Optimal Adjustment Algorithm for p Coordinates and the Starting Point in Interior Point Methods

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1 Amercan Journal of Operaon Reearch 9- do:.436/aor..4 Publhed Onlne December (hp:// 9 Opmal Adumen Algorhm for p Coordnae and he Sarng Pon n Ineror Pon Mehod Abrac Carla T. L. S. Ghdn Aurelo R. L. Olvera Jar Slva Inue of Mahemac Sac and Scenfc of Compuaon (IMECC) Sae Unvery of Campna (UNICAMP) São Paulo Brazl Federal Unvery of Mao Groo do Sul (UFMS) Mao Groo do Sul Brazl E-mal: {carla aurelo}@me.uncamp.br arm@gmal.com Receved Sepember 5 ; reved Ocober 6 ; acceped Ocober 3 Opmal adumen algorhm for p coordnae a generalzaon of he opmal par adumen algorhm for lnear programmng whch n urn baed on von Neumann algorhm. I man advanage are mplcy and quc progre n he early eraon. In h wor o accelerae he convergence of he neror pon mehod few eraon of h generalzed algorhm are appled o he Mehrora heurc whch deermne he arng pon for he neror pon mehod n he PC ofware. Compuaonal epermen n a e of lnear programmng problem have hown ha h approach reduce he oal number of eraon and he runnng me for many of hem ncludng large-cale one. Keyword: Von Neumann Algorhm Mehrora Heurc Ineror Pon Mehod Lnear Programmng. Inroducon In 948 von Neumann propoed o Danzg an algorhm for fndng a feable oluon o a lnear program wh a convey conran reca n he form ([]): P = e = (.) where R m P n R n e R n he vecor of all one and he column of P have norm one.e. P for n. Th algorhm whch wa laer uded by Epelman and Freund [3] ha nereng propere uch a mplcy and fa nal advance. However no very praccal for olvng lnear problem becaue convergence rae low. Gonçalve [4] preened four new algorhm baed on von Neumann algorhm and among hem he opmal par adumen algorhm ha he be performance n pracce. The opmal par adumen algorhm nher he good propere of von Neumann algorhm. Alhough proved n [5] ha h algorhm faer han von Neumann algorhm neverhele mpraccal o olve lnear programmng problem becaue convergence rae alo very low. In [6] he dea preened by Gonçalve Sorer and Gondzo [5] o develop he opmal par adumen algorhm wa generalzed for p coordnae and hen he opmal adumen algorhm for p coordnae aroe. The value of p bounded by he number of varable of he problem. The new algorhm no uable o olve lnear programmng problem unl achevng opmaly. Thu he dea o eplo mplcy and quc nal progre durng he early eraon and o ue ogeher wh an neror pon mehod o accelerae convergence. Knowng ha he arng pon grealy nfluence he performance of he neror pon mehod n h wor he opmal adumen algorhm for p coordnae appled whn he Mehrora heurc [7] whch deermne he arng pon for he mehod neror pon n he PC ofware o ha he arng pon can be obaned even beer. Th approach dfferen o he clacal warm-arng approach whch ue a nown oluon for ome problem (e.g. relaaon n he columngeneraon) o defne a new arng pon for a cloely relaed problem or perurbed problem nance ([8-]). The paper organzed a follow. In Secon he defnon of he problem gven and von Neumann algorhm decrbed. In Secon 3 he opmal par adumen algorhm recalled. In Secon 4 he opmal adumen algorhm for p coordnae propoed Copyrgh ScRe.

