PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE OPTIMIZATION PROBLEM

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1 Joual o Mathematcal Sceces: Advaces ad Applcatos Volume 6 Numbe 00 Pages 77-9 PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE OPTIMIZATION PROBLEM DAU XUAN LUONG ad TRAN VAN AN Depatmet o Natual Sceces Quag Nh Teache Tag College Quag Nh Vetam e-mal: dauualuog@gmal.com Depatmet o Mathematcs Vh Uvest 8 Le Dua Vh Ct Vetam Abstact I ths pape we appl the pealt ucto method to the multobjectve optmzato poblem ode to tasom a costaed poblem eeed to as the ogal poblem to a sequece o smple costaed o ucostaed poblems eeed to as the pealzed poblems. We show that a cluste pot o a sequece o wea ecet solutos o the pealzed poblems s a wea ecet soluto o the ogal poblem. Moeove ude ceta assumptos o the easble ego D ad the objectve ucto we ca show that eve pealzed poblem has a wea ecet soluto ad that a sequece o wea ecet solutos o the pealzed poblems alwas has at least oe cluste pot.. Itoducto The pealt ucto method s ote emploed to tasom a costaed poblem to a sequece o smple costaed (o eve 00 Mathematcs Subject Classcato: 90C9 90C5 65K05 65K0. Kewods ad phases: multobjectve optmzato olea optmzato pealt ucto wea ecet soluto. Receved Septembe Scetc Advaces Publshes

2 78 DAU XUAN LUONG ad TRAN VAN AN ucostaed poblems so that a sequece o solutos o the smple costaed poblems coveges to a soluto o the costaed poblem. Thee have bee etesve studes o how to appl the pealt ucto method to the olea optmzato poblem (see o stace [ ]. The pealt ucto method has bee also emploed to solve the multobjectve optmzato poblem (see [ ]. I [9] the wea ecet solutos o the multobjectve optmzato poblem MOP( D wee studed togethe wth the epoetal pealt uctos. It was show [9] that s a cluste pot o a sequece o wea ecet solutos o the pealzed poblems ad s easble (.e. D the s a wea ecet soluto o the ogal poblem. The easblt o was assumed [9]. We ae hee usg the eteo pealt ucto ad ae able to show the ollowg: ( s a cluste pot o a sequece o wea ecet solutos o the pealzed poblems the s easble; ( such a pot s a wea ecet soluto o the ogal poblem; (3 ude ceta assumptos mposed o the objectve ucto eve pealzed poblem has a wea ecet soluto ad eve sequece o such wea ecet solutos has a cluste pot whch tu s a wea ecet soluto o the ogal poblem. The pape s ogazed as ollows. Necessa detos otatos ad some bass esults o the multobjectve optmzato poblem ae gve Secto. Secto 3 pesets ou ma esults. The pape s cocluded Secto 4.. Pelmaes T Fo ( R ad ( R we adopt the ollowg covetos: T < < </ :. Let deote the Eucldea om o amel

3 PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE 79. The e poduct o ad s deed as. T R Let R { ( : 0 }. Let D be a oempt subset o R. We heceoth assume that D s closed ad cove. We cosde the ollowg D-costaed multobjectve optmzato poblem MOP( D : m ( ( ( ( D eeed to as the easble ego o MOP(. whee : R R ae abta uctos o R. D s D I D the s called a easble pot o MOP( D. I D R the MOP( D s called a ucostaed multobjectve optmzato poblem. Deto.. The pot D s called a wea ecet soluto o MOP( D thee does ot est D satsg ( < (. Thus D s a wea ecet soluto o MOP( D ad ol o all D thee ests some de such that ( (. Suppose that F : R R s a mappg whose age s a set o eal matces. The D-costaed vecto vaatoal equalt poblem s deed as ollows: VVIP( D F : Fd D such that F( ( </ 0 D.

