A Tabu Search Method for Finding Minimal Multi-Homogeneous Bézout Number

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1 Joural of Matheatics ad Statistics 6 (): , 010 ISSN Sciece Publicatios A Tabu Search Method for Fidig Miial Multi-Hoogeeous Bézout Nuber Hassa M.S. Bawazir ad Ali Abd Raha Departet of Matheatics, Faculty of Sciece, Uiversity Techology Malaysia, Malaysia Abstract: Proble stateet: A hootopy ethod has prove to be reliable for coputig all of the isolated solutios of a ultivariate polyoial syste. The ulti-hoogeeous Bézout uber of a polyoial syste is the uber of paths that oe has to trace i order to copute all of its isolated solutios. Each partitio of the variables correspods to a ulti-hoogeeous Bézout uber. It is a crucial proble to fid a partitio with the iiu ulti-hoogeeous Bézout uber sice the size of the space of all the partitios icreases expoetially. Approach: This study preseted a ew ethod by producig the Tabu Search Method (TSM) as a powerful techique for fidig iiu ulti-hoogeeous Bézout uber. Results: A copariso is ade betwee the ew ethod ad soe recet ethods. It is show that our algorith is superior to the latter, besides beig siple ad efficiet i the ipleetatio. Coclusio: Furtherore the preset study exteded the applicability of the Tabu search ethod. Key words: Multi-hoogeeous Bézout uber, polyoial syste, hootopy ethod, local search ethod, Tabu search ethod INTRODUCTION The developet of hootopy cotiuatio ethods started aroud the id seveties with the study of Garcia ad Zagwill (1979). Recetly these ethods have evolved to becoig reliable ad efficiet uerical algoriths for approxiate all of the isolated solutios of polyoial systes. For a survey (Li, 1987; Watso, 1986). Cosider a polyoial syste of equatios: F(x) = (f 1 (x), f (x),,f (x)) (1) where, x = (x 1,x,...,x ) C. The classical hootopy ethod for polyoial syste is based o the classical Bézout uber, i.e., the total degree TD, TD = d, where d i is the degree of the ith equatio f i. TD is a upper boud of the uber of the isolated solutios of (1) ad hece the uber of curves oe has to trace i the hootopy. However, TD is ofte far larger tha the uber of isolated solutios of the syste (1). Hece a hootopy goes through exhaustive coputatios, icludig tracig uecessary curves. Morga ad Soese (1987) proposes the ultihoogeeous Bézout theory. It is show that the ultihoogeeous Bézout uber also gives a upper boud for the uber of isolated solutios of a i= 1 i polyoial syste. Each partitio of the variables usually gives a differet ulti-hoogeeous Bézout uber. It is desired to fid a partitio whose ultihoogeeous Bézout uber is the sallest aog all possible variable partitios. I fact the iial ultihoogeeous Bézout uber is usually saller (soeties far saller) tha the Bézout uber, TD. Thus a saller uber of paths is followed i the ulti-hoogeeous hootopy ethod. Wapler (199) presets a exhaustive search ethod o fidig the optial boud. However, uerical experiets show that while it works well for sall systes, it is costly whe icreases. Li ad Bai (000) provides a local search ethod for iiizig ulti-hoogeeous Bézout ubers; as with ay other local search ethods, it gives a local iiu rather tha the (global) iiu over all possible hoogeizatios. Li et al. (003) presets the so-called fissio ad assebly operatios to geerate the partitios fro each other i order to iiize hoogeeous Bézout ubers, but the search techique is still local i ature ad it oly works for sall systes. Ya et al. (008) provides a geetic algorith for fidig iial ulti-hoogeeous Bézout ubers; the algorith depeds heavily o rado choices fro the populatio space ad coputes their fitess fuctios ad keep the iiu oes, repeatig this Correspodig Author: Hassa M.S. Bawazir, Departet of Matheatics, Faculty of Sciece, Uiversity Techology Malaysia, Malaysia 105

