A PAIR OF HIGHER ORDER SYMMETRIC NONDIFFERENTIABLE MULTIOBJECTIVE MINI-MAXMIXED PROGRAMMING PROBLEMS

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1 Iteatoal Joal of Cote Scece ad Cocato Vol. 3, No., Jaa-Je 0,. 9-5 A PAIR OF HIGHER ORDER SYMMERIC NONDIFFERENIABLE MULIOBJECIVE MINI-MAXMIXED PROGRAMMING PROBLEMS Aa Ka ath ad Gaat Dev Deatet of Matheatcs, det Acade of echolog, F/A, Chadaa Idstal Estate, Bhbaeswa-4, Odsha, Ida, E-al: a_tath06@edffal.co Deatet of Matheatcs, ABI, Cttac, Odsha, Ida, E-al: tath_ath@edffal.co ABSRAC A a of hghe ode setc o dffeetable a ed ogag oble whee each obectve fcto cotas sot fcto of coact cove set R, s folated. Ude hghe ode F-covet assto, wea, stog ad covese setc dalt theoes elated to a oel effcet solto ad selfdalt ae oved Kewods: effcet solto, self dalt, Fcovet, sot fcto, a-ed, Hghe-ode setc. INRODUCION Setc dalt olea ogag oble was fst todced b Do [] who defed a atheatcal ogag oble ad t s dal to be setc f the dal s the al obles. Late Datzg, Esbeg ad Cottle [] ad Mod [3] folated a a of setc dal ogas fo scala fcto f(, ) that s cove the fst vaable ad that s cocave the secod vaable esectvel. Balas[4] geealzed dalt fo lea ad olea ed tege ogag obles. Ude the weae covet asstos osed o f, Mod ad We [5] folated a a of a setc dal ogas. Chada ad Ka folated a a of setc dal a tege Pogas, whch soe al ad dal vaables ae costaed to belog to the set of teges fo abta coes. K ad Sog [6] also, folated two a of olea ltobectve ed tege ogas fo abta coes ad establshed the dalt theoes. Mod [7] fst folated secod ode setc dal odels, todced the cocet of secod ode cove fcto ad oved secod ode setc dalt theoes. Becto ad Chada [8] establshed the secod ode sedo covet ad sedo- cocavt asstos. Dev [9] folated a a of secod ode setc dal ogas ad establshed dalt eslts volvg secod ode ve fctos. Pade [0] todced secod ode -ve fcto fo ltobectve factoal ogag oble ad establshed wea ad stog dalt theoes. Mod ad Schecte [] costcted two ew setc dal as whch the obectve fcto cota a sot fctos of coact cove set R ad ae theefoe odffeetable. Ude the secod ode F-sedo covet asstos, Ho ad Yag [] gave the secod ode setc dalt. Hghe ode dalt olea ogas have bee stded b soe eseaches. Magasaa [3] folated a class of hghe ode dal obles fo olea ogag oble. Mod ad Zhag obtaed dalt eslts fo vaos hghe ode dal ogag obles de hghe ode vet assto, sch as hghe ode te-, hghe ode sedo te-, ad hghe ode qas te - codtos. Msha ad Reda [4] gave vaos dalt eslts whch clded Magasaa hghe ode dalt ad Mod- We hghe ode dalt. Che [5] also dscssed the dalt theoes de the hghe ode F-covet fo a a of odffeetable ogas. Che fst gave a a of odffeetable ltobectve fctos cotas a sot fcto of coact cove set R ad dscssed the setc dalt fo ltobectve a ed tege ogag obles.. PRESEN WORK I ths chate, a