4 80 DAU XUAN LUONG ad TRAN VAN AN I D R the VVIP( D F s called a ucostaed vecto vaatoal equalt poblem. Let F : R R be the compoet uctos o F.e. F ( s the -th ow o the mat F (. The T F( ( ( F ( F (. Theeoe D D s a soluto o VVIP( D F ad ol o all T ( F ( F ( </ 0 o othe wods thee ests some de such that F ( 0. The ollowg theoem establshes the elatoshp betwee the wea ecet solutos o a multobjectve optmzato poblem ad the solutos o the coespodg vecto vaato equalt poblem ude the assumpto o the covet ad deetablt o the objectve ucto. Theoem. ([]. Let be cove ad deetable.e. each compoet o s cove ad deetable. The s a wea ecet soluto o MOP( D ad ol s a soluto o VVIP( D whee s the total devatve o. The ollowg esults o the estece o solutos o a multobjectve optmzato poblem ae useul o us late. Theoem.3 ([8]. Let be cove ad deetable. Futhemoe suppose that D s ubouded ad thee ests a D such that lm D ( a > 0. The MOP( D has a wea ecet soluto.

5 PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE 8 Deto.4 ([]. The ucto F : R R s sad to be mootoe o R each o ts compoets s mootoe.e. ( F ( 0 o all. Theoem.5 ([]. Let F : R R be a cotuous ad mootoe ucto o R. Assume that ( D s bouded o ( F s wea coecve o D amel thee ests a vecto vecto a D such that F s R ad a T lm s F( a. D The VVIP( D F has a soluto. Combg Theoems..3 ad.5 we deduce the ollowg sucet codto o the solvablt o a multobjectve optmzato poblem. Coolla.6. Let be cove ad deetable. Assume that oe o the ollowg holds: ( D s bouded ( D s ubouded ad thee ests a vecto a D such that s R ad a vecto lm s D ( a (3 D s ubouded ad thee ests a vecto a D such that lm D ( a > 0. The MOP( D has a wea ecet soluto.

6 8 DAU XUAN LUONG ad TRAN VAN AN Poo. Due to the covet o t ollows that s cotuous ad mootoe (see o stace Coolla 5.5. []. Let R F : R be the mappg wth compoets. The F s cotuous ad mootoe. I D s bouded b Theoem.5 VVIP( D F has a soluto. Usg Theoem. we deduce that s also a wea ecet soluto o MOP( D. Suppose that D s ubouded. I the secod popet holds the F s wea coecve o D. Theeoe b Theoem.5 VVIP( D F has a soluto. Hece aga b Theoem. s a wea ecet soluto o MOP( D. I the thd popet holds the b Theoem.3 MOP( D has a wea ecet soluto. 3. The Multobjectve Optmzato Poblem ad the Pealt Fuctos Fstl we dee ad stud the pealzed poblems. 3.. The pealzed poblems Deto 3.. Let D be a oempt subset o R. A ucto P : R R s called a pealt ucto o D t satses P( 0 P( > 0 D D. (3. I ths pape we assume that P s chose so that t s ot ol cove but also deetable. Fo stace D s deed as D { R : g ( 0 j m} (3. j whee g j : R R j m ae cotuous uctos we ca tae m P( [ ma { 0 g ( }]. (3.3 j j

7 PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE 83 It s staghtowad to ve that P deed as above ot ol s cove ad deetable o R but also satses (3.. Note that t s wellow that P s cove the o eve ad R. P( P( P( Now a set K D. Fo t > 0 we dee the ollowg pealzed poblem ( ( MOP : m K K whee tp. The ego K ca well be R ad such a case the pealzed poblem has o costat. Net we stud the estece o solutos o the pealzed poblem ( MOP K. Lemma 3.. Let : R R be cove ad deetable. Futhemoe assume that oe o the ollowg codtos holds: ( K s bouded ( K s ubouded ad thee ests a vecto a D such that s R ad a vecto lm s K ( a (3 K s ubouded ad thee ests a D such that lm K ( a > 0. The ( MOP K has a wea ecet soluto. Poo. It s clea that each tp s cove ad deetable. I K s bouded the b Coolla.6 ( MOP K has a wea ecet soluto.