2 9 C. T. L. S. GHIDINI ET AL. and ome heorecal reul are decrbed. Secon 5 dcue warm-arng n neror pon mehod. Numercal epermen are hown n Secon 6. In Secon 7 he concluon are drawn and fuure perpecve are uggeed.. Von Neumann Algorhm Conderng he problem of fndng a feable oluon of he e of lnear conran (.) geomercally he column P can be vewed a pon lyng on he m-dmenonal hyperphere wh un radu and cener a he orgn. Th problem can be decrbed a agnng nonnegave wegh o column P n uch a way ha afer beng re-caled cener of gravy he orgn. Fr he algorhm fnd he column P of P whch mae he large angle wh he redue b - = P - and hen he ne redual b gven by proecng he orgn on he lne egmen connecng b o P. See Fgure. VON NEUMANN S ALGORITHM Gven: wh e =. Compue b = P. = Do {. Compue: arg mn P b. n v P b. If v hen STOP. The problem (.) nfeable. 3. Compue: u = b u v. v 4. Updae: b b P e. Fgure. Illuraon of von Neumann algorhm. where e he vecor of he canoncal ba wh n poon. 5. Soppng Creron: E = b b b = +. } whle (E > ε)} where ε a predeermned percenage. Noe ha b = P for all. The nal appromaon arbrary nce e = hen = /n for n wa ued. For any gven eraon column P he column whch mae he large angle wh he vecor b. Furhermore f v P b hen < v < (u ) v + hu < λ <. Moreover he new redual maller han he prevou one.e. u < u a can ealy be een n Fgure he rangle b b ha hypoenue u = b and de u = b. The effor per eraon of von Neumann algorhm domnaed by mar-vecor mulplcaon needed when elecng column P whch O(nz(P)) where nz(p) he number of he enre of P. Th effor can be reduced gnfcanly a he mar P pare. For more deal of h algorhm ee []. 3. Opmal Par Adumen Algorhm In h PhD he [4] Gonçalve uded von Neumann algorhm and nroduced four new algorhm baed on. Among hem empha gven o he wegh-reducon algorhm and he opmal par adumen algorhm. The opmal par adumen algorhm wa he one ha performed beer n pracce. The wegh-reducon algorhm wa propoed a an aemp o mprove he effcency of von Neumann algorhm. I baed on he dea ha he redual b can be moved cloer o orgn ncreang he wegh for a gven column P and decreang he wegh for anoher column P. In parcular epeced ha he new redual b cloer o orgn han he redual b f he wegh n column P + ncreaed when P + ha he large angle wh he redual b and he wegh n column P decreaed when P + ha he malle angle wh he redual b. Th correpond o mang redual b move n he drecon P + P. The new redual b he pon ha mnmze he dance from orgn o h lne. Noce ha he mnmzaon of h dance ubec o he mamum poble decreae of. Snce for all hen can be decreaed unl vanhe. Fgure a geomerc lluraon of how he algorhm wor by eraon. Copyrgh ScRe.

3 C. T. L. S. GHIDINI ET AL Solve he problem: mn b.. for. (3.) Fgure. Illuraon of wegh-reducon algorhm. 3.. Opmal Par Adumen Algorhm The opmal par adumen algorhm a generalzaon of he wegh-reducon algorhm degned o gve he mamum poble freedom o wo of he wegh (ee [5]). In a way we can ay ha prorze only wo varable by eraon becaue fnd he opmal value for wo coordnae and adu he remanng coordnae accordng o hee value. Smlar o he wegh-reducon algorhm he opmal par adumen algorhm ar by denfyng vecor P + and P whch mae he large and malle angle wh he vecor b. Aferward value are found where for all and. Thee value mnmze he dance beween b and he orgn whle afyng he convey and he non-negavy conran. The oluon of h opmzaon problem ealy compued eamnng he Karuh- Kuhn-Tucher condon (KKT). The man dfference beween he wegh-reducon algorhm and he opmal par adumen algorhm ha only he wegh of P + and P are changed n he fr algorhm whle n he econd algorhm all oher wegh are alo changed. OPTIMAL PAIR ADJUSTMENT ALGORITHM Gven: wh e =. Compue b = P. = Do {. Compue: arg mn P b... n P b arg ma... n v = P b. If v hen STOP. The problem (.) nfeable. where b b P P P P. 4. Updae: b b P P P P u = b and = 5. Soppng Creron: E = b b b = +. } whle (E > ε)}. In eraon column P + he column whch mae he large angle wh he vecor b and column P he column whch mae he malle angle wh he vecor b uch ha >. 3.. Subproblem Soluon In order o olve ubproblem (3.) fr he ubproblem rewren a: mn b.. for. where b b P P P. P (3.3) Defne: g(λ λ ) = λ + λ g (λ λ ) = λ g (λ λ ) = λ. Denoe he obecve funcon by f(λ λ ) and le be a feable oluon. Con derng he convey of he obecve funcon and conran f an opmal local oluon hen alo an opmal global oluon. Thu me he KKT conran gven by: g f. Copyrgh ScRe.

