Discrete Particle Swarm Optimization for TSP: Theoretical Results and Experimental Evaluations

Transcription

1 Dscrete Partcle Swarm Optmzaton for TSP: Theoretcal Results and Expermental Evaluatons Matthas Hoffmann, Mortz Mühlenthaler, Sabne Helwg, Rolf Wanka Department of Computer Scence, Unversty of Erlangen-Nuremberg, Germany Abstract. Partcle swarm optmzaton (PSO) s a nature-nspred technque orgnally desgned for solvng contnuous optmzaton problems. There already exst several approaches that use PSO also as bass for solvng dscrete optmzaton problems, n partcular the Travelng Salesperson Problem (TSP). In ths paper, () we present the frst theoretcal analyss of a dscrete PSO algorthm for TSP whch also provdes nsght nto the convergence behavor of the swarm. In partcular, we prove that the popular choce of usng sequences of transpostons as the dfference between tours tends to decrease the convergence rate. () In the lght of ths observaton, we present a new noton of dfference between tours based on edge exchanges and a new method to combne dfferences by computng ther centrod. Ths leads to a more PSO-lke behavor of the algorthm and avods the observed slow down effect. () Then, we nvestgate mplementatons of our methods and compare them wth prevous mplementatons showng the compettveness of our new approaches. 1 Introducton The problem. Partcle Swarm Optmzaton (PSO) s a popular metaheurstc desgned for solvng optmzaton problems on contnuous domans. It was ntroduced by Kennedy and Eberhard [11, 5] and has snce then been appled successfully to a wde range of optmzaton problems. Snce the structure of the PSO algorthm s relatvely smple, PSO has to some extent been open for theoretcal studes of the swarm behavor. Clerk and Kennedy [4], Trelea [17], and Jang et al. [9] provde analyses of the convergence behavor of partcle swarms, whch offer some nsghts on how to select the swarm parameters, and the ntal behavor of a swarm has been analyzed n [8]. Inspred by the performance of PSO on contnuous optmzaton problems, several approaches have also been proposed for applyng PSO to dscrete problems, such as functon optmzaton on bnary domans [12], schedulng problems [1], and the Travelng Salesperson Problem (TSP) [2, 18, 6, 14, 15, 19].

2 The TSP s one of the classcal problems n dscrete optmzaton. A wealth of methods specfcally talored for solvng TSP has been developed and mathematcally and expermentally nvestgated. A comprehensve overvew of ths lne of research can be found n [7]. But the TSP s also well suted to be approached by (meta-)heurstc methods lke PSO. For dscrete PSO, new nterpretatons of movement and velocty are necessary. The frst approach to adaptng the PSO scheme to TSP s due to Clerc [2, 3]. However, t turns out that ths dscrete PSO (DPSO) by tself s not as successful as the orgnal PSO for contnuous problems. Consequently, subsequent approaches to solvng TSP by PSO typcally rely on downstream optmzaton technques such as k-opt [15, 14] and Ln-Kernghan [6] appled after one PSO teraton to mprove the qualty of the soluton obtaned by PSO. Unfortunately, whereas these hybrd algorthms are evaluated expermentally by beng run on benchmark nstances, they are hard to analyze mathematcally, and so far, no theoretcal nsghts were ganed about the partcles behavor n dscrete PSO at all. In fact, the downstream optmzaton even conceals the performance of plan DPSO. Our contrbuton. In ths paper, we present the frst theoretcal analyss of the dscrete PSO algorthms of Clerc [2] and Wang et al. [18], whch, to some extent, also apples to the approach of Sh et al. [14]. In partcular, we provde for the frst tme theoretcal evdence for why the convergence behavor of these DPSO algorthms for the TSP s qute dfferent from what we would expect from the classcal PSO for contnuous problems. The key nsght s that n later stages of the optmzaton process, the partcles are not lkely to converge towards the best soluton found so far. In fact, we prove that the dstance to the best soluton even remans more or less constant. In the lght of the theoretcal fndngs, we then propose a novel nterpretaton of partcle moton avodng the convergence problem mentoned above. Our method s smlar to computng the mdpont of a dscrete lne. Addtonally, we ntroduce a new representaton of the velocty of a partcle, whch s based on exchangng edges n a potental soluton of a TSP nstance. We evaluate our proposed DPSO wth respect to seven nstances from the TSPlb [13] wth 52 to 105 ctes. In these expermental evaluatons, our focus s on the DPSO performance of dfferent velocty representatons because we are n ths context manly nterested n the performance of the plan DPSO approaches wthout subsequent local mprovement phases. Our results ndcate that the combnaton of the mdpont-based partcle moton wth the edge-exchange operator outperforms the other operators as well as those methods whch suffer from the dentfed convergence problem. In order to also compare our DPSO to the prevous approaches whch use n the PSO teratons addtonal local mprovement heurstcs, we hybrdze our DPSO teraton wth a 2-OPT local optmzaton appled to the global attractor. Here, we make two observatons: The frst one s that better performance of the plan DPSO results n a better performance of the hybrdzed DPSO. And second, for the frst tme t s clearly documented that the huge performance gans (that