a of hghe ode setc odffeetable ltobectve -a ed ogag obles b todcg a dffeetable fcto s folated, whee each of obectve fctos cotas a sot fcto of a coact cove set R. Fo a dffeetable fcto h : R R R the deftos of the hghe ode F-covet (F-sedo covet, F-sedo covet) wth esect to h ae todced. he all ow othe geealzed vet, sch that, te- vet ad hghe ode te- vet ca be t to the catego of the hghe ode te- vet ca be t to the catego of the hghe ode F-ve fctos b tag ceta aoate tasfoatos of F ad h. Ude these the hghe-ode

2 0 Iteatoal Joal of Cote Scece ad Cocato (IJCSC) F-covet assto, the hghe ode wea, hghe ode stog ad hghe covese setc dalt theoes elated to a oel effcet solto ad self dalt ae oved. 3. NOAION AND DEFINIION hoghot ths chate R ad R ae -desoal ad -desoal Eclda saces esectvel. R + ad R + ae o egatve othats of R ad R esectvel. Let U ad V be two abta sets of teges R (0) ad R (0) esectvel ad C ad C ae closed cove coes R ad R. Let R ad R. Wthot loss of geealt, sose the fst cooets of ad the fst cooets of ae costaed to be tegesad wte (, ) = (,,, ) whee U ad V, C ad C. whee = + ad = +. Fo a eal-valed twce dffeetable fcto g(, ) defed o a oe set R R, deote b g(,) the gadet vecto of g wth esect to at (,),(,) g, the Hessa at wth esect to at (,), slal g (,) ad g (,) ae also defed. Let C be a coact cove set R. he sot fcto of C s defed b s( ) C = a{ } C A sot fcto, beg cove ad evewhee fte, has a sb dffeetal, that s thee ests z R sch that s( )( C )() ) s C + z C fo all C. he sb dffeetal of s( C) s gve b s( ) c = { z C z = s( )}. C Fo a set D R, the oal coe to D at a ot D ad s defed as () { () N = R 0, z }. z D D It s obvos that fo a coact cove set C, () NC ff s( ) C =, o eqvaletl, s( ) C Cosde the followg ltobectve ogag oble (MOP) Mze f () sbect to g() 0, X, whee f : R R, g : R R t ad X R. We deote the set of feasble soltos of (MOP) b P = { X g() 0}. Defto 3.: A ot P s sad to be a effcet solto of (MOP) f thee ests o othe P sch that f ()()\{0}, f R that s f()() f fo all {,, 3,... }, ad at least oe {,,3,... },()(); f f P s sad to be a wea effcet solto of (P) f thee ests o othe P sch that fo all {,,3,... },()(). f > f Defto 3.: P s sad to be a Geoffo oel effcet solto of (P), f s a effcet solto, ad thee ests a eal be M > 0 sch that fo all {,, 3,... }, P ad f ()(), < f the f () f () M[()()] f f fo soe {,,3,... } sch that f ()(). < f Lea 3.: If P s a oel effcet solto of (MOP), thee est a = ( a, a,...) a R ad β = ( β, β,...) β R sch that = = a f ()() + 0, β 0, g 0,( =,) a 0. β a β Defto 3.3: A fcto F : X X R R (whee X R ) s sblea wth esect to the thd vaable f fo all (, ) X X () F(, ; a + a ) F(, ; a ) + F(, ; a ), fo all a, a R. () F(, ; αa) = αf(, ; a), a 0, α 0, fo all a R. Defto 3.4: Sose that h : X R R s a dffeetable fcto, F s sb lea wth esect to the