Polynomial Barrier Method for Solving Linear Programming Problems

Transcription

1 Internatonal Journal o Engneerng & echnology IJE-IJENS Vol: No: 45 Polynoal Barrer Method or Solvng Lnear Prograng Probles Parwad Moengn, Meber, IAENG Abstract In ths wor, we study a class o polynoal ordereven barrer unctons or solvng lnear prograng probles wth the essental property that each eber s concave polynoal order-even when vewed as a uncton o the ultpler Under certan assupton on the paraeters o the barrer uncton, we gve a rule or choosng the paraeters o the barrer uncton We also gve an algorth or solvng ths proble Inde er lnear prograng, barrer ethod, polynoal order-even I INRODUCION he basc dea n barrer ethod s to elnate soe or all o the constrants and add to the obectve uncton a barrer ter whch prescrbes a hgh cost to neasble ponts (Wrght, ; Zboo, etc, 999) Assocated wth ths ethod s a paraeter, whch deternes the severty o the barrer and as a consequence the etent to whch the resultng unconstraned proble approates the orgnal proble (Kas, etc, 999; Parwad, etc, ) Parwad () proposed a penalty ethod or solvng lnear prograng probles In ths paper, we restrct attenton to the polynoal order-even barrer uncton Other barrer unctons wll appear elsewhere hs paper s concerned wth the study o the polynoal barrer uncton ethods or solvng lnear prograng It presents soe bacground o the ethods or the proble he paper also descrbes the theores and algorths or the ethods At the end o the paper we gve soe conclusons and coents to the ethods II SAEMEN OF HE PROBLEM hroughout ths paper we consder the proble aze subect to A = b, () n where A R n, c, R, and b R Wthout loss o generalty we assue that A has ull ran We assue that proble () has at least one easble soluton In order to solve ths proble, we can use Kararar s algorth and spel ethod (Durazz, ) Parwad ( and ) also has ntroduced a penalty ethod or solvng pral-dual lnear prograng probles But n ths paper we propose a polynoal barrer ethod as another alternatve ethod to solve lnear prograng proble () hs wor was granted and supported by the Faculty o Industral Engneerng, rsat Unversty, Jaarta Parwad Moengn s wth the Departent o Industral Engneerng, Faculty o Industral echnology, rsat Unversty, Jaarta 44, Indonesa (eal: parwad@hotalco, parwad@trsatacd) IJE-IJENS February IJENS III POLYNOMIAL BARRIER MEHOD For any scalar, we dene the polynoal barrer uncton B (, ) or proble (); B (, ) : R n R by B (, ) = c - ( A b ), () where s an even nuber Here, A and b denote the th row o atrces A and b, respectvely he postve even nuber s chosen to ensure that the uncton () s concave Hence, B (, ) has a global au We reer to as the barrer paraeter hs s the ordnary Lagrangan uncton n whch n the altered proble, the constrants replaced by ( A b ) A b ( =,,) are he barrer ters are ored ro a su o polynoal order o constraned volatons and the barrer paraeter deternes the aount o the barrer he otvaton behnd the ntroducton o the polynoal order- ter s that they ay lead to a representaton o an optal soluton o proble () n ters o a local unconstraned au Sply statng the denton () does not gve an adequate presson o the draatc eects o the posed barrer In order to understand the uncton stated by () we gve an eaple wth soe values or Soe graphs o B (, ) are gven n Fg 3 or the trval proble aze ( ) subect to, or whch the polynoal barrer uncton s gven by B (, ) = ( ) Fg, and 3 depct the one-densonal varaton o the barrer uncton o =, 4 and 6, or three values o barrer paraeter, that s =, = and = 6, respectvely he y-ordnates o these Fg represent B (, ) or =, = 4, = 6, respectvely Clearly, the soluton * = o ths eaple s copared wth the ponts whch aze B (, ), t s clear that * s a lt pont o the unconstraned azers o B (, ) as he ntutve otvaton or the barrer ethod s that we see unconstraned azers o B (, ) or value o ncreasng to nnty hus the ethod o solvng a sequence o azaton proble can be consdered

2 Internatonal Journal o Engneerng & echnology IJE-IJENS Vol: No: 46 Fg 3 he polynoal barrer uncton or = 6 () () (6) Fg he quadratc barrer uncton or = () () (6) Fg he polynoal barrer uncton or = 4 he polynoal barrer ethod or proble () conssts o solvng a sequence o probles o the or aze B(, ) subect to, (3) where s a barrer paraeter sequence satsyng or all, he ethod depends or ts success on sequentally ncreasng the barrer paraeter to nnty In ths paper, we concentrate on the eect o the barrer paraeter he ratonale or the barrer ethod s based on the act that when, then the ter ( A b ), when added to the obectve uncton, tends to nnty A b and equals zero A b or all n hus, we dene the uncton : R (, ] by c ( ) A b A b or all, orall he optal value o the orgnal proble () can be wrtten as * = sup c = sup ( ) Ab, = supl P(, ) (4) On the other hand, the barrer ethod deternes, va the sequence o nzatons (3), l supp(, ) (5) hus, n order or the barrer ethod to be successul, the orgnal proble should be such that the nterchange o l and sup n (4) and (5) s vald Beore we gve a guarantee or the valdty o the nterchange, we nvestgate soe propertes o the uncton dened n () Frst, we derve the concavty behavor o the polynoal barrer uncton dened by () s stated n the ollowng stated theore heore (Concavty) he polynoal barrer uncton B (, ) s concave n ts doan or every Proo It s straghtorward to prove concavty o B (, ) usng the concavty o theore s proven c and - ( A b ) hen the () () (6) he local and global behavor o the polynoal barrer uncton dened by () s stated n net the theore It s a consequence o heore IJE-IJENS February IJENS

3 Internatonal Journal o Engneerng & echnology IJE-IJENS Vol: No: 47 heore (Local and global behavor) Consder the uncton B (, ) whch s dened n () hen (a) B (, ) has a nte unconstraned azer n ts doan or every and the set M o unconstraned azers o B (, ) n ts doan s copact or every (b) Any unconstraned local azer o B (, ) n ts doan s also a global unconstraned azer o B (, ) Proo It ollows ro heore that the sooth uncton B (, ) acheves ts au n ts doan We then conclude that B (, ) has at least one nte unconstraned azer By heore B (, ) s concave, so any local azer s also a global azer hus, the set M o unconstraned azers o B (, ) s bounded and closed, because the au value o B (, ) s unque, and t ollows that M s copact heore has been vered As a consequence o heore we derve the onotoncty behavors o the obectve uncton proble (), the barrer ters n B (, ) and the au value o the polynoal barrer uncton B (, ) o do ths, or any we denote and (, ) B as a azer and au value o proble (3), respectvely heore 3 (Monotoncty) Let { } be an ncreasng sequence o postve barrer paraeters such that and hen (a) c s non-ncreasng (b) (c) A b ) B(, ) s non-ncreasng as ( s non-ncreasng Proo Let and denote the global azers o proble (3) or the barrer paraeters and, respectvely By denton o and as azers and, we have c ( A b c - ( A b ), (6a) c - ( A b c - + ( A b ), (6b) c - + ( A b c - + ( A b ) (6c) We ultply the rst nequalty (6a) wth the rato /, and add the nequalty to the nequalty (6c) we obtan Snce c part (a) s establshed, t ollows that c c c and o prove part (b) o the theore, we add the nequalty (6a) to the nequalty (6c) to get ( A b ) ( A b ), thus ( A as requred or part (b) b ) ( A Usng nequaltes (6a) and (6b), we obtan c ( A b + ( A b ) b ) Hence, part (c) o the theore s establshed c - We now gve the an theore concernng polynoal barrer ethod or lnear prograng proble () heore 4 (Convergence o polynoal barrer uncton) Let { } be an ncreasng sequence o postve barrer paraeters such that Denote (a) and and (, ) as B as n heore 3 hen A b as c as (b) * (c) B(, ) * Proo By denton o as c (, ) and (, ) B, we have B B(, ) or all (7) IJE-IJENS February IJENS

4 Internatonal Journal o Engneerng & echnology IJE-IJENS Vol: No: 48 Let * denotes the optal value o the proble () We have * = c sup = sup B(, ) Ab, B(, ) = c - Let be a lt pont o { } A b ( A b * By tang the lt neror n the above relaton and by usng the contnuty o c and A b, we obtan Snce c - l n ( A b * (8) ( A b ) and that we ust have and ( A b ), t ollows A b or all =,,, 9) otherwse the lt neror n the let-hand sde o (8) wll equal to + hs proves part (a) o the theore n Snce { R } s a closed set we also obtan that Hence, s easble, and * c () Usng (8)-(), we obtan Hence, and * - l n ( A b c - l n ( A b * l sn ( A b ) = * = c, whch proves that s a global au or proble () hs proves part (b) o the theore o prove part (c), we apply the results o parts (a) and (b), and then tang o the denton B(, ) Soe notes about ths theore wll be taen Frst, t assues that the proble (3) has a global au hs ay not be true the obectve uncton o the proble () s replaced by a nonlnear uncton However, ths stuaton ay be handled by choosng approprate value o We also note that the constrant o the proble (3) s portant to ensure that the lt pont o the sequence { } satses the condton Hence, by tang the supreu o the rght-hand sde o (7) over and A b, we obtan IV ALGORIHM he plcatons o these theores are rearably strong he polynoal barrer uncton has a nte unconstraned azer or every value o the barrer paraeter, and every lt pont o a azng sequence or the barrer uncton s a constraned azer o a proble () hus the algorth o solvng a sequence o azaton probles s suggested Based on heores 4, we orulate an algorth or solvng proble () Algorth Gven A = b,, the nuber o teraton N and n Choose R such that A = b and I the optalty condtons are satsed or proble () at, then stop 3 Copute (, ) azer B a B(, ) 4 Copute (, ) 5 I azer B a B(, ) and or = and the, the or B(, ) B(, ) or A b or = N; then stop Else + and go to step 4 V INERIORPOIN ALGORIHM hs secton revews the nterorpont algorth called Kararar s algorth or ndng a soluton o lnear prograng proble he step o ths algorth can be suarzed as ollows or any teraton (Parwad, ) Step Gven the current ntal tral soluton (,,, n ), set D Step Calculate Step 3 Calculate c p Pc~ ~ A AD 3 and ~ n c Dc ~ ~~ ~ P I A ( AA ) A and Step 4 Identy the negatve coponent o c p havng the largest absolute value, and set equal to ths absolute value hen calculate IJE-IJENS February IJENS

5 Internatonal Journal o Engneerng & echnology IJE-IJENS Vol: No: 49 c p ~, where s a selected constant between and Step 5 Calculate D as the tral soluton or the net teraton (step ) (I ths tral soluton s vrtually unchanged ro the precedng one, then the algorth has vrtually converged to an optal soluton, so stop) ~ VI NUMERICAL EXAMPLES hs secton we gve ve eaples to test the Algorth and we copare the results wth Kararar s algorth Consder the ollowng probles (Parwad, ), or =,, 3 Eaple Maze subect to 3 7, Eaple 3 6,, or =, Maze subect to, 4 3,, or =, Eaple Maze 3 subect to 3 6, Eaple Maze subect to 3 6, Eaple 5,, or =, Maze subect to 4, 3,, or =, able I reports the results o coputatonal or Algorth ( = ), Algorth ( = 4) and Kararar s Algorth he rst colun o able I contans the eaple nuber and the net two coluns o each algorth n ths table contan the total teratons and the tes (n seconds) o each algorth Proble No ABEL I ALGORIHM ( = ), ALGORIHM ( = 4) AND KARMARKAR S ALGORIHM ES SAISICS Algorth ( = ) Algorth ( = 4) Kararar s Algorth otal Iteratons 8 5 e (Secs) otal Iteratons e (Secs) otal Iteratons e (Secs) able I also shows that n ters o copleton te or the th nuercal eaples, the Algorth ( = ) s better than Algorth ( = 4), but both are stll less than the Kararar s algorth In ters o the nuber o teratons requred to coplete the ve nuercal eaples shows that Algorth ( = ) loos better than Kararar s algorth and Algorth ( = 4) VII CONCLUSION As entoned above, the paper has descrbed the barrer unctons wth barrer ters n polynoal order- or solvng proble () he algorths or these ethods are also gven n ths paper he Algorth s used to solve the proble () We also note the portant thng o these ethods whch do not need an nteror pont assupton REFERENCES [] Durazz, C () On the Newton nteror-pont ethod or nonlnear prograng probles Journal o Optzaton heory and Applcatons 4() pp 739 [] Kas, P, Klaszy, E, & Malyusz, L (999) Conve progra based on the Young nequalty and ts relaton to lnear prograng Central European Journal or Operatons Research 7(3) pp 934 [3] Parwad, M, Mohd, IB, & Ibrah, NA () Solvng Bounded LP Probles usng Moded Logarthc-eponental Functons In Purwanto (Ed), Proceedngs o the Natonal Conerence on Matheatcs and Its Applcatons n UM Malang (pp 35-4) Malang: Departent o Matheatcs UM Malang [4] Parwad, M () Polynoal penalty ethod or solvng lnear prograng probles IAENG Internatonal Journal o Appled Matheatcs 4(3), pp IJE-IJENS February IJENS

6 Internatonal Journal o Engneerng & echnology IJE-IJENS Vol: No: 5 [5] Parwad, M () Soe algorths or solvng pral-dual lnear prograng usng barrer ethods Internatonal Journal o Matheatcal Archve (), pp 8-4 [6] Parwad, M () Eponental ethods or conve prograng under lnear equaton constrants Internatonal Journal o Matheatcal Archve (), pp 8-9 [7] Wrght, SJ () On the convergence o the Newton/logbarrer ethod Matheatcal Prograng, 9(), 7 [8] Zboo, RA, Yadav, SP, & Mohan, C (999) Penalty ethod or an optal control proble wth equalty and nequalty constrants Indan Journal o Pure and Appled Matheatcs 3(), pp IJE-IJENS February IJENS