3 s acheved when usng local optmzaton n comparson to the plan DPSO) ndcate that the qualty of the solutons found by the DPSO algorthms wth local optmzaton s almost completely determned by the qualty of the local optmzaton. In the prevous work [14, 18, 19], t s not dfferentated between the contrbuton of the PSO and the addtonal local mprovement methods. The remander of ths paper s organzed as follows. In Sec. 2, we provde relevant background nformaton on TSP and PSO. Sec. 3 contans our theoretcal analyss of the partcle convergence behavor of dscrete PSOs for permutaton problems. In Sec. 4, we propose the new dscrete PSO for the TSP whch uses a centrod-based approach for the partcle movement. Sec. 5 provdes the expermental results whch ndcate that our proposed approach outperforms other dscrete PSOs for the TSP. 2 Prelmnares 2.1 The Travelng Salesperson Problem The Travelng Salesperson Problem (TSP) s a classcal combnatoral optmzaton problem. An nstance I = (n, dst) of the TSP wth n ctes {1,..., n} conssts of the dstances between each par of ctes, gven as an n n-nteger matrx dst. The task s to fnd a tour wth mnmum length vstng each cty exactly once, ncludng the way back to the ntal cty. A tour s gven as a permutaton π of the ctes {1,..., n}, where π() denotes the th vsted cty. Hence the set of optmal solutons of an nstance I = (n, dst) s ( argmn dst π(n),π(1) + ) dst π(),π(+1), π S n 1 <n where S n s the symmetrc group on {1,..., n} wth the usual composton as group operaton. The operaton s used for explorng the search space. Note that n ths formulaton the search space conssts of all elements of S n, so one specfc cycle of length n s represented by many permutatons. The advantage of usng all elements of S n for the proposed dscrete PSO s that the partcles can move around freely n the search space wthout the danger of encounterng permutatons whch do not correspond to a vald tour. The decson varant of TSP ( gven an nteger L, s there a tour n I of length at most L? ) s NP-hard and hence, TSP can presumably not be solved exactly n polynomal tme. 2.2 (Dscrete) Partcle Swarm Optmzaton Introduced by Kennedy and Eberhard [11, 5], partcle swarm optmzaton (PSO) s a populaton-based metaheurstc that uses a swarm of potental solutons called partcles to cooperatvely solve optmzaton problems. Typcally, the search space of a problem nstance s an n-dmensonal rectangle B R n, and the objectve functon (often also called ftness functon) s of the form f : R n R. PSO works n teratons. In teraton t, each partcle has a poston x (t) B

