OPTIMALITY AND SECOND ORDER DUALITY FOR A CLASS OF QUASI-DIFFERENTIABLE MULTIOBJECTIVE OPTIMIZATION PROBLEM

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1 Yugoslav Joural of Oeraos Research 3 (3) Nuber, -35 DOI:.98/YJOR394S OPTIMALITY AND SECOND ORDER DUALITY FOR A CLASS OF QUASI-DIFFERENTIABLE MULTIOBJECTIVE OPTIMIZATION PROBLEM Rsh R. SAHAY Deare of Oeraoal Research, Uversy of Delh-7, Ida raasahay@gal.co Guee BHATIA Deare of Maheacs, Uversy of Delh-7, Ida guee7@yahoo.co. Receved: Јаuary 3 / Acceed: Jue 3 Absrac: A secod order Mod-Wer ye dual s reseed for a o-dffereable ulobecve ozao roble wh suare roo ers he obecve as well as he cosras. Oaly ad dualy resuls are reseed. Classes of geeralzed hgher order η bovex ad relaed fucos are roduced o sudy he oaly ad dualy resuls. A fracoal case s reseed a he ed. Keywords: Hgher order η bovexy, Src zers, Secod order dualy. MSC: 6A5, 9C9, 9C46.. INTRODUCTION The oo of secod order dualy was frs roduced by Magasara [5]. The ovao behd he cosruco of a secod order dual was he alcably he develoe of algorhs for cera robles. The secod order dual has couaoal advaage over he frs order dual as rovdes a gher boud for he value of he obecve fuco whe aroxaos are used. Oe ore advaage of secod order dualy s ha, f a feasble o for he roble s rovded ad frs order dualy does o hold he, oe ca use a secod order dual o ge a lower boud for he value of ral obecve fuco [3]. Recely, several auhors [, ad 4] have suded secod order dualy for varous classes of ozao robles.

2 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy Uder he assuo ha eas, varaces ad covaraces of he rado varables are kow, Sha [8] esablshed a way ha a sochasc lear rograg roble leads o a deersc olear rograg roble, where he fucos volve suare roos of osve se-defe uadrac fors. I s geerally dffcul o solve such robles because of o-dffereably of suare roo ers volved. However, s useful o sudy he dualy asecs of such robles, whch ay easly lead o he soluo of hese robles. Frs order dualy for varous fors of scalar as well as ulobecve ozao robles, volvg suare roo ers of cera osve se-defe uadrac fors have bee suded by ay auhors (see, for exale [6,7,4,6,8 ad 9]). Praccal alcaos of hese robles ca be foud ul-facly locao robles ad orfolo seleco roble. I hs aer, a secod order Mod-Wer ye dual for a ulobecve ozao roble volvg suare roo ers obecves as well as he cosrag fucos s reseed, ad dualy resuls are esablshed. For hs urose, we roduce classes of geeralzed hgher order η bovex ad relaed fucos. The resuls of hs aer are ore geeral ha he corresodg resuls already exsg leraure [4, 5]. Le X be a oey subse of. PRELIMINARIES R edowed wh he Eucldea or. Defo. A fuco f : X R s sad o be locally Lschz f for each bouded subse B of X, here exss a cosa l such ha for all x, y B. f ( x) f ( y) l x y, Defo. The drecoal dervave of a fuco f : X R a a o x B he dreco d R s defed as ( ) ( ) ( ; ) l f x + α d f x d = f x. α α Defo.3 ([, 7]) A fuco f : X R s sad o be uas-dffereable a a o x f f ossesses a drecoal dervave a x X for each dreco d R such ha f ( xd ; ) s covex wh resec o d. I s kow ha f f ( x), =,,..., are dffereable, he he fucos θ ( x) = f ( x) + ( x B x), =,,..., are uas-dffereable. Le Lx ( ) be he se of drecos,.e.