thd aget. We sa that () f s sad to be hghe ode F-covet X wth esect to h, f fo all (,)()() X R f f F(, ;()( f,)) + {( h,)} h () f s sad to be hghe ode F-sedo covet X wth esect to h, f, fo all (, ) X R. we have F(, ;()( f,)) + 0 h f ()()(,) f {( h,)} h () f s sad to be hghe -ode F-qas-cove X wth esect to h, f fo all (, ) X R, We have f ()()(,) f + {( h,)} h F(, ;()( f,)) + {( h,)} 0 h Defto 3.5: Let f : R R R ad h : X R R be dffeetable fcto whee X R, F : X X R R be sb lea wth esect to ts thd aget. We sa that () f(., ) s hghe-f-cove at X, wth esect to soe fcto h, f fo all (, ) X R ad fo fed Y R we have f (,)(,)( f, :(,)( F,)) f + h + h(,)(,). h () f (., ) s sad to be hghe-ode F-sedo-cove at X wth esect to h, f fo fed Y R ad fo all (, ) X R we have

3 A Pa of Hghe Ode Setc Nodffeetable Mltobectve M-aed Pogag Pobles () f(., ) s sad to be hghe-ode F-qas-cove at X wth esect to h f fo (, ) X R ad fo fed Y R we have f (,)(,)( f,)(,) h h F(, :(,)( f,)) + 0 h If f (., ) s hghe ode F cove (F sedo-cove o F qas-cove) at wth esect to h f fo all (, ) X R ad fo fed Y R the f(., ) s hghe ode F cocave (F sedo-cocave o F qas-cocave) at wth esect to h fo all (, ) X R ad fo fed Y R. Rea : ( ) Whe h(,) = {()} f ad F( ; ;)( a,) = η, a whee h : X X R, the hgheode F covet (hghe ode F-sedo-covet, hghe ode F-qas-covet) edces to η-bovet (η-sedo-bovet, η-qas-bovet) [9]. () Whe h(,) = {()}, f the hghe ode F covet (hghe-ode F-sedo-covet, hghe ode F-qas-covet) edces to the secod ode F sedo-vet, F qas-vet [7]. ( ) Whe h(,) {()( =,)} f + ad F( ; ;)( a,) = η, a whee a :\{0}, X X : R + η X X R ae ostve fctos ad : X R R s dffeetable fcto, the the hghe ode F covet (hghe ode F-sedo covet, hghe ode F-qascovet) fcto becoes the hghe-ode te (hghe ode sedo- te-, hghe ode qas-te-) fcto. Fo ow o, sose that the sb lea fcto F satsfes the followg codto F(; ; a) + a 0, fo all a R +. () Defto 3.6: A eal valed fcto φ(,,... l ) wll be called addtvel seaable wth esect to f thee est eal valed fctos ξ( ) deedet of,,... l ad (, 3,... l ) sch that φ(,,... l ) = ξ( ) + ξ(, 3,... l ) 4. HIGHER-ORDER SYMMERIC DUALIY I ths secto, we cosde twce dffeetable fctos f : R R R, g : R R R R, h : R R R R ad coact cove sets C R ad D R fo =,,. We folate the followg hghe ode setc odffeetable ltobectve Ma ed tege setc al ad dal obles. Pal oble (MOP). (( f,) + s ()( C,,)()(,.) z, + h h (( f,) ()(,,)()(,.) s C z h h + + a a,,... sbect to λ f(,)(, z,) + 0 h () = ()(,)(,,) 0f z h = λ + (3) U, V, R, R, z D, R, =,,..., λ > 0, λ e =, Dal Poble (MOD) (( f,) v s ()( v, D,)()(,,) w, + g v g v..., a v,..., (( f,) ()(,,)()(,.) v s v D + w + g v g v sbect to (4) λ f(,)( v, w,) + 0, g v (5) = ()(,)(,,) 0, f v w g v = λ + (6) U v V R v R w C,,,,,, =,,..., λ > 0, λ = (7) R = Sce the obectve fctos of (MOP) ad (MOD) cota