4 and a velocty v (t) R n. Whle movng through the search space, the partcles evaluate f at x (t). Each partcle remembers ts best poston p so far (called local attractor) and the best poston p glob of all partcles n the swarm so far (called global attractor). In teraton t, the poston and velocty of each partcle s updated accordng to the followng movement equatons: v (t+1) x (t+1) = a v (t) = x (t) + r loc b loc (p x (t) ) + r glob b glob (p glob x (t) ) (1) + v (t+1). (2) The parameters a, b loc, b glob R are constant weghts whch can be selected by the user. The nerta a adjusts the relatve mportance of the nerta of the partcles, and the so-called acceleraton coeffcents b loc and b glob determne the nfluence of the local and the global attractor, resp. In every teraton, r loc and r glob are drawn unformly at random from [0, 1]. Partcles exchange nformaton about the search space exclusvely va the global attractor p glob. p x (t) s the local attracton exerted by the local attractor on partcle, and p glob x (t) s the global attracton exerted by the global attractor. The PSO algorthm was orgnally desgned for solvng optmzaton problems on contnuous domans. Inspred by the success and conceptual smplcty of PSO, several approaches have been proposed to adapt the PSO dynamcs to dscrete problems ncludng the TSP [2, 6, 15, 14]. In order to adapt PSO s movement equatons (1) and (2) to the dscrete doman of the TSP, Clerk suggests n [2] the followng modfcatons (or new nterpretatons) of the terms nvolved: The partcle s poston x (t) s a permutaton π of the ctes,. e., π = (c 1 c 2... c n ) whch corresponds to the tour c 1 c 2 c n c 1. The dfference x y between two postons x and y (also called the attracton of x to y) s represented by a shortest sequence of transpostons T = t 1,..., t k such that y T = x. Transposton t = (c m c r ) exchanges the two ctes c m and c r n a gven round-trp. The length of a dfference s the length of the sequence T of transpostons. The multplcaton s T of a dfference T = t 1,..., t k wth a scalar s, 0 < s 1, s defned as t 1,..., t s k. For s = 0, s T =. Here, we omt the cases s > 1 and s < 0 snce they do not occur n our proposed PSO. The addton T 1 + T 2 of dfferences T 1 = t 1 1,..., t 1 k and T 1 = t 2 1,..., t 2 l s defned as T 1 + T 2 = t 1 1,..., t 1 k, t2 1,..., t 2 l. The addton of a dfference and a poston s defned as applyng the transpostons of the dfference to the poston. A small example s presented after the proof of Theorem 1. In [2], [18] and [14], the dfference between two postons x and y n the search space s represented as a lst of transpostons that transform round-trp x nto round-trp y. In [6], ths representaton s restrcted to adjacent transpostons. In our new approach n Sec. 4, we replace the transposton by a representaton whch successvely exchanges two edges n a round-trp. Hence, the dfference of two postons x and y s a sequence of edge exchanges of mnmal length whch transforms x nto y.

5 3 Theoretcal Analyss In ths secton, we prove that under certan condtons the prevously developed varants of DPSO mentoned n Sec. 2 behave counterntutvely when compared to the classcal contnuous PSO snce the convergence rate n DPSO s slowed down. More specfcally, we show that transpostons whch occur both n the local attracton and n the global attracton cancel each other and prevent the partcle from movng closer to the global and local attractor. Ths s qute dfferent from the behavor observed n contnuous PSO where common components n these attractons even result n an amplfed attractve force. The phenomenon that the local and the global attracton n prevous approaches have a lot of transpostons n common n the later stages of the optmzaton process can be observed expermentally. Evaluatng the two attractons p x (t) and p glob x (t) for sample runs (see Fg. 1), we see that (n ths example) on average about 30% of the transpostons occur n both attractons. Summng the partcles n the rght half of the bns n Fg. 1, we can conclude that for roughly 20% of the partcles, more than a half of the transpostons are shared by the two attractons. We analyzed the DPSO methods from [2, 18] that fracton of partcles (%) % 10-20% 20-30% 30-40% 40-50% fracton of common transpostons n the local and global attractors Fg. 1. Smlarty of local and global attracton on Clerc s DPSO [2], averaged over 100 runs on the TSP nstance berln52, consderng all partcles n teratons 990 to % 60-70% 70-80% 80-90% % use transpostons for representng dstances between partcles n what we call the Long Term Dscrete PSO model (LTD). In ths model, we assume that the followng four condtons hold: Dfferences between postons are represented by sequences of transpostons. p = p glob =: p, for all partcles. a = 0 r loc and r glob are unformly dstrbuted. When the full swarm converges to a common best soluton p, all local and global attractors are dentcal. If p = p glob for a certan partcle, then t has vsted the