4 94 C. T. L. S. GHIDINI ET AL. g ( ) for. g( ) for. for. where µ are Lagrange mulpler. Problem (3.3) olved by elecng a feable oluon among all poble ha mee he KKT condon. Th done by analyzng he followng cae: a) λ = λ = ; b) λ = ; < λ < ; c) λ = ; λ = ; d) < λ < ; λ = ; e) < λ < ; < λ < ; λ + λ = ; f) λ = ; λ = ; g) < λ < ; < λ < ; λ + λ = ; For each cae above he nown value are replaced n he KKT equaon and he reulng lnear yem olved. 4. Opmal Adumen Algorhm for p Coordnae The opmal adumen algorhm for p coordnae developed generalzng he dea ued n [5] for he opmal par adumen algorhm. Inead of only wo column o be ued o formulae he problem any amoun of column can be ued and hu mporance wll be gven o any dered amoun of varable. The raegy o prorze he varable free. In h wor we choe o ae p/ column mang he large angle wh he vecor b and he remanng p/ column ha mae he malle angle wh he vecor b. If p odd hen one more column added o he e of vecor ha mae he large angle wh he vecor b for eample. The rucure of he opmal adumen algorhm for p coordnae mlar o he opmal par adumen algorhm. I begn by denfyng he column ha mae he large angle wh he vecor b hen column ha mae he malle angle wh he vecor b are deermned where + = p and p he number of column ha are prorzed. Ne he opmzaon ubproblem olved and he redual and he curren pon are updaed. OPTIMAL ADJUSTMENT ALGORITHM FOR p COORDINATES Gven: wh e =. Compue b = P. = Do {. Compue: P P he veor b. whch mae he large angle wh P P whch mae he malle angle wh he vecor b and uch ha where + = p. v mnmum P b. If v hen STOP. The problem (.) nfea- ble. 3. Solve he problem: mn b.. for for. (4.4) where b b P P P P 4. Updae: b b P P u = b P P ; ;. 5. Soppng Creron: E = b b b = +. } whle (E > ε)}. In Sep of he algorhm f v > hen all column P of mar P are on he ame de of he hyper-plane hrough he orgn and perpendcular o drecon b. Th mean ha any conve combnaon of he column P reulng n he orgn can be found. In h cae he Copyrgh ScRe.