3 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy 3 { θ } Lx ( ) = d R : ( xd ; ) <, =,,..., ad T ( x ) be he age coe o X a x,.e. X TX ( x ) = { d R : { dk} d, αk, x + αkdk X} Lea. [3] Le θ( x) = ( θ( x),..., θ ( x) ), where : X R θ, =,,..., are locally Lschz ad ossess drecoal dervaves a each o each dreco. If src zer of θ ( x) o X, he X Lx ( ) T ( x) = φ. x s a Reark. [3] Le δθ ( x ) be he sub-dffereal of fuco θ, =,,..., a x, he we have for each w δθ ( x ), =,,..., wd θ ( x ; d) for all d R. Fro lea., follows ha for all w δθ ( x ), =,,..., he syse wd <, =,,..., has o soluo T ( x ). X We shall eed he followg geeralzed Schwarz eualy he seuel. Lea. [] Le B be a syerc osve se-defe arx ad x, z R, he ( xbz ) ( xbx ) ( zbz ), where eualy holds f ad oly f Bx = λbz for soe λ. Evdely, f we have ( x Bz) ( x Bx). ( zbz ) =, Lea.3 [9] Le ϕ ( x) = ( xbx). The ϕ ( x) s covex ad w δϕ( x) f ad oly f w= Bz, z Bz, x Bz = ( x Bx). Mulobecve ozao robles are ecouered ay areas of hua acvy cludg egeerg ad aagee. For ay eresg alcaos ad develoe of ulobecve ozao, oe ay refer o [8]. I hs aer, we sudy he followg ulobecve ozao roble; (MOP) Mze ( θ( x),..., θ ( x))

4 4 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy subec o G ( x), =,,...,. Where θ ( x) = f ( x) + ( x B x), =,,..., G x g x x C x ( ) = ( ) + ( ), =,,...,. f : X R, =,,...,, g : X R, =,,... are wce dffereable; ad B, =,,..., ad C, =,,..., are osve se-defe syerc arces. The fucos θ ( x), =,,..., ad G ( x), =,,..., are uas-dffereable fucos. Thus (MOP) ay be referred as a uas-dffereable ulobecve ozao roble. Le S be he se of all feasble soluos of (MOP). Here zao eas fdg a src zer. Defo.4 A o θ ( x) θ( x ), </ x ha s here exss o x S such ha θ ( x) < θ( x ). S s sad o be src zer for (MOP) f for all x S To exlore he alcably of oaly ad dualy resuls several auhors [6,7,4,6,8 ad 9] have suded he above ye of ulobecve ozao robles by weakeg he covexy assuos. We ove a se furher hs dreco ad roduce he classes of geeralzed hgher order η -bovex ad relaed fucos as follows: Le ηψ, : X X R be vecor valued fucos, b: X X R + ad φ : R Rare real valued fucos. Defo.5 The fuco θ : X R s sad o be geeralzed η -bovex of order ( ) a x S wh resec o ags b, φ, η ad ψ f here exs a vecor ad a cosa k Rsuch ha for all x S bxx (, ) φθ [ ( x) θ( x) + r θ( x) r] η θ + θ + ψ ( xx, )[ ( x) ( x) r] k ( xx, ). r R Reark. If k >, he he fuco θ s called srogly geeralzed η - bovex of order. If k <, he he fuco θ s called weakly geeralzed η - bovex of order. If k = ad addo b =, φ = I (dey a), we oba he defo of η bovex fucos [4].

