wosp: multi-optima particle swarm algorithm · 2016. 12. 2. · 3thewavesofswarmparticles (wosp)...

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WoSP: A Multi-Optima Particle Swarm Algorithm

Tim HendtlassCentre for Intelligent Systems and Complex Processes

Swinburne University of TechnologyP. 0. Box 218 Hawthorn, Australia 3122

[email protected]

Abstract. When investigating multi-optima problems, aparticle swarm algorithm should not converge on asingle optima but ideally should explore many optimaby continual searching. The common practice of onlyevaluating each particle's performance at discreteintervals can, at small computational cost, be used toadjust particle behaviour in situations where theswarm is 'settling' so as to encourage the swarm toexplore further. An algorithm is proposed that, bymaking each wave of particles partially independent, issuitable for multi optima problems.

1 IntroductionThe Particle Swarm Optimisation (PSO) algorithm isbased on observed aggregations of biological creatures,such as birds, and attempts to model their balancedbehaviour between exploration and exploitation, thushoping to replicate their good search behaviour.

As an optimisation tool, PSO has proved extremelyeffective on a wide range of problems. However, thecommon form of the PSO algorithm replicates thebiological behaviour in which particles typically settlearound a single optimum. This is not suitable behaviourfor investigating problems with multiple optima. In suchcases the desired behaviour is for the swarm to convergeon an optimum but then to move off to seek other optimaonce the value of this optimum is established.

It is possible to use discrete evaluation to alter thebehaviour of PSO so as to automatically achieve furtheraggressive exploration after each optimum has beenexplored. This paper describes a way in which thisbehaviour can be achieved. Results for three problemspaces are given that show the desirable properties of thisalgorithm.

2 The Basic PSO AlgorithmThe PSO algorithm uses a number of particles (the swarm)each of which describes a possible solution to the problembeing explored. Each particle moves through problem

space so that the set of solutions represented by the swarmcontinuously changes. The movement of each particle isinfluenced by the current situation of all other particles inthe swarm, together with some past history of the

X__B x-S

VT+I =%(MVT +rand G( )+rand L(-)) (1)

performance of the swarm as a whole.

Formally, the new velocity of a particle at time T+t isVT+t and is given by:

where VT is the velocity of this particle at time T, M is the

momentum, X is the current position of the individualand rand is a random number in the range (0, 1). G and Lset the relative attention to be placed on the positions Band S respectively. B is the best position found by anyindividual in the swarm so far, and S a position derivedby comparing the relative performance of this individualwith the performances of a number of other swarmmembers. M, G and L are bounded to the range (0, 1).The parameter t is the time between updates. The factor Xprovides velocity constriction. For more information onbasic PSO see [1,2,3,4].

Traditionally finding the position S involves defininga neighbourhood of the particle and only considering theeffect of other particles within that neighbourhood. Analtemate way of calculating the position S in an effective(but computationally modest) way that takes account ofthe performance of all swarm members has beenintroduced in [5]. All other swarm members areconsidered to influence the position S for a given particlebut the magnitude of the influence decreases with bothfitness and distance from the given particle for aminimization problem'.

For a maximisation problem it would increase with fitness butdecrease with distance.

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Equation 1 differs slightly from the conventional

version, in that the unit vectors from X to B and from

X to S are used rather than the vectors themselves. Thisprovides a less greedy version of the equation compared to

the conventional as the magnitude of the attraction to the

points B and S no longer depends on the distance fromthese points. This becomes an important change when, as

in this work, particles are required to explore at distancesfar from currently known optima.

The parameter t is usually taken to be one basic timeinterval and so is often omitted from the equation. Thealgorithm described in this paper depends on a finite valuebeing used for t.

3 The Waves of Swarm Particles (WoSP)Algorithm

3.1 Adding a short-range forceIn order to alter the behaviour of particles when they are

settling close together, it is desirable to introduce a short-range interaction between particles. One possibility is tointroduce a gravitational style attraction that produces a

force of attraction of particle i towards particle j, whosemagnitude is inversely proportional to some power p ofthe distance between them. This short-range force (SRF)will produce a velocity component vi, of one particle

towards the other that can be represented byK

v= (2)dp

where d.. is the distance between the particles i andj and

K is a constant. When combined with non-continuousevaluation this short-range force can have a useful effect.

3.2 The Effect of Discrete EvaluationOne effect of discrete evaluation on the basic PSOalgorithm is that particles may move directly over placesof interest but not recognise them, unless the instant oftransit happens to coincide with a fitness evaluation. Theactual distance travelled between evaluations depends on

the velocity of the particle and it is important that particlesslow in regions containing a fitness peak. The basic PSOtends to do this as a consequence of drawing particlesback that have 'overflown' the current best position.

The short-range force just introduced will have littleeffect while the swarm is dispersed for either continuous2or discrete evaluation. However, as particles approacheach other the magnitude of the short-range force willincrease significantly, producing a substantial increase inthe velocity of the particles towards each other. Fordiscrete evaluation, by the time of the next evaluation,particles may have passed each other and be at such a

2 Truly continuous evaluation is obviously impossible for any

computer based PSO algorithm. Here the practical meaning ofthe word 'continuous' implies that the interval betweenevaluations is sufficiently short so that no particle has moved a

significant distance between evaluations.

distance apart that the short-range attraction that mightbring them back together is far too weak to do this. As a

result, the particles will continue to move rapidly apartwith almost undiminished velocity, exploring beyond theirprevious positions. This process is shown in Figure 1.