6 global attractor at least once. We assume the nerta a of the partcles beng 0 snce n our experments, the performance of the PSO algorthm even becomes worse f the nerta weght s set to a hgher value. r loc and r glob are qute often unformly dstrbuted n practce. Ths assumpton s also made n the mathematcal analyss n [17]. For Theorem 1, we assume b loc = b glob, whch allows for a closed and smple representaton. After ts proof, we deal wth the more general case whch can be analyzed analogously and present a small example. Theorem 1. Let s [0, 1], and let b loc = b glob = b. The probablty that n the LTD model a certan partcle reduces ts dstance to p n an teraton by a factor of at least b s, s (1 s) 2. Proof. As a = 0, the two movement equatons (1) and (2) can be reduced to one: x (t+1) = x (t) + r loc b (p x (t) ) + r glob b (p x (t) ) Let d be the number of transpostons n the attracton (p x (t) ). Snce we multply the dfference wth r loc b and r glob b, resp., we apply the frst r loc b d and then the frst r glob b d transpostons to x (t). Both dfferences have a common part consstng of the frst mn(r loc, r glob ) b d transpostons. By applyng the frst r loc b d transpostons, for each transposton an element of x (t) reaches the place that t also has n p. However, when applyng the transpostons of the second dfference, the common part of both dfferences s appled twce and the elements of the permutaton that were already at the rght place move now to another place. To brng the elements back to the orgnal place we have to apply the nverse of the common part. Snce the nverse of the common part has exactly the same number of transpostons as the common part, the dstance to p s only reduced by the transpostons that are not common n both dfferences and so are only appled once. The number of the transpostons that are appled only once s r loc r glob b d. Only these transpostons contrbute to the convergence towards p because the other transpostons move the partcle further away from p when they are appled a second tme. Therefore, we call transpostons that are appled only once effectve transpostons. The probablty that the fracton of effectve transposton s at least b s s, s gven by ( rloc r glob b d ) P b s = P( r loc r glob s). d Snce r loc and r glob are unformly dstrbuted, we may conclude (see also Fg. 2 choosng b loc = b glob = b): P( r loc r glob s) = (1 s) 2

7 If b loc b glob, we analogously get the followng expresson for the probablty q s of the fracton of effectve transpostons beng larger than s: q s = P( r loc b loc r glob b glob s) ( = P r glob b loc r loc s ) ( + P r glob b loc r loc + s ) b glob b glob 1 ( { { }} { { }}) = mn 1, max 0, b loc r loc s b glob + mn 1, max 0, b loc r loc +s b glob dr loc 0 r glob 1 s b glob s b loc b glob s b loc b glob +s b loc Fg. 2. The shaded area denotes q s the probablty that the partcle reduces ts dstance to p by at least 25% s ( ) Our analyss drectly apples to Clerc s DPSO [2]. The algorthm proposed by Wang et al. [18] works a bt dfferent wth respect to the scalng of the attractons. In [18], Wang et al. proposed to scale the attractons by b loc, b glob [0, 1] keepng each transposton wth probablty b loc and b glob, resp., n the attracton. So n the LTD model, the movement equatons (1) and (2) reduce to x (t+1) = x (t) + b glob (p x (t) ) + b loc (p x (t) ). A transposton becomes an effectve transposton f t s kept n exactly one of the two attractons. Therefore, effectve transpostons occur wth probablty b loc (1 b glob ) + b glob (1 b loc ) = b loc + b glob 2b glob b loc. Ths s also the expected value of the fracton of effectve transpostons. The coeffcents b loc and b glob are ntended to adjust the weght of the local and the global attractor. Intutvely, f the attractors should exert a large nfluence on the partcles, b loc and b glob are set to 1. Ths works fne n the classcal PSO for contnuous problems. In the dscrete case however, whenever the LTD model apples, the local and global attractons do not pull the partcles closer to the attractors at all. 1 The probablty q s can be vsual- zed lke shown n Fg. 2, where q s amounts to the shaded area. Consder the followng small example. Let x (t) = ( ) and p = ( ). Then p x (t) = ((2 3) (3 5) (4 7) (6 8) (8 9)). Wth b = 0.8, we have b (p x (t) ) = ((2 3) (3 5) (4 7) (6 8)). Wth r loc = 0.75 and r glob = 0.5, we get r loc b (p x (t) ) = ((2 3)(3 5)(4 7)) and r glob b (p x (t) ) = ((2 3) (3 5)), and fnally x (t+1) = x (t) + r loc b (p x (t) ) + r glob b (p x (t) ) = ( ). The transposton (4 7) s the only effectve transposton. By Theorem 1, r loc