2 J. Math. & Stat., 6 (): , 010 choice util soe stoppig criterio is satisfied. This ethod depeds o o-cotrolled oves i search of feasible solutios, so i soeties it revisits a large uber of cadidates ad soeties it ecouters a ifiite cyclig loop. The coputatio of iiu Bézout uber is a NP-hard proble (Malajovich ad Meer, 007); cosequetly the topic of iiizig Bézout uber is very iportat because it gives the uber of the paths that eed to be traced i a hootopy ethod. I this study we will preset a heuristic techique based o the Tabu search. The Tabu search exhibits several stregths, listed as follows (Glover ad Kocheberger, 003): Coverges ear the optial solutio Ca be cosidered as a cotrolled rado walk i the space of feasible solutios Uses the short ter eory to prevet the reversal of recet oves Uses the log ter frequecy eory to reiforce attractive copoets Uses a Tabu list to prevet cyclig back to previously visited solutios. Tabu list records the recet search history, a key idea that ca be liked to Artificial Itelligece cocepts The ulti-hoogeeous Bézout uber: Cosider the ultivariate polyoial syste (1). Let Z = {z 1, z,,z } be a -partitio of the ukows X = {x 1, x,,x } where { } z = z,z,...,z, j = 1,,..., j j1 j jk j Defie the degree atrix of the syste F(x) = 0 as the followig: d11 d1 d 1 d1 d d D = d1 d d the degree polyoial f D (y), ad deoted by B, where k = (k 1,k,,k ), with k j = #(z j ), j = 1,,, ad k j= 1 j =. The followig well-kow exaple shows the sigificace of the differece betwee the iial ulti-hoogeeous Bézout uber ad the classical Bézout uber (Li et al., 003). Exaple: Cosider the atrix eigevalue proble: = λ = C λ C. Ax x,x (x 1,x,...,x ), Oe ca view it as a polyoial syste of variables (x,x,...,x, λ) C : 1 Ax = λx η T x = 1 where, η C is a radoly chose vector. Clearly, the classical Bézout uber of this syste is TD =. But we all kow that this eigevalue proble oly has solutios, coutig ultiplicities. Sice >>, the hootopy ethod will be very costly. By takig the partitio Z = {z 1 = X,z = {λ}} we fid the - hoogeeous Bézout uber exactly. We seek to iiize Bézout uber ad sice each partitio Z of the set of ukows X gives oe uber the the search space will be the space of all partitios of set of eleets. The total uber of all possible partitios is deoted by B() ad called Bell uber. It is the uber of all possible ways of puttig distict balls ito idetical boxes, where soe of the boxes could be epty. The followig recursive relatioship holds for Bell ubers (Li et al., 003): 1 ( ) B() = 1 B(k), B(0) = 1 k = 0 k There is a estiatio of Bell uber give by (Li ad Bai, 000): where, d ij is the degree of polyoial f i w.r.t. the variable z j. The degree polyoial of F with respect to < B() <! the partitio Z is defied as: This eas that Bell uber icreases expoetially f D(y) = d i 1 j 1 ijy = as grows. For exaple, B(4) = 15, B(5) = 5, B(10) = = j 115,957 ad B(15) = 1,38,958,545. It is ot ecessary to copute Bézout uber by The -hoogeeous Bézout uber of F w.r.t. the expadig the degree polyoial because we eed just k k1 k k partitio Z equals the coefficiet of y = y1 y...y i the coefficiet of oe desired ooial. Wapler 106

