Multiobjective Duality in Variational Problems with Higher Order Derivatives

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$Maoo Deare of Maheacs Jayee sue of Egeerg ad echology Gua da Deare of Sascs Uversy of Kashr Sragar da E-al: {husa ruaa_research}@yahoo.$

1 Coucaos ad Newor do:0.436/c.00.0 Publshed Ole May 00 (h:// Mulobjecve Dualy Varaoal Probles wh Hgher Order Dervaves Absrac qbal Husa Ruaa G. Maoo Deare of Maheacs Jayee sue of Egeerg ad echology Gua da Deare of Sascs Uversy of Kashr Sragar da E-al: {husa Receved Deceber 6 009; revsed Arl 5 00; acceed Arl A ulobjecve varaoal roble volvg hgher order dervaves s cosdered ad oaly codos for hs roble are derved. A Mod-Wer ye dual o hs roble s cosruced ad varous dualy resuls are valdaed uder geeralzed vey. Soe secal cases are eoed ad s also oed ou ha our resuls ca be cosdered as a dyac geeralzao of he already esg resuls olear rograg. Keywords: Mulobjecve Varaoal Proble; Effcecy Dualy Pseudovey Quasvey Nolear Prograg. roduco Calculus of varao s a owerful echque for he soluo of varous robles aearg dyacs of rgd bodes ozao of orbs heory of varaos ad ay oher felds. he subjecs whose orace s fas growg scece ad egeerg rarly cocer wh fdg oal of a defe egral volvg a cera fuco subjec o fed o boudary codos. [] Coura ad Hlber quog a earler wor of Fredrchs [] gave a dual relaosh for a sle ye of ucosraed varaoal roble. Subsequely Haso [3] oed ou ha soe of he dualy resuls of aheacal rograg have aalogous varaoal calculus. Elorg hs relaosh bewee aheacal rograg ad he classcal calculus of varaos Mod ad Haso [4] forulaed a cosraed varaoal roble as a aheacal rograg roble ad usg Valee s [5] oaly codos for he sae reseed s Wolfe ye dual varaoal roble for valdag varous dualy resuls uder covey. Laer Becor Chadra ad Husa [6] suded Mod-Wer ye dualy for he roble of [4] for weaeg s covey requree. [7] Chadra Crave ad Husa suded oaly ad dualy for a class of o-dffereable varaoal roble wh o-dffereable er he egrad of he objecve fucoal whle [8] hey derved oaly codos ad dualy resuls for a cosraed varaoal roble havg ers wh arbrary ors he objecve as well as cosraed fucos. Recely Husa ad Jabee [9] suded a wder class of varaoal roble whch he arc fuco s wce dffereable by eedg he oo of vey gve [0]. hey obaed Frz Joh as well as Karush- Kuh-ucer ecessary oaly codos as a alcao of Karush-Kuh-ucer oaly codos suded varous dualy resuls for Wolfe ad Mod ad Wer ye odels. sgle objecve rograg we us sele o a sgle objecve such as zg cos or azg rof. However geerally ay real world robles ca be defed wh ulle coflcg crera e.g. he robles of ol refery schedulg roduco lag orfolo seleco ad ay ohers ca be odelled as ulobjecve rograg robles. Dualy resuls are very useful he develoe of uercal algorhs for solvg cera classes of ozao robles. Dualy for ulobjecve varaoal roble has bee suded by a uber of auhors oably Becor ad Husa [] Che [] ad ay ohers ced hese refereces. Alcaos of dualy heory are roe hyscs ecoocs aagee sceces ec. Sce aheacal rograg ad classcal calculus of varaos have udergoe deede develoe s fel ha uual adaao of deas ad ech- Coyrgh 00 ScRes.

2 ques ay rove useful. Movaed wh hs dea hs eoso we roose o sudy oaly crera ad dualy for a wder class of ulobjecve varaoal robles volvg hgher order dervave. hese resuls o oly geeralze he resuls of Husa ad Jabee [9] ad Becor ad Husa [] bu also rese a dyac geeralzao of soe of he resuls ulobjecve olear rograg already esg.. vey ad Geeralzed vey Cosder he real erval vey was roduced for fucos varaoal robles by Mod Chadra ad Husa [0] whle Mod ad Sar [3] defed vey for fucoals sead of fucos. Here we roduce eeded fors of defos of vey ad varous geeralzed vey for fucoal varaoal robles volvg hgher order dervaves. ab ad he cou- ously dffereable fuco : R R R R where s wce dffereable wh s frs ad secod order dervaves ad resecvely. f... he grade vecors of f wh resec o ad resecvely deoed by DEFNON. (vey): f here ess vecor fuco uu R wh 0 ad u ad D 0 for u such ha for a scalar fuco he fucoal Φ d sasfes uu Φ -Φ D D d Φ s sad o be ve ad o wh resec o η. Here D s a dffereao oeraor defed laer. DEFNON. (Pseudovey): Φ s sad o be seudove ad wh resec o f D. HUSAN E AL. 39 D d0 les Φ uu Φ. DEFNON 3. (Quas-ve): he fucoal Φ s sad o quas-ve ad wh resec o η f uu Φ Φ les D D d0 3. Varaoal Proble ad Oaly Codos Before sag our varaoal roble ad dervg s ecessary oaly codo we eo he followg coveos for vecors ad y -desoal Euclda sace R wll be used hroughou he aalyss of hs research. y y. y y. y y bu y y s he egao of y For y R y ad y have he usual eag. hs seco we rese he followg varaoal roble whose oaly codos wll be derved ad dualy wll be vesgaed he subseque secos: (VPE) Mze f d... f d a0 b () a 0 b () g 0 (3) h 0 (4) where f : R R R R g : R R R R ad h: R R R R are couously dffereable fucos ad X desgaes he sace of ecewse fucos : R os- sessg dervaves ad wh he or D D gve by where he dffereao oeraor D s Coyrgh 00 ScRes.

3 40. HUSAN E AL. where α s gve boudary value; hus dscoues. u D u s ds he resuls o follow we use C R a d D ece a d he sace of couous fucos : R wh he ufor or su o deoe ; he aral dervaves of g ad h are ad arces resecvely; suerscr deoes ar rasose. We requre he followg defo of effce soluo for our furher aalyss. DEFNON 4. (Effce Soluo): A feasble soluo s effce for (VPE) f here es o oher feas- P... ble for (VPE) such ha for soe f d f d ad f d f d for all j P j. relao o (VPE) we roduce he followg se of robles P for each r... he sr of r [4] wh a sgle objecve r P r Mze f d b a 0 b a 0 g 0 h 0 f d f d... r he followg lea ca be roved o he les of Chaog ad Haes [4]. LEMMA : s a effce soluo of (VPE) f ad oly f s a oal soluo of Pr for each r.... d (P 0 ) Mze a b 0 b a 0 g 0 h 0 where : R R R R. PROPOSON. [9]: (Frz Joh Oaly Codos) f s a oal soluo of (P 0 ) ad h as X o he subsace of C R he here ess Lagrage uller R he ecewse so oh y : R ad z : R such ha D y g z y g z h h D y g z h 0 y g 0 y 0 y z 0. f he he above oal y codos wll reduce o he Karush-Kuh-ucer ye oaly codos ad he soluo s referred o as a oral soluo. We ow esablsh he followg heore ha gves he ecessary oaly codos for (VPE). HEOREM : (Frz-Joh Codos): Le be a h effce soluo of (VPE) ad as X o he subsace of C R R ad he ecewse sooh : z : R such ha he here es y R ad g z D f y g z h 0 f y g z h D f y h (5) y g 0 (6) y 0. (7) y z 0. (8) PROOF: Sce s a effce soluo of (VPE) by Lea s a oal soluo of ( P r ) for each r.... Fro Prooso follows ha here es scalars r r r ad ecewse sooh fuco y: R ad z : R such ha Coyrgh 00 ScRes.

4 . HUSAN E AL. 4 rr r r jr j lr l f f y g z h j l r rr r r jr j lr l D f f y g z h j l r rr r r jr j lr l D f f y g z h j l r y g 0 0 r r r r r r 0 y y y r r r r r r r r lr z y y y z z Sug over r we have 0. r jr j lr l f y g z h r r j r l r jr j lr l D f y g z h r r j r l 0 r jr j lr l D f y g z h r r j r l y g 0 r r r r ; y y 0 r r r r r r y y z r r Seg l r ; ; lr z 0. r r j jr r r l lr z z we have y y ad gz h D fy g D f y g z h 0 f y z h y g 0 y 0 y z Mod-Wer ye Dualy hs seco we cosder he followg varaoal roble volvg hgher order dervaves by suressg he equaly cosra (VPE). (VP) Mze f d... f d a 0 a 0 b b g 0 We forulae he followg Mod-Wer ye dual o he roble (VP) ad esablsh varous dualy resuls uder vey defed he recedg seco. (M-WD) Maze... f uuu d f uuu d a 0 b (9) f y g D f y a 0 b (0) g D f y g 0 () y g uuu d0 () y 0 (3) 0. HEOREM. (Wea Dualy): Le X be feasble for (VP) ad u y be feasble for (M-WD) f for allfeasble u y f uuu d s seudove a d y (4) g uuu d s quas-ve wh resec o he sae η. he Coyrgh 00 ScRes.

5 4. HUSAN E AL. f d f uuud. PROOF: he relaos g 0 y 0 ly y g d y g uuu d hs because of he quas-vey of y guuu d les ha u u y gu 0 y g D y g D b y gud y gu a u Dy g d D y g D Dy g d u b u a (By egrao by ars) Usg he boudary codos whch gves D 0 a a b y Dy gud Dy gu d b gu D y gud a (By egrao by ars) Usg he boudary codos whch gve a a b d D 0 y gud Dy u u g d D y g d0 u u u y g Dy g D y g d0 Fro Equao () hs yelds f D f D f d0 hs by egrao by ars ad he usg boudary codos gves f D f D f d0 hs vew of sedovey of f d les ha f u f d For hs follows u u d. f d f uuud. HEOREM 3. (Srog Dualy): f s a feasble soluo for (VP) ad assue ha s a effce soluo ad for a leas oe P sasfes a regulary codo for [7] for P. he here ess oe R y R such ha y s effce fo r (VD). Furher f he assuos of heore are sa sfed he y s a effce soluo of (VD). PROOF: Sce s effce soluo by Lea s a oal soluo of P. By Prooso hs les ha here e ecewse sooh y : R ss such ha ad f Df D f f Df D f y g Dy g D y g 0 f y g D f y g D f y g 0 (5) (6) y g d 0 (7) y 0 (8) y 0 (9) Fro (7) we have y g d 0 (0) Equaos (6) (7) ad (8) ly ha y s feasble for (M-WD). he equaly of objecve fuc- Effcecy of y s edae fro he alcao of heore. oal of ral ad dual robles s obvous fro her forulaos. As [4] by eloyg cha rule calculus ca be easly see ha he eresso f y g D f y g D f y g ay be regarded as a fuco of varable s yyy 3 ad where D ad y D y. ha s we ca wre Coyrgh 00 ScRes.

6 yy y f y g D f y g D f y g order o rove coverse dualy bewee (VP) ad (M-WD) he sace X s ow relaced by a saller sace X of ecewse sooh hrce dffereable fuco : R wh he or 3 D. he roble (M-WD) wre as Mze D D ay ow be brefly f d f d a 0 b a 0 b yyy 0 y g d 0 y 0 Cosder y y y 0 as defg a ag : X YR B where Y s a sace of ecewse wce dffereable fuco ad B s he Baach Sace. order o aly heore o he roble (M-WD) he fe desoal equaly us be resrced. he followg heore we use o rerese he Frèchè dervave y y y y. HEOREM 4. (Coverse Dualy): Le D be a effce soluo wh X yy ad R ad have a (wea*) closed rage hyohess. Le f ad g be wce couously dffereable. Assue ha (H ) f d be seudove ad y gd be quas-ve wh resec o sae. (H ) D D 3 D 0.. HUSAN E AL (H 3 ) f Df D f... are learly deede. he s a effce soluo of (VP). Proof: Sce y where X ad havg a clo sed rage s a effce soluo of (M-WD) by heore les ha here es R R R ad ecewse sooh : R R ad : R R sasfyg he followg codos. f Df D f y g y g D y g D D D y D 3 0 () y D yg 0 D () f Df D f 0 (3) y gd 0 (4) 0 y 0 (5) 0 ad 0 (6) Sce 0 0 whch les 0 hs yelds fro (3) f Df 0 (7) D f Usg he equaly cosra () () we have f Df D f 3 0 (8) D D D Posullyg Equao () by (7) ( 8) we ge 3 ad usg D D 0 hs by hyohess (H ) les Also fro (8) we have f Df D f 0 0 hs because of lear deedece of f Df D f... gves 0 Now suose 0 he fro () ad (9) we have 0 ad 0 resecvely. hs les (9) 0 whch s he Coyrgh 00 ScRes.

7 44 coradco o 0. Hece 0 ad by (9) we have 0. he relao ( ) cojuco wh 0 ad 0 gves g 0 hs les he feasbly of for (VP) ad s efof heore fcecy s evde fro ad alcao. 5. Naural Boudary Values he dualy resuls obaed he recedg secos ca easly be eeded o he ulobjecve varaoal robles wh aural boudary values raher ha fed ed os. Pral (P ) Mze f d... f d Dual (D ) Maze g 0 f d... f d f y g D f y g D f y g 0 y g 0 a a ad y g 0 a a ad y Nolear Prograg b b f he robles (P ) ad (D ) are deede of he hey wll reduce o he followg ulobjecve ol- ear rograg robles suded [5] (NP): Mze f (ND): Maze f g 0. f y g 0 0 y 0.. HUSAN E AL. 7. Refereces [] R. Coura ad D. Hlb er Mehods of Maheacal Physcs Wley New Yor Vol [] K. O. Fredrchs E Verfrahre der Varaos-Rechug das Mu ees egrals Mau ees Adere Ausdruces Dazusella Göge Nachrche 99. [3] M. A. Haso Bods for Fucoally Cove Oal Corol Probles Joural of Maheacal Aalyss ad Alcaos Vol. 8 No. February [4] B. Mod ad M. A. Haso Dualy for Varaoal Probles Joural of Maheacal Aalyss ad Alcaos Vol. 8 No. May [5] F. A. Valee he Proble of Lagrage wh Dffereal equales as Added Sde Codos Corbuos o he Calculus of Varaos Uversy of Chcago Press [6] C. R. Becor S. Chadra ad. Husa Geeralzed Cocavy ad Dualy Couous Prograg Ulas Maheaca Vol [7] S. Chadra B. D. Crave ad. Husa A Class of Nodffereable Couous Prograg Probles Joural of Maheacal Aalyss Alcaos Vol. 07 No. Arl [8] S. Chadra B. D. Crave ad. Husa Couous Prograg Coag Arbrary Nors Joural of Ausrala Maheacal Socey (Seres A) Vol. 39 No [9]. Husa ad Z. Jabee O Varaoal Probles volvg Hgher Order Dervaves Joural of Aled Maheacs ad Coug Vol. 7 No. - March [0] B. Mod ad S. Chadra ad. Husa Dualy of Varaoal Probles wh vey Joural of Maheacal Aalyss ad Alcaos Vol. 34 No. Seeber [] C. R. Becor ad. H. Husa Dualy for Mulobjecve Varaoal Probles Joural of Maheacal Aalyss ad Alcaos Vol. 66 No. May [] X. H. Che Dualy for Mulobjecve Varaoal Probles wh vey Joural of Maheacal Aalyss ad Alacos Vol. 03 No. Ocober [3] B. Mod ad. Sar Dualy wh vey for a Class of Nodffereable Sac ad Couous Prograg Probles Joural of Maheacal Aalyss ad Alcaos Vol [4] V. Chaog ad Y. Y. Haes Mulobjecve Decso Mag: heory ad Mehodology Norh Hollad New Yor 983. [5] R. R. Egudo ad M. A. Haso Mulobjecve Dualy wh vey Joural of Maheacal Aalyss ad Alacos Vol. 6 No. Seeber Coyrgh 00 ScRes.

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

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