8 4 A new DPSO for the TSP 4.1 Centrod-based partcle movement: a new nterpretaton of addton As descrbed n Sec. 2.2, concatenaton s used n [2] and [18] as addton of dfferences,. e., attractons. In Sec. 3, we showed that ths approach has the dsadvantage that after some tme the expected progress becomes consderably slow. Now we propose a new method of combnng the attractons that avods ths dsadvantage. Instead of composng two attractons to a long lst of operators, we look at the destnatons, whch are the ponts the dfferent weghted attractons lead to, and compute the centrod of those destnatons. In our approach, we use no nerta (. e., we set a = 0), but only the attracton to the local and to the global attractor, each weghted n accordance wth equaton (1). Snce we have only two attractors, the centrod can be calculated easly by computng the dfference between the destnatons, scalng them by one half and addng the result to the frst destnaton. The PSO movement equatons can now be expressed wth the destnaton ponts of the attracton to the local attractor, to the global attractor and a random velocty: d loc = x (t) d glob = x (t) + r loc b loc (p x (t) ) + r glob b glob (p glob x (t) ) v rand = r rand b rand (p rand x (t) ) x (t+1) = d glob (d loc d glob ) + v rand The random movement n the end ensures that the swarm does not converge too fast. A graphcal, contnuous representaton s depcted n Fg. 3. x (t+1) p glob v rand d glob p d loc x (t) Fg. 3. Centrod-based partcle movement The advantage of ths model s that t takes the spatal structure of the search space nto account. Snce the centrod s the mean of the destnatons, our factors b loc and b glob can be transferred easly to the classcal PSO by dvdng them by 2.

9 4.2 Edge recombnaton: a new nterpretaton of velocty In [2], [18] and [14], the dfference between two postons b and a n the search space,. e., the velocty or the attracton of b to a s expressed as a lst of transpostons that transforms one sequence of ctes nto the other. Here, we propose a new method that s based on edge exchanges. Edge exchanges are a common technque used n local search methods for the TSP [7]. The dea s to mprove a soluton by exchangng crossng edges, whch results n a shorter tour. For an example, see the transformatons from Fgures 4(a) to (c). A generalzaton of ths operaton s the edge recombnaton operator. Gven a lst l = (c 1 c 2... c n ) of ctes representng a round-trp, the edge recombnaton operator edger(, j) nverts the sequence of ctes between ndces and j: l edger(, j) = (c 1... c 1 c j c j 1 c j 2... c +2 c +1 c c j+1... c n ) In our approach, we use ths operator to express the dfference between two partcles b and a. Instead of a lst of transpostons, the dfference (or velocty, or attracton) s now a lst of edge recombnaton operators that yelds b f the operators are appled to a. For example, the dfference between a = ( ) and b = ( ) s b a = (edger(5, 6) edger(3, 6)) (see Fg. 4). v 1 v 1 v 1 v 6 v 6 v 6 v 2 v 2 v 2 v 5 v 5 v 5 v 3 v 3 v 3 v 4 (a) ( ) edger(5,6) v 4 (b) ( ) edger(3,6) v 4 (c) ( ) Fg. 4. Vsualzaton of two edge recombnatons The problem of fndng the mnmum number of edge exchanges that s needed to transform one permutaton nto the other s NP-complete [16]. Therefore, we use n our experments the smple and fast approxmaton algorthm GetLstOfEdgeExchanges from [10] that s smlar to the algorthm that fnds the mnmum number of transpostons. The soluton found by the approxmaton algorthm GetLstOfEdgeExchanges has n the worst case 1 2 (n 1) tmes more edge exchanges than the optmal soluton [10]. 5 Expermental Results In our experments, we have compared our new approaches to the prevous exstng ones. We have on purpose ntally not done any local optmzaton between