5 C. T. L. S. GHIDINI ET AL. 95 problem nfeable. 4.. Subproblem Soluon Ung Ineror Pon Mehod For each eraon of he opmal adumen algorhm for p coordnae neceary o olve ubproblem (4.4). Th ubproblem olved fndng a oluon n he pove orhan ubec o a lnear equaon yem of order (p + ). A way o olve o verfy all poble feable oluon a n he par adumen algorhm. The drawbac ha come naurally n olvng he ubproblem n h way ha he number of poble cae o be verfed grow eponenally wh he value of p a hown ne. In fac o olve he ubproblem (4.4) fr he varable λ a defned n (4.5) removed from he problem. Thu he ubproblem (4.4) wa rewren a: where mn b.. for for. b b P P P Defnng: P g g h (4.5) (4.6) and denong he obecve funcon by f he KKT condon o e f h ubproblem are gven by: f g g h h g g for h h for g for h for g for h for. (4.7) Then he ubproblem (4.6) olved by elecng a feable oluon among all he poble ha afy he KKT condon (4.7). For h all poble cae of poble value for he varable and mu be analyzed. The cae o be condered are: Cae: and Cae: one wh and oher equal o zero whch gve a combnaon p of C cae bu agan we have o conder he cae where and. Thu we have C p poble. Cae: and and and oher equal o p zero whch gve a combnaon of C cae bu agan we have o conder he cae where p and. Thu we ha ve C poble. The oal number of cae when p coordnae are modfed : p p p p 3 p p C C C C Therefore h raegy neffcen even for value of p whch are no oo large. In order o deal wh uch a dffculy he ubproblem (4.4) approached dfferenly and olved ung neror pon mehod. Th done a follow: Fr he ubproblem rewren n mar form: Copyrgh ScRe.

6 96 C. T. L. S. GHIDINI ET AL. where mn W.. a W wp P P P wb P P a a a The KKT equaon of problem (4.8) are gven by: (4.8) (4.9) WW a a. (4.) where γ and µ are Lagrange mulpler of equaly and n equaly conran repecvely and (p + ) (p + ) he dmenon of he mar W W. Now he neror pon mehod appled o hoe equaon. The lnear yem arng a each eraon of he neror pon mehod appled o (4.) are: where WW a Id r U d r a d r 3 U r a W W dag dag r r3 a. (4.) By performng ome algebrac manpulaon he drecon dµ dλ and dγ are gven by: dµ r Ud 4 d W W U r W W U ad a W W U ad a W W U r r (4.) where r 4 = r + Λ r. Defnng l = (W W + Λ U) a and l = (W W + Λ U) r 4 o compue he drecon he followng lnear yem mu be olved: WW U l a WW U l r (4.3) Noe ha (W W + Λ U) a pove defne mar of order p + and boh yem can be olved ung he ame facorzaon. 4.. Theorecal Propere of he Opmal Adumen Algorhm for p Coordnae Theorem and decrbed below and proved n [6] enure ha he opmal adumen algorhm for p coordnae converge n he wor cae wh he ame convergence rae of von Neumann algorhm and ha from he heorecal pon of vew for larger value of p he algorhm more robu and wll perform beer. Theorem The decreae n b obaned by an eraon of he opmal adumen algorhm for p coordnae wh p n where n he column number of mar P n he wor cae equal o he decreae obaned by an eraon of von Neumann algorhm. Theorem The decreae n b obaned by an eraon of he opmal adumen algorhm for p coordnae n he wor cae equal o ha obaned by an eraon of he opmal adumen algorhm for p coordnae wh p p n where n he P column number. On he oher hand no advable o chooe a very large value of p nce here a co of buldng and updang mar W n (4.3). Such a co neglgble for mall value of p. However become noceable for larger value. 5. Mehrora Heurc and Sarng Pon n Ineror Pon Mehod I well nown ha he arng pon can nfluence he performance of neror pon mehod. On he oher hand a he opmal adumen algorhm for p coordnae ha a mall compuaonal co per eraon a naural dea o ue h algorhm o fnd a good arng pon for neror pon mehod or mprove he curren one. In h wor few eraon of he opmal adumen algorhm for p coordnae are performed o mprove he arng pon nroduced by Mehrora heurc [7]. Th heurc con of he followng ep: ) Ue lea quare o compue he pon: y= AA Ac z=c A y =AAA b where y and z are he prmal varable dual varable and dual lac repecvely. ) Fnd value δ and δz uch ha and z z are non-negave: ma.5 mn z ma.5 mn. 3) Deermne and z uch ha he pon and z Copyrgh ScRe.