3 J. Math. & Stat., 6 (): , 010 (199) gives a recursive relatio for coputig Bézout uber by lettig a partitio, say - partitio Z, fixed, k = (k 1, k,, k ) the cardialities of the sets i Z, D the correspodig degree atrix whose etries are d ij : 1, if i = + 1 b(d, k,i) = dij b(d, k e j1,k j,i + 1) otherwise () j 0 where, e j is the jth row of the idetity atrix of degree ad the -hoogeeous Bézout uber is b (D,k,1). MATERIALS AND METHODS eighborhood of P, deoted by N(P), will geerally be defied as a subset of the set of all possible cases. I our ethod we will focus o a siple type of eighborhoods, as i the followig defiitio. Defiitio 1: Let P = {z 1,z,,z } be a partitio of the set of the variables X = {x 1,x,,x } of the polyoial syste. let N(P) ɺ be the set of all feasible solutios geerated fro P by splittig oe z i 's at a tie ito two parts, or ergig two z i 's at a tie ito oe part. We defie a typical eighborhood of the partitio p as N (P) which has ɺ eleets chose radoly fro N(P) ɺ where: The ethod used by Wapler (199) is exhaustive, i the sese that it searches over all the partitios. It oly works well i sall systes. The local search ethod of Li ad Bai (000) reduces the uber of visited solutios copared to Wapler ethod but soeties fails to obtai the optial solutio. The a priori cost show i Li et al. (003) shows that the assebly ethod reduces the uber of visited solutios to 3, i.e., i a polyoial tie ad the fissio ethod reduces the uber of visited solutios to - i.e., uch less tha that of Wapler, but still exhibits a expoetial growth. A Geetic Algorith (GA) for iiizig ultihoogeeous Bézout uber preseted by Ya et al. (008) is heuristic I ature; this algorith shows soe attractive results copared to local search ethods, but it depeds heavily o o-cotrolled rado walk through the feasible solutios which akes it costly, especially for large-scale systes. Tabu Search Method (TSM): Tabu search ethod is a heuristic based o a good cotrolled rado walk through attractive feasible solutios ad coverges ear the optial solutio. TSM is well-suited for hard optiizatio probles. The crucial cocept i TSM is the defiitio of the eighborhood of a fixed feasible solutio, because each proble has its differet ature. I our ethod let P = {z 1,z,,z } be a give feasible solutio, where z i X, z i z j for i j ad =. The eighborhood of P #(z ) i= 1 i ca be give i ay ways. Fro a give feasible solutio such P we will geerate aother feasible solutio by oe of the followig: firstly, split oe of z i 's or ore to oe or ore parts, secodly, erge two or ore of z i 's ito oe part, thirdly, ove oe eleet or ore fro oe set z i to aother set z j. Each type of geeratio ca give so ay feasible solutios. The 107, if #(N(P)) ɺ > ɺ = #(N(P)), ɺ otherwise I the followig we establish our algorith for iiizig ulti-hoogeeous Bézout uber usig TSM, where L the tabu list for storig partitios ad L B for storig the correspodig Bézout ubers. Algorith 1: Fidig the iiu ultihoogeeous Bézout uber usig TSM. S0: Iput criterio ubers M, Mɺ > 0. Set i = 0, L = ad L B =. Go to step S1. S1: Choose rado partitio P, add P to the tabu list L, copute B(P) ad add it to L B. Go to step S. S: If B(P) M or i M ɺ declare the result: the partitios i L whose Bézout uber is the iiu over all the values i L B ad stop. It is cosidered as a approxiatio of the iiu Bézout uber over all partitios. Otherwise let i : = i + 1 ad go to step S3. S3: Geerate N(P) ɺ, the N(P) as i Defiitio 1, copute B(N(P)) = {(B(s) : s N(P)}. Go to step S4. S4: Let Y = {y N(P) : B(y) = i(b(n(p)))}. Pick P ɺ Y such that Pɺ L, go to step S5. S5: Let P : = P ɺ, add P to L ad add B(P) to L B. Go to step S. Note that B(P) is the ulti-hoogeeous Bézout uber of the partitio p coputed usig (). We ote that choosig the eighborhood with the uber of eleets ot ore tha the size of the syste decreases the coputatio cost. Fro other viewpoit the radoess ad the fact that optial solutio ca be reached i so ay ways pay off what ca arise fro decreasig the size of the eighborhood.