the sot fcto s( C ) ad s(v D ), =,, 3,., the ae o-dffeetable ltobectve ogag obles. Rea : () If U = φ, V = φ the (MOP) ad (MOD) becoe the obles cosdeed b X. Che [5]. () If h (,,)()( =,),, f = g(, v,) = ()(,) f, v = ad =, the (MOP) ad (MOD) ca be chaged to the followg obles. Pal: a [(( f,) + s () C, ()( z +,) ] f Sbect to (( f,)(,) z + 0, f Dal: () [(,(,))] 0, f z + f U V z D R,, 0,,. a [(( f,) v s ()()( v D,))] + w f v v v,

4 Iteatoal Joal of Cote Scece ad Cocato (IJCSC) Sbect to (( f,)( v,) + w0, + f v () [(,)(,) ] 0, f v + w + f v U, v V, v 0, w D, R whch ae the geealzed fos Ho ad Yag []. I the seqel we shall establsh the wea stog ad covese dalt theoe de the hghe ode F- covet asstos. Fo ths we sose that the fcto F : R R R R ad G: R R R R ae sb lea ad satsf the codto (a) F(, ; :) a + a 0, a R, + (b) G( v, :) a + a 0, a R + Also sose that the followg codto ae satsfed: () he fcto f (., v) + (.) w ae hghe ode F-cove at wth esect to g (, v, ) ad () he fcto f (,.) + (.) z ae hghe ode G-cove at wth esect to h (,, ) fo =,,.... heoe 4. (Wea Dalt): Asse that ad f (, ) ad h (,, ) ae addtvel seaable wth esect to o ad g (, v, ) s addtvel seaable wth esect to o v. Fo each feasble solto (,, λ, z, z,... z,,,... ) of (MOP) ad each feasble solto (, v, λ, w, w,... w,,,... ) of (MOD), the the followg eqalt eqaltes caot hold sltaeosl. () Fo all {,, 3,... } f + s C Z + h h (,)( )()(,,)() [(,,)] f v s v D + w + g v v (A) (,)( )()(,,) [(,,)] g () Fo at least oe {,, 3,... } f + s C Z + h h (,)( )()(,,)() [(,,)] f v s v D + w + g v h (B) (,)( )()(,,) [(,,,)] Poof: Sce f (, ) ad h (,, ) ae addtvel seaable wth esect o (sa wth esect to ), t holds ad Z = f (,)()( = f,), + f h (,,)()( =, h,), + h f (,)( =,) f h = h = (,,)(,,),,, 3... hs (MMP) ca be ewtte as, (()()( f,)() + h + f + s C z + h (,,)()(,,),... h ()(, z,)()( + h,,) h a,...,(()()( f,) + + h + f + s C Sbect to λ f (,)(, z,) + 0, h (8) = = ()(,)( λ,,) f 0, z + h (9) U, V, R, R, z D, R, (0) =,,..., λ > 0, λ e =, So (MOP) ca be wtte as Z = a[(()()(),... f + h + φ...()()()] f + h + φ Sbect to (8), (9) ad (0) whee φ () {( = f,)(,,)( + h ) + s C, ()()( z,,)} g Slal, (MOD) ca also be wtte as Z = a[(()()(),... f + h + ψ v v...()()()] f + h + ψ v Sbect to λ (,)(,,) 0, f v + w + g v = = ()(,)( λ,,) f 0, v + w + g v U, V, R, v R, w C, R, =,,..., λ > 0, λ e =, () () (3) Fo a gve ad v, the obles (MOP) ad (MOD) ae eactl the a of hghe ode setc dal o-dffeetable ltobectve ogag obles b X. Che [5]. Hece vew of the asstos, heoe- b X. Che [5] becoes alcable ad theefoe, we have fo each feasble solto (,, λ, z, z,... z,,,...) of (MOP) ad each feasble solto (, v,, w, w,... w,,,...) λ of (MOD), {(( f,)()( v, s,)() v [( D,,,)] w g v g v } = λ + +, {(( f,)()(, s,)() [( C,,,)] z h h } λ + + = ths les that the coclso holds.

