (33), δµ g( x) = Thus, ( x, λ, µ ) is a feasibe souio for (NVUD) Furher, if (,, ) β ik, here exiss a feasibe souio (,, ) wih respec o ω for (NVUD), he for ay such ha or, x λ µ is o a wea axiizer of order y λµ of (NVUD) f( y) ( δ ) µ g( y) f( x) ( δ ) µ g( x) β ω( yx, ) i K, λ λ f( y) ( δ ) µ g( y) f( x) β ω( yx, ) i K λ 8

13 S K Sueja e a i K, yx, ik, so ha we have Sice, β β ω f ( y) ( δ ) µ g( y) f ( x) i K, λ which coradics Theore 4 (Theore 42) Hece (,, ) ω for (NVUD) 5 Cocusio x λ µ is a wea axiizer of order wih respec o I his paper, we iroduced he cocep of a higher-order (wea) iiizer for a osooh vecor opiizaio probe over coes Furherore, o sudy he ew souio cocep, we defied ew geeraized casses of coe-osooh (F, ρ)-covex fucios ad esabished severa sufficie opiaiy ad duaiy resus usig hese casses The resus obaied i his paper wi be hepfu i sudyig he sabiiy ad covergece aaysis of ieraive procedures for various opiizaio probes Refereces [] Becor, CR, Chadra, S ad Becor, MK (988) Sufficie Opiaiy Codiios ad Duaiy for a Quasicovex Prograig Probe Joura of Opiizaio Theory ad Appicaios, 59, [2] Magasaria, OL (969) Noiear Prograig McGraw-Hi, New Yor [3] Via, JP (983) Srog ad Wea Covexiy of Ses ad Fucios Maheaics of Operaios Research, 8, hp://dxdoiorg/287/oor8223 [4] Haso, MA ad Mod, B (982) Furher Geeraizaio of Covexiy i Maheaica Prograig Joura of Iforaio ad Opiizaio Scieces, 3, hp://dxdoiorg/8/ [5] Preda, V (992) O Efficiecy ad Duaiy for Muiobjecive Progras Joura of Maheaica Aaysis ad Appicaios, 66, hp://dxdoiorg/6/22-247x(92)933-u [6] Acza, T ad Kisie, K (26) Sric Miiizers of Order i Nosooh Opiizaio Probes Coeaioes Maheaicae Uiversiais Caroiae, 47, [7] Acza, T (2) Characerizaio of Vecor Sric Goba Miiizers of Order 2 i Differeiabe Vecor Opiizaio Probes uder a New Approxiaio Mehod Joura of Copuaioa ad Appied Maheaics, 235, hp://dxdoiorg/6/jca2429 [8] Croe, L (978) Srog Uiqueess: A Far Crierio for he Covergece Aaysis of Ieraive Procedures Nuerische Maheai, 29, hp://dxdoiorg/7/bf39337 [9] Sudiarsi, M (989) Sufficie Codiios for he Sabiiy of Loca Miiu Pois i Nosooh Opiizaio Opiizaio, 2, hp://dxdoiorg/8/ [] Bhaia, G ad Sahay, RR (23) Sric Goba Miiizers ad Higher-Order Geeraized Srog Ivexiy i Muiobjecive Opiizaio Joura of Iequaiies ad Appicaios, 23, 3 hp://dxdoiorg/86/29-242x-23-3 [] Care, FH (983) Opiizaio ad Nosooh Aaysis Wiey, New Yor [2] Sudiarsi, M (997) Characerizaios of Sric Loca Miia for Soe Noiear Prograig Probes Noiear Aaysis, Theory, Mehods & Appicaios, 3, hp://dxdoiorg/6/s x(97)352- [3] Ward, DW (994) Characerizaios of Sric Loca Miia ad Necessary Codiios for Wea Sharp Miia Joura of Opiizaio Theory ad Appicaios, 8, hp://dxdoiorg/7/bf22778 [4] Naha, C ad Mohapara, RN (22) Nosooh ρ ( ηθ, ) -Ivexiy i Muiobjecive Prograig Probes Opiizaio Leers, 6, hp://dxdoiorg/7/s [5] Crave, BD (989) Nosooh Muiobjecive Prograig Nuerica Fucioa Aaysis ad Opiizaio,, hp://dxdoiorg/8/ [6] Cabii, R ad Carosi, L (2) Mixed Type Duaiy for Muiobjecive Opiizaio Probes wih Se Cosrais I: Jiėez, MA, Garzȯ, GR ad Lizaa, AR, Eds, Opiaiy Codiios i Vecor Opiizaio, Beha Sciece Pubishers, Sharjah,

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