4 Algorith 1 geerates #(N(P)) ɺ feasible solutios ad coputes fitess fuctio for just rado # (N(P)) #(N(P)) solutios i step S3; this procedure is repeated o ore tha ties i oe loop ad previous work is doe M ɺ ties i oe ru of the algorith; so the uber of evaluated solutios is M ɺ or O( ), i.e., polyoial tie. RESULTS We ipleet our ethod o twety ultivariate polyoial systes with differet sizes; these systes are cited i (Li ad Bai, 000). Table 1 provides basic iforatio about the used systes. Table ad 3 suarize the ipleetatio of Local Search Method (LSM) (Li ad Bai, 000), Fissio Method (FM), Assebly Method (AM) (Li et al., 003), Geetic Algorith (GA) (Ya et al., 008) ad Tabu Search Method (TSM) i two aspects, firstly the covergece to the global optial solutio (MB) ad secodly, how ay feasible solutios eeded to be visited to arrive at the Optial Solutio (OS). The colu of percetage shows the percetage of the uber of visited solutios to the whole populatio. The result is the average of 10 ties ru of the progra. Soe sybols used are listed below: TD = Total degree (classical Bézout uber) MB = Miiu Bézout uber P = The uber of all partitios vp = The uber of visited solutios (partitios) OS = The optial solutio (global or local) # = The uber of the syste as cited by (Li ad Bai, 000) * = The solutio is ot global J. Math. & Stat., 6 (): , DISCUSSION O the oe had, the result i Table shows that LSM, FM ad AM ay fail to reach the global solutio; oreover, each of these approxiates a solutio through visitig so ay feasible solutios. O the other had, the result i Table 3 shows that GA achieves the optial solutio by visitig a less uber of feasible solutios coparig to LSM, FM, ad AM. However GA is still iefficiet sice it goes through exhaustive searchig i the space of feasible solutios whereas TSM achieves the optial solutio through a cosiderably shorter path of the solutios as show i Table 3. Table 1: Basic iforatio of the used systes i TSM Sr. No. TD MB NP Table : Nuerical results (1) LSM FM AM Sr. No. VP % OS VP % OS VP % OS * * * * Error report Error report

5 J. Math. & Stat., 6 (): , 010 Table 3: Nuerical results () GA TSM Sr. No. VP % OS VP % OS CONCLUSION We have preseted a heuristic ethod based o Tabu search ethod. Two aspects of the perforace are clear i the uerical results. Firstly where the local search ethod, fissio ethod ad assebly ethod ay fail to achieve the optial solutio, as show i Table, while our Tabu search ethod obtai the optial solutio with less uber of visited solutios, see Table 3. As for the heuristic ethod geetic algorith, oe ca reach the optial solutio but by costly o-cotrolled walk through the feasible solutios as show i Table 3. TSM is easy, copetitive ad efficiet to ipleet so it ca deal well with large scale systes. This study exteds also the applicatio fields of Tabu search ethod. REFERENCES Garcia, C.B. ad W.L. Zagwill, Fidig all solutios to polyoial systes ad other systes of equatios. Math. Progra., 16: x363/fulltext.pdf Glover, F. ad Gary A. Kocheberger, 003. Hadbook of Metaheuristics. 1st editio, Kluer Acadeic Publishers, ISBN: , pp: Li, T. ad F. Bai, 000. Miiizig ultihoogeeous Bézout uber by a local search ethod. Math. Cop., 70: Li, T., Z. Li ad F.S. Bai, 003. Heuristic ethods for coputatig the iial ulti-hoogeeous Bézout uber. Appl. Math. Cop., 146: DOI: /S (0) Li, T.Y., Solvig polyoial systes. Math. Itel., 9: h345/fulltext.pdf Malajovich, G. ad K. Meer, 007. Coputig ultihoogeeous Bézout ubers is hard. Theory Coput. Systes, 40: DOI: /s y Morga, A. P. ad A. J. Soese, A hootopy for solvig geeral polyoial systes that respects -hoogeeous structures. Applied Math. Coput., 4: DOI: / (87) Wapler, C.W., 199. Bézout uber calculatios for ulti-hoogeeous polyoial systes. Applied Math. Coput., 51: DOI: / (9)90070-H Watso, L.T., Nuerical liear algebra aspects of globally coverget hootopy ethods. SIAM. Rev., 8: Ya, D., J. Zhag, B. Yu, C. Luo ad S. Zhag, 008. A geetic algorith for fidig iial ultihoogeeous Bézout uber. Proceedig of the 7th IEEE/ACIS Iteratioal Coferece o Coputer ad Iforatio Sciece, May 008, IEEE Coputer Society, USA., pp: DOI: /ICIS