On Optimal Termination Rule for Primal-Dual Algorithm for Semi- Definite Programming

Transcription

1 Avalable ole at wwwelagareearchlbrarco Pelaga Reearch Lbrar Advace Aled Scece Reearch 6:4-3 ISSN: CODEN USA: AASRFC O Otal Terato Rule or Pral-Dual Algorth or Se- Dete Prograg BO Adejo ad E Ogala Deartet o Matheatcal Scece Kog State Uvert Agba _ ABSTRACT I th artcle we coare three revou terato rule or ral-dual hort te athollowg algorth or e-dete rograg rooed earler b Motero Adejo ad Adejo ad Sgh deedetl whch were baed o aale carred out deedetl b Fral ad Sgh et al or Kararar algorth or lear rograg Here we develo a ore ecet terato rule whch o leetato ave at leat 95% terato over that o Motero ad at leat 8% terato over that o Adejo ad Sgh Keword: Se-dete uer celg ucto ral-dual ethod ath-ollowg ethod NP-hard roble _ INTRODUCTION Not log ater Kararar [ 5 ] 984 develoed the rt ever rojectve teror ot algorth that guarateed terate that le the teror o the eable et t wa recogzed that teror ot ethod IPMS or LP ca be ued the ae wa or a atr vero o lear rograg roble LP a well th atr vero Kow a e-dete rogra SDP SDP are ute ote egeerg dcle tattc te ad cotrol gal roceg etc Roughl eag a SDP the ae a LP ecet that the cotrat are atr eualte tead o a et o calar eualte Thu SDP a eteo o lear rograg LP where the cooet we eualte betwee vector are relaced b atr eualte or euvaletl the rt orthat relaced b the coe o otve edete atrce Se-dete rograg ue everal tadard roble eg lear ad uadratc rograg ad d a alcato egeerg ad cobatoral otzato Although e-dete rogra are uch ore geeral tha lear rogra the are ot uch harder to olve Mot teror- ot-ethod or lear rograg have bee geeralzed to e-dete rogra A lear rograg thee ethod have oloal wort-cae colet ad eror ver well ractce I cotrol theor the are ver oular thee da ad or a NP-hard roble SDP ca be ued to obta eagul lower or uer boud Pelaga Reearch Lbrar 4

2 BO Adejo et al Adv Al Sc Re 6:4-3 I a e-dete rogra SDP we ze a lear ucto o a varable a atr eualt: where ze c T R ubject to ubject to F F F The roble data are the vector c R ad etrc atrce F F R The eualt g F ea that F otve e-dete e z T F z or all z R We call the eualt F a lear atr eualt LMI Now we let R deote -deoal Eucldea ace whle R deote the et o all atrce wth real etre I S deote the et o all real etrc atrce the c wth the Frobeu atrce er roduct o c ad deed a T c Trace c c I we dee a ucto ace ad the or j j j : S R a Q Q Q the S a ored lear that dee the ace called the Frobeu or I or a Q Q le that Q otve e-dete whle Q > le that Q otve dete the S { Q } Q c A S ad b R a SDP ca be tated a P ze ubject to c A S > whle S { Q > } Q b The dual D o the SDP ca be tated a Now or D aze b T where R ad ubject to S S A S C 3 Alzadeh 3 eteded otetal reducto ethod develoed b Ye 8 or LP to SDP Th ha led to the eteo o a teror ot IP ethod to e-dete rograg [] Pral-Dual Algorth or SDP Pelaga Reearch Lbrar 5

3 BO Adejo et al Adv Al Sc Re 6:4-3 Let P { S / A b } F be the teror o the eable et or the ral roble 4 ad F D S R / A S C S be the teror o the eable et or the dual roble 4 the we aue the ollowg: F P F D ad the atrce A A A that dee the SDP are learl deedet S S S X R to The et o ral-dual otal oluto cot o the oluto the ollowg te: I A b A S C S where the lat euato I the coleetart euato To etrze I Zhag 7 troduced a geeral etrzato ag: II H : R S deed a T [ PMP PMP ] H R where P R oe o-gular atr Baed o etrc ag II the te o euato I ca be wrtte a III A b A S C H S Coeuetl the earch drecto S S S R oluto o the ollowg te: IV A b A A S C S H S µ I H at a ot S the A Pelaga Reearch Lbrar 6