7 C. T. L. S. GHIDINI ET AL. 97 z are cenralzed: e z e z n z z n z e z e z 4) Compue he arng pon a follow: y =y z z ze z e.. Remar 5. Any lnear programmng problem can be reduced o problem (.). For deal of h ranformaon ee [4]. Uually o mprove he performance of neror pon mehod modfcaon are nroduced afer Sep 4 of he algorhm decrbed above unle our propoal whch con of performng a few eraon of he opmal adumen algorhm for p coordnae before cenralzng.e. before Sep. One eplanaon for h ha f he algorhm hen ued afer o cenralze he pon hee are mproved. However he cenraly whch mporan for he neror pon mehod may be lo. We have adaped he opmal adumen algorhm for p coordnae n he PC code [3] amng o mprove he arng pon gven by Mehrora [7]. The arng pon for he opmal adumen algorhm for p coordnae deermned by olvng he lea quare n Sep. 6. Compuaonal Epermen In he followng epermen all eng wa performed on an Inel Core Duo T75 GB RAM GHz and 5 GB hd and Ubunu 3-b operang yem. We ued 76 e problem o compare he performance of he orgnal PC and he modfed PC (PCMod) whch ue he opmal adumen algorhm for p coordnae a a warm-arng approach. Mo eed problem have free acce uch a NETLIB problem QAPLIB problem and KENNINGTON problem ([45]). Oher problem whch are no avalable publcly were obaned from Gonçalve [4]. The e problem are preened n Table. Table. Te problem. Problem Row Column Problem Row Column Problem Row Column NETLIB Mcellaneou Gonçalve 8bau3b 4 66 nug bl agg nug bl agg nug co czprob cre-a co degen cre-b cq dfl cre-c e eamacro cre-d 4 86 e f ffff en e fd 5 54 en e fp en e m aro-r oa f or modz oa for perold oa for plo oa for plo el for plowe 7 84 chr5a for cfm chrb for eba cr for hp8l 47 3 cr for ocfor rou ge woodp 7 78 pd nl pd pd pd pd pd pd pd pd pd Copyrgh ScRe.

8 98 C. T. L. S. GHIDINI ET AL. 6.. Choce of p For he opmal adumen algorhm for p coordnae o wor properly an approprae choce of parameer p eena l. Thu everal compuaonal epermen were done o deermne a heurc ha wor well n any lnear programmng problem. Wh reul obaned n he e became clear ha he value of p mu be choen dependng on he ze of he problem. Recallng ha m he number of row and n he number of column of he lnear problem conran mar he heurc ha howed beer reul a follow: < (m + n) p = 4; < (m + n) p = 8; < (m + n) 4 p = ; 4 < (m + n) 6 p = 4; 6 < (m + n) p = Soppn g Creron The number o f eraon o f h e opmal adumen algo- rhm for p coordnae o be perfor med an mporan parameer o be deermned nce drecly nfluence he performance of he PC. Th algorhm acheve beer reul when he oluon deermned wh good accuracy. However n ome cae he number of eraon needed for convergence of he algorhm can be very large mang mpraccal o ue. In hee cae a mamum number of eraon hould be adoped. Therefore he oppng creron for he opmal adumen algorhm for p coordnae ued he followng: mamum number of eraon () or he relave error of he redual norm maller han a gven olerance ( 4 ) (he one ha occur fr) Reul The performance of he PC and PCMod wa compared wh repec o he oal number of eraon and he run me. The reul are preened n Table. The oal number of neror pon eraon (Column Ieraon) and he oal runnng me n econd (Column Tme) of wo veron of he PC are howed n Table. The PCMod he veron ha ue he opmal adumen algorhm for p coordnae n Mehrora heurc and PC he one ha doe no adop. Column p how he value of p ued by he opmal adumen algorhm. Column IAu gve he number of eraon performed by he opmal adumen algorhm for p coordnae. Accordng o he reul he PCMod ae le me o oban he opmal oluon n 55.3% of he eed problem and h e PC n 34.