10 Table 1. Comparson of dfferent dstance representatons and movement types wthout local optmzaton (left) and wth an addtonal 2-OPT-based local optmzaton of the global attractor (rght). Move type Compostoston Centrod CentrodCentrod Compo- Centrod CentrodCentrod Transposton Dstance Repr. Transposton Adj. Trans- edger Transposton poston Problem berln % 194.6% 70.5% 22.5% (7542) ± ±531.6 ±863.0 ± pr % 317.7% 156.5% 88.9% (108159) gr % 430.4% 220.8% 128.5% (55209) ± ± ± ± kroa % 529.2% 238.0% 111.2% (21282) ± ± ± ± kroc % 537.4% 256.2% 133.9% (20749) ± ± ± ± krod % 503.1% 239.0% 127.7% (21294) ± ± ± ± ln % 575.8% 305.3% 188.5% (14379) ± ± ± ± Adj. Transposton edger Transposton 24.2% 186,2% 8,2% 7.0% ±362.6 ± ±241.5 ± % 229.1% 5.8% 4.7% ± ± ± ± ± ± ± ± % 368.1% 9.7% 6.3% ± ± ± % 401.9% 7.4% 5.5% ± ± ± ±620.1 ± % 435.9% 8.2% 7.1% ± ± ±728.4 ± % 368.9% 7.9% 7.1% ± ± ±589.7 ± % 475.5% 18.0% 7.1% ± ± ±626.2 ± two teratons to see the clear mpact of exchangng the exstng approaches wth ours. The swarm we use conssts of 100 partcles and we use 1000 teratons to optmze the functon. Each confguraton s run 100 tmes to compute the mean error and the standard devaton. The entres n Table 1 provde data n the followng format: Problem name (optmal value) relatve error maxmal value found by the algorthm mean value ± standard devaton best soluton found by the algorthm In Table 1, the left four result columns present our results obtaned wth the proposed DPSO varants wthout local optmzaton. In order to make our results also comparable to other approaches, we have added the local optmzaton method from Sh et al. n [14]. These results are shown n the rght four columns

11 of Table 1. Smlarly to our proposed approach, the method of Sh also avods the convergence problems analyzed n Sec. 3, but seems to result n a smaller relatve error. In every four columns block, the frst column shows the results of the method representng dfferences as transpostons and usng a smple composton to combne the dfferences. The other three columns show the results obtaned by the centrod-based method from Sec The centrod-based approach s combned wth varous representatons of dfferences, namely adjacent transpostons, transpostons and the edge recombnatons ntroduced n Sec Our centrod-based approach avods the counter-ntutve convergence behavor explaned by the theoretcal analyss n Sec. 3. The experments show that ths method s a better choce than the smple composton of dfferences. Another crucal factor s the choce of the representaton of partcle veloctes. Our expermental results show that transpostons are better than adjacent transpostons and that the proposed edge recombnaton method performs best. Fnally, a comparson between dfferent approaches of dscrete PSO can only be sgnfcant, f the actual contrbuton of the PSO algorthm s not obfuscated by an addtonal local search procedure. Ths s why we show the results of the pure PSO wthout local optmzaton, whch can serve as a reference for future DPSO varants for the TSP. 6 Conclusons In our theoretcal analyss of dscrete PSO for the TSP we showed that the convergence behavor the convergence behavor dffers sgnfcantly from what we expect from a PSO for contnuous problems. Our analyss can be appled manly to the DPSO varants of Clerc n [2] and Wang n [18] but can also be extended to the approaches of Sh et al. [14]. The convergence behavor can be observed whenever the local and the global attractor of a partcle nearly concde. If ths s the case the transpostons occurrng n respectve veloctes cancel each other out. Ths slows down the partcles and prevents convergence. We proposed a new model for partcle moton whch from a theoretcal pont of vew does not suffer from the aforementoned convergence problem. Ths s backed by our experments, whch show a clear mprovement of the DPSO performance wth ths model. Addtonally we ntroduced a representaton for partcle veloctes, whch s based on edge exchanges n a tour. Our evaluaton shows that the edge exchange-based representaton produces better results than tradtonal approaches from the lterature. References 1. D. Anghnolf and M. Paolucc. A new dscrete partcle swarm optmzaton approach for the sngle-machne total weghted tardness schedulng problem wth sequence-dependent setup tmes. European Journal of Operatonal Research, 193:73 85, 2009.

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