5 A Pa of Hghe Ode Setc Nodffeetable Mltobectve M-aed Pogag Pobles 3 Rea 3: () Fo ow o, wthot loss of geealt we ca asse that f (, ), h (,, ), ad g (,, ) ae addtvel seaable wth esect to, =,,. () Fo the ocess of the oof theoe-, we ca also obta that ( A) ad ( B) caot hold sltaeosl f sb lea fctos F ad G satsf the codto (a) ad (b) ad fo each feasble solto (,, λ, z, z,... z,,,... ) of (MOP) ad each feasble solto (, v, λ, w, w,... w,,,... ) of (MOD), oe of the followg codtos holds. () f (,., v) + (.) w s hghe ode F-sedo cove at wth esect to g (, v, ) ad f (,,.) (.) z s hghe ode G-sedo-cocave at wth esect to h (,,). () f (,..,)(.) v + w s hghe ode F-qas-cove at wth esect to g (, v, ) ad f (,,.) (.) z s hghe ode G-qas-cocave at wth esect to h (,, ). he followg eslt dcates that de soe codtos, a oel effcet solto of (MOP) s also the oes of (MOD) ad the two obectve vales ae coesodgl eqal. heoe 4. (Stog Dalt): Let (,, λ, z, z,... z,,,...) be a effcet solto of (MMP), f : R R R s twce dffeetable at (,), h: R R R R, : g R R R R, s twce dffeetable at (,,), s twce dffeetable at (,,), fo =,, 3,.... Asse that the followg codtos hold; h (,, 0) = 0,( g,, 0) 0, = h () (,, 0) = 0,(, h, 0) 0 = h (,, 0)( =, h, 0),,, 3... = ; () fo all {,, 3,... }, we have the Hessa at h (,,), s ostve defte o egatve defte; () the set of vectos { f (,)(, z,) + } h s leal deedet, = (v) Fo soe a R ( a > 0) ad R, 0, =,,3,... we have = a ()(,)(, f,) 0. z + h he = 0, =,, 3... ; Ad thee ests sch that w C (,,,,,... λ w w w, = 0, = 0... = 0) s a feasble solto of (MOD). Ftheoe, f the hotheses heoe 3. ae satsfed ad ()(), h = g =,, 3,..., the (,,,,,... λ w w w, = 0, = 0... = 0) s a effcet solto of MOD), ad the two obectve vales ae eqal. Poof: If (,, λ, z, z,... z,,,...) a oel effcet fo (MMP) the (,, λ, z, z,... z,,,...) s also effcet fo (MP). hs, de the codto ths theoe we obta fo theoe- X. Che[5] that thee est w C, =,, 3,..., sch that (,, λ, w, w,.... w, = =...) = 0 = s a feasble solto of (MD). It s obvos that t s also feasble fo (MMD). Ftheoe, f the hotheses of hghe ode F-covet theoe- ae satsfed, the the obectve vales of (MP) ad (MD) ae eqal b [5], that s f (,)()( + s,,)() C [(,,,)] z + h h, = (( f,)()(, s,)() [( D, +,)] w + g g, =,, 3, 4... Note that h()(),() = g0, h = = 0, we have f (,)()( + s,,)() C [(,,)] z + h h, ((,)()(,,)() [(,,)],,,... g () = f s D + w + g g = Fo theoe-, (,, λ, w, w,... w, =, =... =.) 0= s a effcet solto of (MMD). It s sla to the ethod of the oof of theoe X. Che-004[5] that t s also a oel effcet solto of (MMD). Slal, we have the followg covese Dalt. heoe 4.3: (Covese Dalt): Let (, v, λ, w, w,.... w, =... =.) 0= be a oel effcet solto of (MMD), f : R R R s twce dffeetable at (,), v g: R R R R s twce dffeetable at (, v,), h: R R R R s dffeetable at (, v,) f the followg codtos hold () h (, v, 0) = 0,( g,, 0) v 0, = v, 0) g (, = 0, g (, v, 0)( =, h, 0), v,, 3,... = ; () Fo all {,, 3,... }, the Hessa at s ostve defte o egatve defte. (,,) g v () he set of vectos { f (,)( v +, w,) + g v } s leal deedet. =