5 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy 5 Reark.3 If r =, k =, we oba he defo of uvexy [5]. If b =, k =, = ad φ = I, he he defo of geeralzed hgher order η bovexy reduces o he defo of vexy []. We ow rese he followg obvous lcaos of he above defo. Defo.6 The fuco θ : X R s sad o be geeralzed η seudo bovex of order ( ) a r x R ad a cosa k R S wh resec o ags b, φ, η ad ψ f here exs a vecor such ha for all x S η θ + θ + ψ ( xx, )[ ( x) ( x) r] k ( xx, ) les bxx (, ) φθ [ ( x) θ( x) + r θ( x) r] or euvalely bxx (, ) φθ [ ( x) θ( x) + r θ( x) r] < les η ( xx, )[ θ( x) + θ( x) r] + k ψ( xx, ) < Defo.7 The fuco θ : X R s sad o be geeralzed η -srcly seudo bovex of order ( ) a x S wh resec o ags b, φ, η ad ψ f here exs a vecor r R ad a cosa k R such ha for all x S η θ + θ + ψ ( xx, )[ ( x) ( x) r] k ( xx, ) les bxx (, ) φθ [ ( x) θ( x) + r θ( x) r] >. Defo.8 The fuco θ : X R s sad o be geeralzed η - uas bovex of order ( ) a r x R ad a cosa k R S wh resec o ags b, φ, η ad ψ f here exs a vecor such ha for all x S bxx (, ) φθ [ ( x) θ( x) + r θ( x) r] les η ( xx, )[ θ( x) + θ( x) r] + k ψ( xx, ). 3. OPTIMALITY We ow derve he followg ecessary oaly codos for (MOP). Theore 3. If x s a src zer for (MOP) ad assue ha Abade cosra ualfcao a x, where y R, v R, =,,..., + λ R+ G I sasfes he I = { : G ( x ) = }. The, here exs, λ = ad z R, =,,... such ha =

6 6 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy λ f( x ) + λbz + y g ( x ) + ycv = = = = = (3.) yg x = = (3.) ( ),,,..., zbz, =,... (3.3) ( vcv), =,,..., (3.4) x Bz = ( x Bx ), =,..., (3.5) vcx = ( x Cx ), =,,..., (3.6) Proof Sce f, =,,..., are dffereable fucos, B, =,,..., are osve se-defe arces, we have fro [9] ha he fucos θ ( x) = f( x) + ( x Bx), =,,..., are uas-dffereable, hece locally Lschz ad have drecoal dervaves θ ( x; d) for all d R, =,,...,. Therefore θ, =,,..., sasfy he codos of Lea.. Fro Lea. ad Abade cosra ualfcao, follows ha he syse ρ d <, =,,..., wd, I, s cosse for all ρ δθ( x ), =,,..., ad w δ G( x ), I. Therefore by basc alerave heore [3], here exss λ, =,,..., o all zero ad y, I such ha: λρ + yw = (3.7) = I for all ( ρ,..., ρ ) = ρ δθ( x ) ad I, we ca rewre (3.7) as w δ G ( x ), I. Seg y = for all o λρ + yw = (3.8) = = yg( x) =, =,,..., (3.9) Bu δθ ( x ), =,,..., s he se

7 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy 7 for soe { f x + Bz zbz x Bz = x Bx } z ( ) :, ( ), R. Slarly δ G ( x ) s he se { g x + Cv vcv vcx = x Cx I} ( ) :, ( ),, Hece fro (3.) ad (3.), we have (3.) (3.) λ f x λbz y g x ycv = = = = ( ) + + ( ) + = yg x ( ) =, =,,..., zbz, =,... ( vcv), =,,..., x Bz = ( x Bx ), =,..., vcx = ( x Cx ), =,,...,. 4. DUALITY We ow roose he followg secod order Mod-Wer ye dual for (MOP). (MD)Maxze ( f( u) + zbu r f( u) r,..., f u z B u r f u r subec o ( ) + ( ) ) λ( f() u + Bz + f()) u r + y( g () u + Cv + g ()) u r = = = (4.) y( g ( u) + vcu r g ( u) r) (4.) zbz, =,... (4.3) ( vcv), =,,..., (4.4) y, =,...,, λ, =,...,, λ = =

8 8 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy Theore 4. (Weak dualy) Le x be feasble for (MOP) ad ( u, λ, y, z, v, r) be feasble for (MD). Furher suose ha. ( f (.) + Bz), =,,..., be geeralzed η seudo bovex of order a u wh resec o b, φ, η ad ψ, where b > for all =,,...,.. y ( g () + C v ), =,,..., be geeralzed η uas bovex of order a u wh resec o b, φ, η ad ψ.. a φ ( a), =,..., ad a< φ ( a) <, =,..., v. λ k + k. = = The f ( x) + ( x Bx) < / f( u) + u Bz r f( u) r (4.5) Proof Le x be ay feasble soluo for (MOP) ad ( u, λ, y, z, v, r) be ay feasble soluo for (MD). The we have y ( g ( x) + ( x Cx) ) y( g ( u) + vcu r g ( u) r), =,..., Usg relao (4.4) ad Lea., we have y ( g ( x) + vcx) y( g ( u) + vcu r g ( u) r), =,...,, whch ca be rewre as y( g ( x) + vcx) y( g ( u) + vcu) + r yg ( u) r, =,,..., (4.6) Sce a φ ( a) ad b ( x, u), =,..., ; (4.6) yelds b x u y g x v C x y g u v C u r y g u r (, ) φ [ ( ( ) + ) ( ( ) + ) + ( ) )] O usg geeralzed η uas bovexy of order a u for y ( g () + C v ) wh resec o b, φ, η ad ψ, =,,...,, we have η x u y g u + y C v + y g u r + k x u =. (, )[ ( ) ( ) ] ψ(, ),,..., The above eualy yelds

9 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy 9 η ( xu, )[ ( yg ( u) + ycv + yg ( ur ) )] ( k) ψ( xu, ) = = (4.7) O usg (4.), he eualy (4.7) yelds ( x, u)[ ( f( u) + Bz + f( u) r] ( k) ( x, u) = = (4.8) η λ ψ Corary o he resul of he heore, le f( x) + ( x Bx) < f( u) + u Bz r f( u) r, =,..., Usg lea., we have f( x) + x Bz < f( u) + u Bz r f( u) r, =,..., (4.9) Sce a< φ ( a) < ad b > for all =,...,, he euales (4.9) lead o b( x, u) φ [( f( x) + x Bz) f( u) u Bz + r f( u) r] <, =,.., Fro geeralzed η seudo bovexy of order for ( f () + Bz) a u wh resec o b, φ, η ad ψ, =,,...,, we have η x u f u B z f u r k x u (, )[ ( ) + + ( ) ] + ψ(, ) <, =,..., Sce λ, =,,..., ad = λ =, we oba ( xu, )[ ( f( u) + Bz + f( ur ) )] + ( k) ( xu, ) < = =. η λ λ ψ Usg hyohess (v), he above eualy yelds ( x, u)[ ( f( u) + Bz + f( u) r)] < ( k) ( x, u) = =, η λ ψ a coradco o (4.8). Hece f x x Bx f u u Bz r f u r ( ) + ( ) < / ( ) + ( ). Theore 4. (srog dualy) Le x be a src zer for (MOP) ad assue ha Abade cosra ualfcao holds a x. The, here exs λ R+, y R+, z R, v R such ha ( x, λ, y, z, v, r = ) s feasble for (MD) ad he corresodg values of (MOP) ad (MD) are eual. Furher, f he assuos of

10 3 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy weak dualy Theore 4. hold, he (MD). ( x,, y, z, v, r ) λ = s a src axzer for Proof Sce x s a src zer for (MOP) ad Abade cosra ualfcao s sasfed a x, he by Theore 3. here exs λ R+, y R+, z R, v R, such ha λ f x λ Bz y g x ycv = = = = y ( ( ) ( g x + x Cx ) ) =, =,..., ( ) + + ( ) + = z Bz, =,... ( ), =,,..., v C v x Bz = ( x Bx ), =,..., v C x = ( x C x ), =,,..., = y, λ, λ =. Hece ( x, λ, y, z, v, r = ) s feasble for (MD) ad he corresodg values of obecve fucos are eual. Weak dualy Theore 4. les ha ( x, λ, y, z, v, r = ) s a src axzer for (MD). Theore 4.3 (src coverse dualy) Le x ad ( u, λ, y, v, z, r ) be src exrea for (MOP) ad (MD) resecvely, such ha ( ( ) ( ) ) ( ( ) ( ) ) λ f x + x Bx = λ f u + u Bz r f u r = = (4.) Furher, suose ha. y( g () + Cv) be geeralzed η uas bovex of order wh resec o b, φ, η ad ψ, =,,..., a u.. λ ( f() + Bz ) be geeralzed η src seudo bovex of order = wh resec o b, φ, η ad ψ a u.. a φ ( a), =,..., ad φ ( a) > a >.