4 BO Adejo et al Adv Al Sc Re 6:4-3 where [ ] the ceterg araeter ad µ j Sj j ad I the dett atr I d deote the eaure o cetralt at a ot S S the d S µ I where [ ] λ [ ] µ λ deote the ege value o the atr Baed o th cetralt codto we dee a Frobeu eghbourhood N o the cetral ath a ollow: { F P F D / d } N µ where a real cotat the terval The algorth that ollow a hort-te ath-ollowg algorth baed o Motero Zhag ued earch drecto obtaed ug euato IV IV ad IV Algorth: Chooe real cotat ad uch that ad let 7 8 σ 4 Let be a otve teger ad F F D atg the codto µ where µ Reeat utl covergece µ µ reached Chooe a o-gular atr P R Coute earch drecto µ ad be a tal tartg ot a the oluto o te IV wth Pelaga Reearch Lbrar 7

5 BO Adejo et al Adv Al Sc Re 6:4-3 8 Pelaga Reearch Lbrar Obta the et ot the oluto euece a v v Set µ ad creae b Theore I ad are oe ed real cotat uch that ad 4 at the ollowg eualte: 7 5 the the euece o terate { } geerated b the algorth the eghbourhood { } µ : 6 d D F F N ate e Proo Fro Motero[ 6 ] [ ] 6 Now a value o the terval 4 wll at the codto o the theore Motero [ 6 ] ce the terval 4 le the terval the eualt 5 euvalet to the eualt 6 Motero [ 6 ] Hece all the reult that are vald or the theore Motero [ 6 ] wll alo be vald or our ow aal Hece l l 3 3 e

6 BO Adejo et al Adv Al Sc Re 6:4-3 9 Pelaga Reearch Lbrar Theore I at ot l a terato the algorth d a oluto to the SDP wth Fro Kararar [ 5 ] a otve teger l l l l l 6 Fro the eualt theore e [ ] e l l [ 7 ] l e l l l l 7 Now eualt 6 true ate l l l Now a Fral [ 4 ] ad Sgh et al [ 7 ] we a dee the uber o terato to d otal oluto a l a

7 BO Adejo et al Adv Al Sc Re 6:4-3 CONCLUSION The algorth wll to the uber o terato reache a I the reache a beore the covergece chec reached we to ad coclude that the SDP ha o oluto The larger the value o atg 5 the ater the covergece o the algorth Motero [ 6 ] rooed ad wth th choce he obtaed a l Adejo [] rooed ad obtaed a l wth a terato rule whch reduced at leat 4% terato coaro to Motero choce 9 Adejo ad Sgh [ ] choe ad a obtaed a l whch urther reduce b 945% the uber o terato coaro to that o Motero [ 6 ] Adejo ad Sgh [ ] terato rule wa better tha that acheved Adejo [ ] b 98% 9% Here our curret terato rule o leetato are 9495% terato over Motero [ 6 ] ad wth 88% Slght roveet o the terato over Adejo ad Sgh [ 3 ] Jutcato The larger the value o that ate 4 the ater the rate o covergece Now Motero [ 6 ] choe Adejo [ ] choe Adejo ad Sgh choe 9 whle here we chooe 99 Sce 99 > 9 > > t le that our choce o 99 wll eure ater covergece tha thoe o 9 Hece or Motero [ 6 ] For Adejo [ ] whle Adejo ad Sgh [ ] wth K a l wth K a l 9 wth K a l Now ercetage % roveet o Adejo [ ] over Motero [ 6 ] 4% % roveet o Adejo ad gh [ ] over Adejo [ ] 945% Pelaga Reearch Lbrar 3

8 BO Adejo et al Adv Al Sc Re 6:4-3 % roveet o Adejo ad Sgh [ ] over Adejo [ ] 983% % roveet o Adejo ad Ogala over Motero [ 6 ] 9495% % roveet o Adejo ad Ogala over Adejo ad Sgh [ ] 88% REFERENCES [ ] Adejo B O 4: Modcato o Soe Poloal-te Algorth or Lear Prograg Uublhed PhD the Deartet o Matheatc Ahadu Bello Uvert Zara [ ] Adejo B O ad Sgh D 7: Ngera Joural o Scetc Reearch NJSR Vol 6 3 [ 3 ] Alzadeh F 995: SIAM Joural o Otzato 5 : 3 5 [ 4 ] Fral J 987: Covergece Kararar Algorth SIAM Joural o Nuercal Aal 44: [ 5 ] Kararar N K 984: Cobatorca 4 : [ 6 ] Motero R D C 997: SIAM Joural o Otzato 7 : [ 7 ] Sgh J N Sgh D ad Raja Shah 995: Joural o Iorato ad Otzato Scece [ 8 ] Ye Y 99: SIAM Joural o Coutg [ 9 ] Zhag Y 998: SIAM Joural o Otzato 8: Pelaga Reearch Lbrar 3