6 4 Iteatoal Joal of Cote Scece ad Cocato (IJCSC) (v) Fo soe α R ( α > 0) ad R, 0 ( =,,3,...) les that = { (,)(,,) 0 } α f v + w + g v he () = 0, =,, 3,... ; () hee ests z C sch that (, v, λ, z, z,,.... z, =, =... =.) 0= s feasble solto of (MMP). Ftheoe, f the hotheses heoe 3. ae satsfed ad g ()(), = h,,... = the (, v, λ, z, z,,.... z, =, =... =.) 0= s oel effcet solto of (MMP), ad the two obectve vales ae coesodgl eqal. 5. HIGHER-ORDER SELF DUALIY A atheatcal ogag oble s sad to be selfdal, f whe the al s ecast the fo of the dal, the ew oble obtaed s the sae as the dal oble. Fst, we gve the followg defto. Defto 5.: he fcto h : I R R R R R s sad to be sew-setc wth esect to ad f fo all ad the doa of h sch that h(,,, ) = h (,,, ) whee U, R ad R ad U s a abta sets of teges R, + = heoe 5. (Self-dalt): If f ad h (MMP) ae sew setc fcto wth esect to ad ad =, U = V, C = D, z = w, = ad h(,, ) = g(,, ), =,, 3,.... he (MMP) s self-dal, that s, the dal oble of (MMP) s tself, ad the oe effcec of (,, λ, z, z,... z,,,...) fo (MMD), ad the covese. Ftheoe, de the codtos of theoe (4.) ad (4.3), f (,, λ, z, z,... z, = =... = 0) s a oel effcet solto of (MMP ), the (,,,,,,... λ z z z, = =... = 0) a oel effcet solto of (MMD), the coo otal vales s zeo ad the ovese. Poof: he oble (MMP)a be eeseted as a a- oble (( f,)()(,,)()( s C,.) +, z h z + h z a,,, z,...,(( f,)()(,,)()( s, C.),.. + z h z + h z sbect to λ f(,,)(, z,) + 0, h = ()(,)(,,) 0, f z h = λ + U V R R z D,,,,, R e, =,,..., λ > 0, λ =, Sce f ad h s sew-setc fcto wth esect to ad C = D, z = w, = ad h(,, ) = g(,, ) = g(,, ), =,, 3,.... t holds (( f,)()(, s,)()( D,,) +,, w + g g a,,, z,...,((,)()(,,)()(,.),.. f s D + w + g g Sbect to λ f(,)( +, w,) + 0, g = ()(,)( λ,,) f0, + w + g = U V R R w D,,,,, R, =,,..., λ > 0, λ e =, whch s the dal oble (MMD). hs (MMP) s self dal. It s obvos that the oe effcec of (,, λ, z, z,... z,,,...) fo (MMP) les the oe effcec of (,, λ, z, z,... z,,,...) fo (MMD),ad the covese. Net, we show that f (,)()( + s,,) C z + h, () [(,,)] h 0 = B theoe (4.), (4.3) ad (5.) we have =,, 3, (8) f (,)()( + s,,)() C [(,,)] z + h h, = (( f,)()(, s,)() [( D, +,)] w + g g = f (,)()( s,,)() C [( +,,)] z h + h, Whee the eqalt s fo the codtos. hs les that (8) holds. 6. CONCLUSION I the above, we folate a a of the hghe-ode setc o-dffeetable ltobectve -a ed ogag oble whch the obectve fctos cota a sot fcto of a coact cove set R o R. Ude the hghe-ode F-covet (hghe-ode F-sedo-covet, hghe-ode F-qas-covet) assto, we gve the hghe-ode wea., hgheode stog, hghe-ode covese dalt, ad self dalt. I o odels, U = φ, V = φ, the (MMP) ad (MMD) becoe the obles cosdeed b X. Che[5]. If h (,, ) = (,)(,,)(, f g v = f v) ad =, U = φ, V = φ the (MMP) ad (MMD) edce to the secod-ode setc odels of Ho ad Yag[].

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