11 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy 3 v. k+ k. = The x = u, ha s, u s a src zer for (MOP). Proof Suose ha x u. Sce x s feasble for (MOP) ad for =,...,, ( u,, y, v, z, r ) λ s feasble for (MD), we have y( g ( x ) + ( x Cx ) ) y( g ( u ) + v Cu r g ( u ) r ). Usg Lea., for =,...,, we have y( g ( x ) + v Cx ) y( g ( u ) + v Cu r g ( u ) r ) (4.) Sce b ( x, u ), euales (4.) alog wh hyohess () yelds b x u y g x v C x y g u v C u r y g u r (, ) φ [ ( ( ) + ) ( ( ) + ) + ( ) )] O usg geeralzed η uas bovexy of order a u for resec o b, φ, η ad ψ, =,,...,, we have y ( g () + C v ) wh η x u y g u y C v y g u r k x u (, )[ ( ) + + ( ) ] + ψ(, ), =,..., The above eualy yelds η ( x, u)[ ( yg ( u) + ycv + yg ( u) r)] ( k) ψ( x, u) = =. Usg he dual cosra (4.) he above eualy, we have ( x, u )[ ( f( u ) + Bz + f( u ) r ] ( k) ( x, u ) = = η λ ψ Usg hyohess (v), we have ( x, u )[ ( f( u ) + Bz + f( u ) r ] + k ( x, u ) = η λ ψ.

12 3 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy Now geeralzed η src seudo bovexy of order a u for he fuco λ ( f() + Bz ) wh resec o, = Sce b φ, η ad ψ les (, ) φ[ λ ( ( ) + ) λ ( ( ) + ) = = bx u f x x Bz f u u Bz + r ( f ( u )) r ] >. λ = bx (, u ) >, he above eualy alog wh hyohess () yelds λ + > λ + = = ( f ( x ) x B z ) ( f ( u ) u B z r f ( u ) r), whch o usg Lea. coradcs (4.). 5. A FRACTIONAL CASE We ow cosder he followg uas-dffereable ulobecve fracoal rograg roble (MOFP) whch he cooes of he obecve fucos are he raos of he fucos ha are he sus of dffereable ers ad suare roo ers of cera osve se-defe uadrac fors, whereas he cosrag fucos are he sae as hose for (MOP). f( x) ( x B ( ) ( ) x) f x x Bx (MOFP) Maxze,..., h( x) + ( x Dx) h( x) + ( x Dx) subec o G ( x), =,,...,, where G x g x x C x ( ) = ( ) + ( ), =,,...,. f : X R, h : X R, =,,..., ad g : X R, =,,... are wce dffereable; ad B, D, =,,..., ad C, =,,..., are osve se-defe syerc arces. Le S be he se of all feasble soluos of (MOFP). We also assue ha f( x) ( x Bx) ad fdg src zer. h ( x) + ( x D x) >, =,,...,. Here zao eas We rese he followg wo dualy odels for (MOFP): (MD) Mze ( σ,..., σ ) subec o = + λ [ f ( u) σ h ( u) B z σ D w f ( u) r σ h ( u) r]

13 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy 33 y( g( u) Cv g( u) r) (5.) = + + = f( u) σh( u) u Bz σu Dw r ( f( u) σh( u)) r + + y( g ( u) vcu) r ( yg ( u)) r = =,,,..., zbz, wbw, =,... ( vcv), =,,..., = (5.) f( u) u Bz σ =, =,,..., h ( u) + u Dw (5.3) y, =,...,, λ, =,...,, λ = = (MD) Mze ( σ,..., σ ) subec o = + λ [ f ( u) σ h ( u) B z σ D w f ( u) r σ h ( u) r] y( g( u) Cv g( u) r) (5.4) = + + = f( u) σh( u) u Bz σu Dw r ( f( u) σh( u)) r, =,,..., y( g ( u) + vcu) r yg ( u) r (5.5) zbz, wbw, =,... ( vcv), =,,..., f( u) u Bz σ =, =,,..., h ( u) + u Dw y, =,...,, λ, =,...,, λ =. =