% of h e problem. More- ove r he oal number of eraon wa lower n 4.8% of he problem for he PCMod and.3% (only one problem) for he PC. In h cae he run me of he PCMod wa lower becaue a beer arng pon wa ued. In almo all problem wh larger runnng me he new approach performed beer han he radonal one. Th reveal a welcome feaure of he propoed approach nce hoe are he mo mporan problem o be olved. Alhough he oal number of eraon dd no decreae n ome problem a beer arng pon wa deermned a he opmal adumen algorhm for p coordnae wa ued afer Sep of he Mehrora heurc and conequenly he runnng me wa reduced. An nereng reul he fac ha wo problem (co9 and nug8) were olved only by PCMod. In anoher epermen he ame e of e problem were olved ung only he PCMod. Fr p = (-coord) wa condered whch repreen he opmal par adumen algorhm and afer he value of p accordng o he oppng creron prevouly decrbed wa deermned. The reul obaned are hown n Table 3. The reul howed ha he PCMod ae le eraon wh value of p greaer han n 4.8% of he e problem. For p = h number appromaely 5.3%. The oal runnng me wa reduced by abou 6.5% of he problem for p-coordnae and only.4% were olved n a lower me a p = wa condered. Agan he problem co9 obaned au opmal only for p = 8. Thee e confrmed ha he performance of he algorhm mproved by ncreang he value of p. Tha happen becaue he redual b ha a greaer reducon from one eraon o anoher and alo becaue he gven pon by he algorhm a a oluon dfferen for each value of p. The obaned pon for p > acheved beer performance n mo cae. I hould be menoned ha he requred oal me o oban a oluon for he opmal adumen algorhm for p coordnae no gnfcan n relaon o he reoluon oal me of he problem. Th me almo null n many of he eed problem. Conderng he pd-8 problem he me pen by he opmal adumen algorhm wa 4 econd whch repreen appromaely.7% of he oal runnng me. 7. Concluon and Fuure Wor In h wor he opmal adumen algorhm for p co- Copyrgh ScRe.

9 C. T. L. S. GHIDINI ET AL. 99 Table. PC PCMod. Dmenon Ieraon Tme Problem Row Column p Au PC PCMod PC PCMod 8bau3b agg agg czprob degen dfl eamacro fffff fd fp maro-r modz perold plo plo plowe cfm eba hp8l ocfor wood p cre-a cre-b cre-c cre-d en en en oa oa oa oa bl bl co co cq e e e e e f or f or f or for for for for for for ge nl el chr5a chrb cr5 cr Copyrgh ScRe.

10 C. T. L. S. GHIDINI ET AL. rou nug nug nug pd pd pd pd pd pd pd pd pd pd pd pd : mea n ha he mehod faled. Table 3. -coordnae p-coordnae. Dmenon I Au Ieraon Tme Problem Row Column p -coord p-coord -coord p-coord -coord p-coord 8bau3b agg agg czprob degen dfl eamacro fffff fd fp maro-r m odz perold plo plo plowe cfm eba hp8l ocfor woodp cre-a cre-b cre-c cre-d en en en oa oa oa oa bl bl co co cq e e e e Copyrgh ScRe.