8 84 DAU XUAN LUONG ad TRAN VAN AN Suppose that the secod codto s satsed. We have s ( t ( a s ( a t s P( a s s ( a t( P( P( a ( a as K. The secod equalt s due to the act that P ( 0 ad P ( a 0 as a D. Theeoe Coolla.6 mples that ( MOP K has a wea ecet soluto. Suppose that the thd codto s satsed. Fo each we have ( a ( a t P( a ( a t( P( P( a ( a. Theeoe lm ( a > 0. K Aga b Coolla.6 we coclude that ( MOP K has a wea ecet soluto. 3.. The covegece theoems Let S ad S ( t deote the soluto sets o MOP( D ad MOP ( K espectvel. Let { t } be a sequece o postve eal umbes whch mootocall teds to as.

9 PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE 85 S o Lemma 3.3. Assume that s cotuous ad that all N. Suppose that s a cluste pot o { }. The D. Poo. We pove ths lemma b cotadcto. Suppose that s the lmt o a subsequece m o ad that D. The { } m { } P ( > 0 ad hece P ( > ε o some ε > 0. Tae D. Sce S( thee ests t m Sce { } m m such that m ( m ( m. m m ( m thee ests a te sequece { } o dces all o whch have the same value sa o all N. To smpl the otato we assume that o all m N. Theeoe o all m N we have m m m m ( m ( m. (3.4 ( Sce P( m P( > ε o all m sucetl lage we have P( m > ε. Hece o m sucetl lage ( ( m m ( ( ( ( m m ( t ( P( m P( m ( m ( t ε m ( ( as m. Ths cotadcts (3.4. Note that hee P ( 0 as D. The ollowg theoem shows that a sequece o wea ecet solutos o the pealzed poblems coveges to a pot the s also a wea ecet soluto o the ogal poblem. Note that thee ests some N such that ( S D the t s eas to ve that s also a soluto o MOP( D.

10 86 DAU XUAN LUONG ad TRAN VAN AN S o Theoem 3.4. Assume that s cotuous ad that all N. The a cluste pot o the sequece { } s a wea ecet soluto o MOP( D. { }. Poo. We assume that s a cluste pot o the sequece Let m be a subsequece o whch coveges to. B { } m Lemma 3.3 we alead have D. { } Suppose o cotadcto that S. The thee ests D satsg ( < (. Sce m S thee ests m m such that m ( m ( m. m m Sce m { } thee ests a te sequece { m } o dces all o whch have the same value sa m o all N. Aga to smpl the otato we assume that the sequece { m } m tsel satses ths popet amel o all m N. Theeoe o all m N we have m m ( m ( m. (3.5 Sce ( < ( we deduce that ( ( < o some ε > 0. Sce m ε as m o sucetl lage m we have ( ( ( m <. ε Hece o m sucetl lage we have m ( m ( m ( ( m t ( P( P( m m

11 PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE 87 < ε t P ( ε m m whch cotadcts (3.5. Theoem 3.5. Let : R R be cove ad deetable. Futhemoe assume that oe o the ollowg codtos holds: ( K s bouded ( K s ubouded ad thee ests a vecto a D such that lm K ( a > 0. Assume also that S o all N. The the sequece { } has at least oe cluste pot ad eve cluste pot o ths sequece s a wea ecet soluto o MOP( D. Poo. Fst ote that b Lemma 3. we have S (. Theeoe the sequece { } stated the theoem s well-deed. I K s bouded the the sequece { } K s also bouded. Theeoe t has at least oe cluste pot. The clam that eve cluste pot o ths sequece s a wea ecet soluto o MOP( D ollows dectl om Theoem 3.4. t Now assume that K s ubouded ad thee ests a D such that lm K ( a > 0. We am to show that the sequece { } s bouded. Fo t > 0 let B ( t be the smallest closed ball R ceteed at the og such that o all B( t ad o all we have ( a > 0

12 88 DAU XUAN LUONG ad TRAN VAN AN o othe wods ( a t P( a > 0. Sce whe s sucetl lage ad ( a > 0 t P( a t( P( P( a 0 we deduce that B ( t has te adus. Net we show that S( t B. Ideed suppose o the cota that thee ests some S \ B. The b deto o B o all we have o equvaletl t ( a > 0 ( ( a < 0. Hece s ot a soluto o ( ( VVIP K. B Theoem. we deduce that S( t a cotadcto. Hece S B. O the othe had o t > t we have B B. Ideed o all B( t ad o all we have whch mples as ( a t P( a > 0 ( a t P( a > 0 (3.6 P ( a P( P( a 0.

