University of Pavia, Pavia, Italy. North Andover MA 01845, USA

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1 Iteatoal Joual of Optmzato: heoy, Method ad Applcato (Pt) (Ole) wwwgph/otma 29 Global Ifomato Publhe (HK) Co, Ltd 29, Vol, No 2, η -Peudoleaty ad Effcecy Gogo Gog, Noma G Rueda 2 * Secto of Geeal ad Appled Mathematc, Faculty of Ecoomc, Uvety of Pava, 27 Pava, Italy 2 Depatmet of Mathematc, Memac College, Noth Adove MA 845, USA NomaRueda@memacedu Abtact: Gve a olea vecto-valued pogammg poblem volvg geealzed peudolea fucto (e η-peudolea fucto) t how that evey effcet oluto popely effcet ude ome boudede codto Key wod: olea pogammg, η-peudocovex fucto, η-peudolea fucto, effcecy, pope effcecy Itoducto A eal-valued dffeetable fucto defed o a ope et D R ad to be peudolea f f ad -f ae peudocovex Hao [5] toduced the cla of fucto f wth the followg popety: f ( y) f( x) f( x) η( y, x) fo all x, y fo a vecto fucto η ( yx, ), a a geealzato of covex fucto Late thoe fucto wee ow a η-covex o vex Hao [5] alo toduced a moe geeal cla of fucto defed a follow: f( x) η( y, x) mple f ( y) f( x) fo all x, y D * Coepodg Autho el: , Fax: , Emal: NomaRueda@memacedu GLOBAL INFORMAION PUBLISHER 55

2 Iteatoal Joual of Optmzato: heoy, Method ad Applcato Such fucto wee late called η-peudocovex o peudovex he eade may coult Mha ad Gog [7] fo a ecet uvey of vex fucto ad the geealzato If η ( yx, ) = y x, the the defto of η-covexty ad η-peudocovexty educe to the defto of covexty ad peudocovexty, epectvely We ote that, ule covex ad peudocovex fucto, the cla of vex fucto ad the cla of peudovex fucto cocde Defto (Aa et al []) A dffeetable fucto f defed o a ope et D R called η-peudolea f f ad f ae η-peudocovex wth epect to the ame η Evey peudolea fucto η-peudolea wth epect to η ( yx, ) = y x, but the covee ot tue Aa et al [] gave a example of fucto f ad η that howed that a fucto f ca be η-peudolea wthout beg peudolea Chew ad Choo [2] obtaed ft ad ecod ode chaactezato of peudolea fucto ad foud codto fo a effcet oluto of a olea vecto-valued pogammg poblem to be pope effcet Kaul et al [4] exteded the cla of peudolea fucto to emlocally peudolea fucto ad dcued codto of effcecy ad popely effcecy fo a multobectve pogammg poblem Late Aa et al [] obtaed the followg ft ode chaactezato of η-peudolea fucto, geealzg ome of the eult obtaed by Chew ad Choo [2] Ft we eed the followg defto, due to Moha ad Neogy [8] he vecto-valued fucto : η D D R, X R, atfe codto C f fo ay x, y η( xx, + λη( yx, )) = λη( yx, ), η( yx, + λη( yx, )) = ( λ) η( yx, ) fo all λ [,] Suppoe that f : D R η-peudolea, wth η atfyg codto C he fo all x, y f( x) η( y, x) = f ad oly f f ( y) = f( x) A dffeetable fucto f : D R η-peudolea f ad oly f thee ext a fucto p, called popotoal fuctoal, defed o D Duch that pxy (, ) > ad f( y) = f( x) + p( x, y) f ( x) η( y, x) fo all x, y D I th pape we ae gog to code the followg multobectve pogammg poblem: maxmze f( x) = ( f( x), L, f ( x)), ubect to g ( x) b, =, L, m, volvg η-peudolea fucto f ad g, ad we ae gog to how that fo a feable pot x the feable ego to be effcet t eceay ad uffcet that the Kuh-uce codto hold fo the fucto λf + L + λ f fo ome potve multple λ, L, λ hat, thee ext multple μ,, L μm uch that ad λ f ( x ) + L+ λ f ( x ) = μ g ( x ) + L + μ g ( x ), m m μ = = L ( g ( x ) b),,, m We ae alo gog to how that evey effcet oluto that atfe a ceta boudede codto popely effcet 56 GLOBAL INFORMAION PUBLISHER

3 η-peudoleaty ad Effcecy 2 Effcecy Code the followg multobectve η-peudolea pogammg poblem: (P) V-maxmze f( x) = ( f( x), L, f ( x)) ubect to g ( x) b, =, L, m, whee the dffeetable fucto f ad g ae η-peudolea o the ope et D R, wth popotoal fuctoal p ad q, epectvely Let X be the et of feable pot fo poblem (P) Defto 2 A feable pot x ad to be a effcet oluto of (P) f f ( x) f ( y) mple f ( x) = f ( y) fo all feable y I othe wod, thee o othe feable y uch that, fo ome =, 2, L,, we have f ( x) < f ( y), f ( x) f ( y), Geoffo [3] toduced the cocept of pope effcecy fo the maxmzato poblem V-maxmze f ( x) ubect to x X R Defto 3 A feable pot x popely effcet f t effcet ad thee ext a eal umbe M > uch that, fo each, we have f ( y) f ( x) M( f ( x) f ( y)) fo ome uch that f ( x) > f ( y) wheeve f( y) > f( x) he followg eult geealze Popoto 32 of (Chew ad Choo [2]) to η-peudoleaty Popoto Code poblem (P) whee the dffeetable fucto f ad g ( =, L, ; =, L, m) ae η-peudolea o the et D R wth popotoal fuctoal p ad q, epectvely Let codto C be atfed fo all x, y D A feable pot x a effcet oluto of (P) f ad oly f thee ext multple λ > ad μ, =, 2, L,, I( x ) = { g ( x ) = b} uch that λ f x = μ g x = I( x ) ( ) ( ) () Poof: Suppoe that thee ext λ ad μ that atfy (), but x ot effcet he thee ext a feable pot y uch that f ( x ) f ( y) fo all ad f ( x ) < f ( y) fo ome he whch a cotadcto Coveely, uppoe that μ ( g ( y) g ( x )) = ( ) (, ) μ g x η y x I( x ) q ( x, y) I( x ) f y f x = ( ) (, ) = >, λ( ( ) ( )) λ f x η y x = = p ( x, y) x a effcet oluto fo (P) Fo, the ytem (2) GLOBAL INFORMAION PUBLISHER 57

4 Iteatoal Joual of Optmzato: heoy, Method ad Applcato g x η x x I x ( ) (, ) ( ) f ( x ) η( x, x ) =,2, L,, +, L, (3) f x η x x > ( ) (, ) ha o oluto x X Suppoe thee ext y uch that g x η y x I x f x y x f x y x > ( ) (, ) ( ) ( ) η(, ) ( ) η(, ) he g( y) g( x ), f( y) f( x ), ad f( y) f( x ) but f( y) f( x ) ce f() y = f( x ) f ad oly f f ( x ) η( y, x ) = heefoe f( y) > f( x ), whch cotadct that x a effcet oluto It follow that (3) ha o oluto By Faa lemma (Magaaa [6]), thee ext λ ad μ uch that λ f ( x ) + f ( x ) = μ g ( x ) I( x ) Summg ove we get () wth λ = + λ ad μ = μ = Chew ad Choo [2] poved that effcecy ad popely effcecy ae equvalet ude ceta codto fo a patcula cae of poblem (P) whe the fucto volved ae peudolea We ae gog to how that the ame tue fo η-peudolea fucto Defto 4 (Chew ad Choo [2]) A feable pot x ad to atfy the boudede codto f the et p x p x x x X f x < f x f x > f x (, x), ( ) ( ), ( ) ( ),, (, ) (4) bouded fom above Popoto 2 Aume the ame hypothee a Popoto he evey effcet oluto of (P) that atfe the boudede codto popely effcet Poof: Let x be a effcet oluto of (P) he t follow fom Popoto that thee ext λ f( x ) = μ g ( x ) heefoe fo ay feable x, = I( x ) λ f ( x ) η( xx, ) = μ g ( x) η( xx, ) = I( x ) λ > ad μ uch that Notce that λ f( x ) η( x, x ) (5) = Othewe, we would obta a cotadcto a (2), Popoto 58 GLOBAL INFORMAION PUBLISHER

5 η-peudoleaty ad Effcecy Sce the et defed by (4) bouded fom above, the followg et alo bouded fom above: λ p( x, x) ( ) x X, f ( ) ( ), ( ) ( ),, x < f x f x > f x (6) λp( x, x) Let M be a potve eal umbe that a uppe boud of the et defed by (6) We ae gog to how that x popely effcet Suppoe that thee ext ad x X uch that f ( x) > f ( x ) he ( ) (, ) f x η x x > (7) Let { } λ η = λ η η < (8) f( x) ( xx, ) max f( x) ( xx, ) f( x) ( xx, ) Fom (5), (7), ad (8), we obta heefoe λ η λ η f( x ) ( x, x ) ( )( f( x ) ( x, x )) λ p ( x, x) f ( x) f ( x ) ( ) ( f ( x ) f ( x)), λp( x, x) ad gve the choce of M, f x f x M f x f x hu, x popely effcet ( ) ( ) ( ( ) ( )) Refeece Aa, Q H, Schable, S, Yao, J-C: η-peudoleaty, Rvta d Matematca pe la Sceze Ecoomche e Socal 22 (999) Chew, K L, Choo, E U: Peudoleaty ad Effcecy, Mathematcal Pogammg 28 (984) Geoffo, A M: Pope Effcecy ad the heoy of Vecto Optmzato, Joual of Mathematcal Aaly ad Applcato 22 (968) Hao, M A: O Suffcecy of the Kuh-uce Codto, Joual of Mathematcal Aaly ad Applcato 8 (98) Kaul, R N, Lyall, V, Kau, S: Semlocal Peudoleaty ad Effcecy, Euopea Joual of Opeatoal Reeach 36 (988) Magaaa, O L: Nolea Pogammg, SIAM, Phladelpha (994) 7 Mha, S K, Gog, G: Ivexty ad Optmzato, Nocovex Optmzato ad It Applcato 88, Spge-Velag, Bel Hedelbeg (28) 8 Moha, S R, Neogy, S K: O Ivex Set ad Pevex Fucto, Joual of Mathematcal Aaly ad Applcato 89 (995) 9-98 GLOBAL INFORMAION PUBLISHER 59