11 C. T. L. S. GHIDINI ET AL. e f or f or f or f or for for for for for ge nl el chr5a chrb cr cr rou nug nug nug pd pd pd pd pd pd pd pd pd pd pd pd : me an ha he meh od faled. ordnae wa ued n conuncon wh he nero r pon mehod o deermne good arng pon enabl ng he me hod o perform beer and conequenly converge fa er. The man advanage of h algorhm are mplcy and q uc nal progre. Numercal epermen on a e of problem howed ha by ung h approach poble o reduce he oal number of eraon and he runnng me for many problem manly large-cale one. In addon wo problem were olved o opmaly only by h approach. Tha ung he algorhm for p coordnae o mprove he arng pon compued by PC lead o a more robu mplemenaon. By ncorporang he opmal adumen algorhm for p coordnae on he code PC more pecfcally Mehrora heurc and performng a few eraon a more robu mplemenaon wa obaned. The compuaonal epermen on a e of lnear programmng problem howed ha h approach can reduce he oal number of eraon and oal run me for everal problem epecally he larger problem and hoe wh hgher runnng me. Moreover he opmal o luon of ome unolved proble m wa found. A fuure wor oph caed heurc for he choce of he num ber of coordnae p wll be developed a afer everal epermen value for p ha reduce he number of eraon by abou 9% of he e problem were found. Alo oher oppng creron for he o pmal ad- be umen algorhm for p coordnae wll be nvegaed nce he number of eraon performed by uch an algorhm drecly nfluence he qualy of he arng pon deermned. New form of choong he p column wll developed. 8. Acnowledgemen Th reearch wa ponored by he Brazlan Agence CAPES and CNPq. 9. Reference [] G. B. Danzg Converng A Convergng Algorhm no a Polynomally Bounded Algorhm Techncal Repor Sanford Unvery SOL Copyrgh ScRe.

12 C. T. L. S. GHIDINI ET AL. [] G. B. Danzg An -Prece Feable Soluon o a Lnear Program wh a Convey Conran n / Ieraon Independen of Problem Sze Techncal Repor Sanford Unvery SOL [3] M. Epelman and R. M. Freund Condon Number Compley of an Elemenary Algorhm for Compung a Relable Soluon of a Conc Lnear Syem Mahe- macal Programmng Vol. 88 No. 3 pp [4] J. P. M. Gonçalve A Famly of Lnear Programmng Algorhm Baed on he Von Neumann Algorhm Ph.D. The Lehgh Unvery Behlehem 4. [5] J. P. M. Gonçalve R. H. Sorer and J. Gondzo A Famly of Ln ear Programmng Algorhm Baed on an Algorhm by Von Neumann Opmzaon Mehod and Sofware Vol. 4 No. 3 9 pp do:.8/ [6] J. Slva Uma Famíla de Algormo para Programação Lnear Baeada no Algormo de Von Neumann Ph.D. The IMECC-UNICAMP Campna 9 (n Poruguee). [7] S. Mehrora On he Implemenaon of a Prmal-Dual Ineror Pon Mehod SIAM Journal on Opmzaon Vol. No pp [8] H. Y. Benon and D. F. Shanno An Eac Prmal Dual Penaly Mehod Approach o Warm Sarng Ineror- Pon Mehod for Lnear Programmng Compuaonal Opmzaon and Applcaon Vol. 38 No. 3 7 pp do:.7/ [9] E. John and E. A. Yldrm Implemenaon of Warm- Sar Sraege n Ineror-Pon Mehod for Lnear Programmng n Fed Dmenon Compuaonal Opmzaon and Applcaon Vol. 4 No. 8 pp do:.7/ y [] A. Engau M. F. Ano and A. Vannell A Prmal-Dual Slac Approach o Warm Sarng Ineror-Pon Mehod for Lnear Programmng Operaon Reearch and Cyber-Infrarucure Vol pp do:.7/ _ [] G. B. Danzg and M. N. Thapa Lnear Programmng : Theory and Eenon Sprnger-Velag New Yor 997. [] D. Berma and J. N. Tl Inroducon o Lnear Opmzaon Ahena Scenfc Belmon 997. [3] J. Czyzy S. Mehrora M. Wagner and S. J. Wrgh PC an Ineror Pon Code for Lnear Programmng Opmzaon Mehod & Sofware Vol. No pp do:.8/ [4] NETLIB Collecon LP Te Se NETLIB LP Repoory. hp:// [5] M. L. Model Hungaran Academy of Scence OR Lab. hp:// Copyrgh ScRe.