13 PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE 89 B deto B s the smallest ball such that o all B the equalt (3.6 holds o all. Theeoe B s cotaed B (. t } Fall as { t s mootocall ceasg we have Theeoe o all N we have B B B. S B B(. Sce the adus o B s te we coclude that the sequece s bouded. We complete the poo. Eample 3.6. Let t { } D T { ( R : 0 0}. Fgue. The easble ego D.

14 DAU XUAN LUONG ad TRAN VAN AN 90 Tae P as (3.3: ( [ { }] [ { }] 0 ma 0 ma P ( ( ( ( ( ( (. III II I 0 D We have ( ( ( ( ( ( ( ( ( ( ( (. III 4 II 4 I 0 0 D P Let ( ( ( ( whee ( e (. e Cleal chose as above s cove ad deetable o. R Let R K ad. 0 D a The we have ( ( e ( e whch ae obvousl postve whe. Theeoe b Theoem 3.5 a sequece o wea ecet solutos o the pealzed poblems ( ( MOP K t has a cluste pot ad eve cluste pot o that sequece s a wea ecet soluto o (. MOP D

15 PENALTY FUNCTIONS FOR THE MULTIOBJECTIVE 9 4. Cocluso We vestgate the elatoshp betwee the set o wea ecet solutos o the ogal multobjectve optmzato poblem ad the sets o wea ecet solutos o the coespodg pealzed poblems. Theoem 3.4 shows that eve cluste pot o a sequece o wea ecet solutos o the pealzed poblems s a wea ecet soluto o the ogal poblem ude ol the cotut o. No covet o deetablt s equed. Howeve Theoem 3.5 eques such popetes o. Seveal authos have establshed esults o the estece o wea ecet solutos o a multobjectve optmzato poblem ude less stct equemets mposed o o stace whe s osmooth ad ocove (see o stace [7 8 ]. Based o these esults oe ma ela the assumptos mposed o the objectve ucto stated Theoem 3.5. Reeeces [] G. Y. Che ad X. Q. Yag The vecto complemeta poblem ad ts equvaleces wth the wea mmal elemet odeed spaces J. Math. Aal. Appl. 53 ( [] G. Y. Che ad B. D. Cave Estece ad cotut o solutos o vecto optmzato J. Math. Aal. Appl. 0 ( [3] R. Comett ad J. P. Dussault Stable epoetal-pealt algothm wth supelea covegece J. Optm. Theo Appl. 83( ( [4] J. P. Evas ad F. J. Gould A estece theoem o pealt ucto theo SIAM J. Cotol. ( [5] X. Q. Huag ad X. Q. Yag Nolea Lagaga o multobjectve optmzato ad applcatos to dualt ad eact pealzato SIAM J. Optm. 3(3 ( [6] X. Q. Huag X. Q. Yag ad K. L. Teo Covegece aalss o a class o pealt methods o vecto optmzato poblems wth coe costats J. Global Optm. 36(4 ( [7] K. R. Kazm Estece o solutos o vecto optmzato Appl. Math. Lett. 9 ( [8] G. M. Lee ad D. S. Km Estece o solutos o vecto optmzato poblems J. Optm. Theo Appl. 8(3 (

16 9 DAU XUAN LUONG ad TRAN VAN AN [9] S. Lu ad E. Feg The epoetal pealt ucto method o multobjectve pogammg poblems Optm. Methods Sotw. 5(5 ( [0] V. H. Ngue ad J. J. Stodot O the covegece ate o a pealt ucto method o epoetal tpe J. Optm. Theo Appl. 7(4 ( [] R. T. Rocaella Cove Aalss Pceto Uv. Pess Pceto New Jese 970. [] L. B. Satos G. Ruz-Gazo M. A. Rojas-Meda ad A. Rua-Lzaa Estece o weal ecet solutos osmooth vecto optmzato Appl. Math. Comput. 00( ( [3] D. J. Whte Multobjectve pogammg ad pealt uctos J. Optm. Theo Appl. 3(3 ( [4] W. I. Zagwll Nolea Pogammg: A Ued Appoach Petce Hall Eglewood Cls NJ 969. g

= y and Normed Linear Spaces

= y and Normed Linear Spaces 304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads