GRADIENT METHODS FOR BINARY INTEGER PROGRAMMING

Transcription

1 Proceedns of the 4st Internatona Conference on Computers & Industra Enneern GRADIENT METHODS FOR BINARY INTEGER PROGRAMMING Chen-Yuan Huan¹, Ta-Chun Wan² Insttute of Cv Avaton, Natona Chen Kun Unversty, No., Unversty Road, Tanan Cty 70, Tawan R.O.C.) TEL: et Abstract: Inteer Prorammn IP) s a common probem cass where decson varabes represent ndvsbty or yes/no decsons. IP can aso be nterpreted as dscrete optmzatons whch are etensvey used n the areas of routn decsons, feet assnment, and arcraft/arcrew schedun. Sovn IP modes s much harder than sovn Lnear Prorammn LP) modes because of ther ntrnscay combnatora nature. In addton, many rea-word probems have a hue probem sze that maes sovn the reated IP-based mode tme-consumn. In ths paper, we propose a radent method for bnary IP that enabe are probems to be soved n a short amount of tme. The aorthm s parttoned nto two phases. In phase of the aorthm, nteraty constrants are modfed and the orna probem s soved by the LP aorthm. In phase, a nteraty constrants are modfed aan and added bac. The Larane mutpers are then ntroduced to form a Larane dua probem. We then use the radent method to qucy search for the optma or neary optma souton. After deveopn the aorthm, we show the effectveness of ths approach by appyn t to varous nds of bnary IP probems wth dscussons about the reducton n computaton tme and the ap between obectve functons. Keywords: Inteer Prorammn; Gradent Methods; Larane Dua Probems.. Introducton Inteer Prorammn IP) probems are based on Lnear Prorammn LP) probems wth etra requrements that each decson varabe be an nteer vaue representn ndvsbty, e.. arcraft, manpower, and factes. It s thus possbe to formuate modes that are more reastc and meannfu. In addton, the nteer varabes used are often restrcted to bnary vaues n order to hep companes mae cruca yes/no decsons, and there are a vast number of appcatons of ths method over very wde rane of rea-word probems. However, IP probems are ntrnscay hard to be soved than that of LP probems. In ths paper, we propose a radent method for bnary nteer prorammn BIP), whch s a totay dfferent approach to the currenty ubqutous branch-and-bound technque. By usn ths method, we can 40

2 Proceedns of the 4st Internatona Conference on Computers & Industra Enneern overcome the nherent weanesses of the branch-and-bound aorthm Taha, 007), by avodn spendn too much tme cacuatn near prorammn reaatons LPRs) and eepn trac of the best nteer feasbe souton at every branchn node. In fact, n the approach presented n ths wor we cacuate the modfed LPR ust once, and then the aorthm proceeds to phase for fast souton searchn.. Computatona Bottenecs The we-estabshed aorthm for sovn IP probems s the branch-and-bound aorthm, whch adopts a very smpe concept, dvde and conquer, to avod compete enumeraton. Ths concept s based on the tree search whch dvdes the orna probem nto a possbe sub-probems and then soves a of these. An mportant feature of the IP mode s ts abty to capture nonneartes and nonconvetes Nemhauser, 994). Athouh most IP probems have a fnte number of feasbe soutons, ths number can row eponentay. Aso, due to the nteer restrctons, the souton space of IP modes are attces rather than conve. Moreover, branch-and-bound aorthms do not offer the same advantaes as the smpe method Dantz, 963) n ts abty to fndn optma soutons. IP probems are potentay eponenta n computatona compety, and, Suden 99) noted that some IPs are ntrnscay hard, no matter what branch-and-bound heurstc s used to fathom the search tree. Uness speca steps are taen, the number of sub-probems branches) can row eponentay. In addton, another crtca probem arses n branch-and-bound aorthms, whch s that there s a fundamentay combnatora nature embedded n IP, so that dfferent node and varabe choces yed dfferent search trees. Ths feature of hh uncertanty may resut n etremey unpredctabe computatona tme. Consequenty, n enera, t s mpossbe to determne the best tree search stratey. One can thus concude that branch-and-bound aorthms possess a hh deree of non-determnsm wth reard to ther performance MITRA et a., 997), and even thouh such aorthms can be proved to termnate wth an nteer souton n a fnte number of steps, t s dffcut to predct the necessary runnn tme. IP modes are thus cassfed as NP-hard, and are much more dffcut sove than LP probems. 3. Aorthm The standard bnary nteer prorammn BIP) probem can be stated as foows: mn. C T ) s.t. a b 0, =,, m, =,, n ) d e = 0, =,, p, =,, n 3) bnary where s the decson varabe, C s the vector of cost coeffcent, and ) and 3) are the constrants to be satsfed for an optma souton. 4

3 Proceedns of the 4st Internatona Conference on Computers & Industra Enneern The proposed aorthm s parttoned nto two phases. In phase of the aorthm, the bnary nteer restrctons are frst reaed as 0. 4) Then the probem s soved by the revsed smpe method or the nteror pont method Nemrovs & Todd, 008), dependn on the probem structure. By the effcent LP aorthm, we sove ths thtened LPR n the hope of fndn a ood or neary ood nta pont qucy as an nput of phase. In phase of the aorthm, the constrants of bnary varabes are frst modfed as foows ) = 0 5) to ensure that the bnary nteer restrctons can have a specfc mathematca representaton to be ncorporated nto the obectve functon wth Larane mutpers. Then, by ntroducn the Larane mutpers to a other constrants, the orna BIP probem has transferred nto a Larane dua probem. The resutant Larane dua functon s where m = p T L, u, v, w ) = C + u a b ) + v d e ) + w )) 6) = u, v, w are Larane mutpers, and u 0. n = To determne the optma souton, the Karush-Kuhn-Tucer KKT) necessary optmaty condtons shoud be satsfed. Snce ths probem has become an unconstraned optmzaton probem. We can thus use the radent method for fast searchn the optma or neary optma soutons. Note that the KKT suffcent condtons for optmaty are not consdered n ths paper. Readers who are nterested n optmaty condtons coud refer to Bazaraa & Shetty, 979). In addton, to further refne the proposed aorthm, we aso tae nto account of conuate radent CG) methods to avod searchn n drectons that have been searched before. Snce the orna search drecton the neatve radent) may have a rapd chane n ts foown search drecton, and ead to a poor performance especay on some are-scae probems. Yuan 009) proposed a hybrd method combnn the FR conuate radent method Fetcher & Reeves, 964) and the WYL conuate radent method We et a., 006). Ths hybrd method has been proved havn the suffcent descent property under the stron Wofe-Powe SWP) ne search rue. The reat performance s attrbuted to the CG update parameter H FR WYL ) = q ) + q ), 7) where q 0, q 0. But, Yua n dd not suest that how to choose the parameters q and q n the aorthm. In phase of the proposed aorthm, we aso adopt the hybrd CG method to update the prma and dua varabes, and aso nvestate the nfuence on the performance of dfferent choces of q and q. The aorthm s descrbed as foows. 4

4 Proceedns of the 4st Internatona Conference on Computers & Industra Enneern Step : Input the prma and dua nta pont 0 and 0, where = u, v, w) Step : Set = 0, d = = L, ), d = = L ) ˆ 0. Choose q 0, q 0. Step 3: If ε ε =0 4 ), then stop; otherwse o to step 4. 0 Step 4: Compute step-sze Step 5: Update = α and + α d αˆ by some ne search rues. + and + = + α Step 6: Compute = L, ), + = L + + ). If ε, then stop. ˆ dˆ Step 7: Update the CG parameters + T + FR ) =, T ) ) ) ) FR = ) + T + T ) ) ) + T T + + WYL WYL ) =, T ) = T ) ) + + FR WYL d = + q ) + q ) ) d, Step 8: Update the search drecton d Step 9: Set = +, and o to Step 4. + = + + FR q ) + q ) ) WYL ) dˆ ) ) 4. Epermenta Resuts The epermenta resuts for the proposed radent methods and the standard branch-and-bound technque are presented n the foown. Tabe ves some statstcs for the testn modes. A codes were wrtten n MATLAB 7.0.0) runnn on a persona computer Inte Core 5 CPU GHz, 3.49 GB RAM). Tabes and 3 summarze the computatona tme and computatona resuts. The frst coumn n both tabes reports the mode name; TBB s the computatona CPU) tme of the branch-and-bound aorthm; TLP s the computatona CPU) tme of the phase procedures; TGD s the computatona CPU) tme of the phase radent method; TT reports the tota tme of the proposed aorthm; and speed-up s cacuated by TBB/TT. Tabe 3 ves the computatona resuts. GD reports the fna radent of when the process has convered; LPRap s the obectve vaue ap between the phase LPR and the branch-and-bound aorthm; GDap s the ap between the proposed aorthm and the eact souton; and Rap s the fna ap between the roundn souton and the eact souton. If the fna souton s suffcenty cose to zero or one e or 0.999), then the souton s round to the nearby bnary vaue. Tabe and Tabe 3 ustrate the vaue of the proposed radent methods. The optma souton s obvousy acheved n a shorter tme than needed for the branch-and-bound aorthm. Athouh n case M-0, the fna ap st has 5.79 %, the branch-and-bound aorthm taes over than 96 hours CPU tme to sove ths probem whe the proposed method ony taes about 4.63 seconds CPU tme to appromate the 43

5 Proceedns of the 4st Internatona Conference on Computers & Industra Enneern Tabe : Mode statstcs bnary nteer prorammn). Mode Number of rows Number of coumns Number of non-zeros T-0 57 T T M Tabe : Computatona tme. Mode TBB s) TLP s) TGD s) TT s) Speed-up T T T M-0 Over Over Tabe 3: Computatona resuts. Mode GD LPRap %) GDap %) Rap %) T e e-3 0 T e e-4 0 T e M e souton. The overa computatona performance of the proposed radent method s snfcanty better than that of the branch-and-bound aorthm. In the speca case M-0, we noted that the phase LPR produces a souton wth the vaue equas to 0.5 on each. Snce the vaue 0.5 ust ocates at the mdde pont of zero and one, any move to the bnary vaue becomes a cruca decson. For ths reason, we modfy aan the constrants of bnary varabes by mutpyn a mutpcatve constant y on 5) as foows. Throuh ths way, we can force the souton convern at the bnary vaue. y ) = 0 8) Tabe 4: Types of dfferent choces of parameters q and q. Type q q

6 Proceedns of the 4st Internatona Conference on Computers & Industra Enneern Iteraton number C9 C4 C8 C5 C7 C0 C Type Fure : Performance of dfferent choces of parameters q and q. To further refne the aorthm, we aso nvestate dfferent choces of the nta pont of dua varabes that can have a reat mpact on the performance. We use modes from Tabe and enerate etra modes by settn dfferent mutpcatve constant. Throuh the eperment, we suest that the nta pont of dua varabes correspondn to the modfed bnary constrants shoud set coser to zero. Another ssue reated to the CG performance s that the best choce of parameters q and q are not yet nown so far. In order to fnd out what parameters q and q shoud be chosen to drve the hybrd CG aorthm performn the best performance, we nvestate dfferent type of choces of parameters. Tabe 4 ustrates the testn types of dfferent choces of the parameters. F. shows the teraton number versus dfferent types of choces of the parameters on each case. Most cases dvered whe usn type 8 and type 9 parameters. Therefore, to further mprove the performance, we coud choose the recommended parameters q = 0.6 and q = Concusons and Future Wor In ths paper we construct a new sover, and demonstrate that the proposed radent method has better performance than the ubqutous branch-and-bound technque. In addton, we have found the best or at east ood enouh choces of the nta vaue of dua varabes and the premnary parameters n the hybrd CG method. These can snfcanty mprove the performance of the aorthm. In the future, more BIP modes shoud be tested for further testn. The dfferent modfed bnary constrants shoud be nvestated to further reduce the ap between the appromate and eact souton. Moreover, dfferent souton searchn routes enerated by asynchronous updatn the prma and dua varabes w be taen nto account. Fnay, the error anayss between an appromate souton and an eact souton w aso be nvestated. References Bazaraa, M.S. & Shetty, C.M. 979). Nonnear prorammn: theory and aorthms. New Yor: Wey. 45

7 Proceedns of the 4st Internatona Conference on Computers & Industra Enneern Dantz, G.B. 963). Lnear prorammn and etensons. Prnceton, NJ: Prnceton Unversty Press Fetcher, R. & Reeves, C.M. 964). Functon mnmzaton by conuate radents. The Computer Journa, 7), MITRA, G., HAI, I. & HAJIAN, M.T. 997). A dstrbuted processn aorthm for sovn nteer prorams usn a custer of worstatons. Parae Computn, 3, Nemhauser, G.L. 994). The ae of optmzaton: sovn are-scae rea-word probems. Operatons Research, 4), 5-3. Nemrovs, A.S. & Todd, M.J. 008). Interor-pont methods for optmzaton. Acta Numerca, 7, Suden, S.J. 99). A cass of drect search methods for nonnear nteer prorammn. God Coast, Bond Unversty. Taha, H.A. 007). Operatons research: an ntroducton. Upper Sadde Rver, N.J.: Pearson Prentce Ha. We, Z., Yao, S. & Lu, L. 006). The converence propertes of some new conuate radent methods. Apped Mathematcs and Computaton, 83), Yuan, G. 009). A conuate radent method for unconstraned optmzaton probems. Internatona Journa of Mathematcs and Mathematca Scences, 009,