14 34 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy Dualy resuls bewee (MOFP) ad s corresodg wo duals ca be esablshed o he sae les as hose obaed he case of ulobecve ozao roble (MOP). 6. CONCLUSION I hs aer, we have suded a secod order Mod-Wer ye dual for a uasdffereable rograg roble wh suare roo ers he obecve as well as he cosrag fucos. For hs urose, we have roduced he oo of geeralzed hgher order η bovexy. We have also cosdered a fracoal case. The resuls ca easly be exeded o secod order Magasara ye dual. I would be eresg o exed he resuls for oher classes of ozao robles, vz. ax rograg roble ad ax fracoal rograg roble. Ackowledgee : The auhors would lke o hak Prof. Davder Bhaa (Red.) ad Dr. Paka Gua, Deare of Oeraoal Research, Uversy of Delh for her kee eres ad couous hel hroughou he rearao of hs arcle. REFERENCES [] Ahad, I., ad Husa, Z., Secod order (F, α, ρ, d) covexy ad dualy ulobecve Prograg, Ifor. Sc., 76 (6) [] Ahad, I., Husa, Z., ad Al-Hoda, S., Secod order dualy odffereable fracoal Prograg, Nolear Aal. Real World Al., () 3-. [3] Bazaraa, M.S., Sheral, H.D., ad shey, C.M., Nolear rograg: Theory ad Algorhs, Joh Wley ad Sos, New York, 993. [4] Becor, C. R., ad Chadra, S., Geeralzed bovexy ad hgher order dualy for fracoal Prograg, Osearch, 4 (987) [5] Becor, C. R., Suea, S. K., ad Gua, S., Uvex fucos ad uvex olear rograg, Proceedgs of he Adsrave Sceces Assocao of Caada, (99) 5-4. [6] Bhaa, D., A oe o dualy heore for olear rograg roble, Maagee Sc., 6 (97) [7] Chadra, S., Crave B. D., ad Mod, B., Geeralzed cocavy ad dualy wh a suare roo er, Ozao 6 (5) (985) [8] Chchuluu, A., ad Pardalos, P. M., A survey of ulobecve ozao, A. Oer. Res., 54 (7) 9-5. [9] Crave, B.D., ad Mod, B., Suffce Frz-Joh ozao codos for odffereable covex rograg, J. Aus. Mah. Soc., Seres B, 9 (976) [] Crave, B. D., O uasdffereable ozao, J. Aus. Mah. Soc., seres A, 4 (986) [] Eseberg, E., Suor of a covex fuco, Bull. Aer. Mah. Soc., 68 (96) [] Haso, M. A., O suffcecy of Kuh Tucker codos, J. Mah. Aal. Al., 8 (98) [3] Haso, M. A., Secod order vexy ad dualy aheacal rograg, Osearch, 3 (993) [4] Jayswal, A., Kuar, D., ad Kuar, R., Secod order dualy for odffereable ulobecve rograg roble volvg (F, α, ρ, d) V ye I fucos, O. Le., 4 () () -6.

15 R.R. Sahay, G. Bhaa / Oaly Ad Secod Order Dualy 35 [5] Magasara, O. L., Secod ad hgher order dualy olear rograg, J. Mah. Aal. Al., 5 (975) [6] Mod, B., Husa, I., ad Prasad, M.V.D., Dualy for a class of odffereable ulle obecve rograg robles, J. If. O. Sceces, 9 (3) (988) [7] Schecher, M., More o subgrade dualy, J. Mah. Aal. Al., 7 (979) 5-6. [8] Sha, S. M., A dualy heore for olear rograg, Maagee Sc., (966) [9] Zhag, J., ad Mod, B, Dualy for a odffereable rograg roble, Bull. Aus. Mah. Soc. 55 (997) 9-44.