At time T, the separation between the particles ismoving into the region in which the magnitude of theshort-range attraction (shown by broad arrows) isbecoming significant. This augments the effect of theirvelocities (shown by thin arrows) so that the particlesmove close together. By time T+t the particles are closeand the short-range effect is large. As a result, the velocityof the particles increases substantially, almost entirely as a

consequence of the short-range attraction. By time T+2twhen the next evaluation is made the particles have passedeach other, and are so far apart the short-range force isweak. Consequently, the particles continue to diverge,retaining at T+2t much of the velocity obtained as a resultof the short-range forces acting at time T+t. The short-range forces will continue to decrease as the particlesmove apart, leaving only the normal swarm factors toinfluence their future movement in the absence of otherclose encounters.

If continuous evaluation could be used, as soon as theparticles pass each other the direction of the short-rangeforce would reverse and the still high values of short-range attraction would rapidly nullify the effects of theacceleration as the particles converged. After the particleshave passed and separated they would be first slowed,then come to rest before once again converging thuseventually entering a stable limit cycle.

Figure 1. A series of 'snapshots' showing the twoparticles (shown as circles), their velocities (thinarrows and forces (thick arrows). The 'snapshots' are

taken starting at some arbitrary time T (at the top)with the lower 'snapshots' being taken progressivelylater. The time interval between 'snapshots' is thebasic time interval for the PSO and is represented by t.

Using discrete evaluation the effect described mightoccur if two particles happen to pass closely in a region ofindifferent fitness but is most likely to happen as particlesconverge on a single optimum. In this latter case the short-range force will have the desirable effect that some of theneighbourhood so engaged will be ejected with significantvelocities, and thus reduce the number of those left to

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converge and therefore reduce the probability of furtherejections.

Hence the effect of the short-range force on the normalsettling behaviour of the PSO is self-limiting with someswarm particles being left to continue exploration in thelocal vicinity. Further discussion of such a short-rangeforce can be found in [6].

3.3 Organising Ejected Particles into WavesSimply ejecting particles from the locations of a knownoptimum will only result in other optima being foundunder very favourable circumstances when the ejectedparticles still have knowledge of the fitness at their pointof ejection. This knowledge will tend to draw them backto that region unless they happen upon an area with evenbetter fitness before the effect of the global attractioncancels their short range force induced velocity. Theprobability of such favourable ejection occurringdecreases rapidly with increased problem dimensionality.

One solution is for ejected particles to 'forget' all aboutthe optimum they were ejected from. This can be achievedby assigning each particle to a wave, and treating eachwave as a "sub swarm" by only allowing particles in a

particular wave to respond to the values of B and Sgenerated by particles in their wave.

Initially all particles belong to wave zero. Every time aparticle is ejected it is promoted by having its wavenumber increased to that of the highest numbered wave(creating a new wave if necessary) and it starts to respondto the other particles in its new wave (if any). Sinceparticles are commonly (but not always) ejected in pairs awave will typically have an initial size of two. When twoparticles start a new wave (say wave N), the initial valueof B will be the best position occupied by one of theseparticles. To reduce the probability that this will be a pointassociated with the optimum the particles are just leaving,every particle is required to move a user specifiedminimum distance away from its promotion positionbefore it is allowed to inform other members of its waveon its current situation. This, together with the highejection velocity, and an active repulsion from thisparticles promotion points until it is at least a specifieddistance (the search scale) from all of them, sharplyreduces the probability that particles will fall back to theoptimum region associated with the wave they just left.

As most particles are promoted to successively higherwaves, it is possible for some particles to get 'left behind'still circling an optimum found long ago but not gettingclose enough to other particles be ejected so as to join anew wave. As such these particles represent a wastedresource.

Cleaning up left over wave remnants can be achievedin two ways. Firstly, when a promotion event takes placethat would leave only one particle in a wave, this lastparticle is also recruited to the wave the promoted particlehas just joined. Secondly, another inter-wave interactionis introduced. Once the value of B for wave N is betterthan the value of B for some earlier (lower numbered)

wave, all remaining members of the earlier wave are

promoted to wave N and start to respond to wave N's Band S values. Assuming that these just promotedparticles do not uncover any region of interest while intransit (or move close enough to another particle so as tocause another promotion event), they will finally arrive inthe region being explored by wave N.

3.4 Detecting a Particle Promotion EjectionIdeally particles should be considered as ejected (andtherefore promoted) when they come close while the waveis settling on an optimum. As a swarm settles the speed ofthe particles deceases. The ratio of the velocity componentintroduced by the effect of the short-range force to theother velocity components is a suitable measure withwhich an ejection can be detected. Promotion occurs anytime this ratio exceeds a user specified value, called thepromotion factor.

4 The WoSP Algorithm in DetailEach particle is in one of two different modes ofbehaviour as determined by its distance from the closest ofall of its previous promotion points. Only if this distance ismore than the scale search parameter is the particle able toreport its fitness and position to its wave. Only reportingparticles are allowed to respond to their waves B and Svalues.

The algorithm proceeds as follows. The new positionof each particle is first calculated, based on the positionsand velocities calculated in the previous iteration andassuming that the velocity remained unchanged for thetime t between the iterations. A check is made of theclosest distance of each particle to any of its promotionpoints and its report status is updated as required. Thefitness of each particle is calculated. Starting from theparticle with the best fitness, and in descending order offitness, those particles allowed to report do so to theirwave, updating that waves best-known point as required.

Each time a wave updates its best position a check ismade to see if this in now fitter than the best position ofsome other lower numbered (earlier) wave. When thisoccurs, all members of earlier wave immediately join thatlatter wave without recording their current position as apromotion point.

The velocity of every particle is now updated. Particlesthat are allowed to report update their velocity as:

VT+t =(MVT+ rand G( )+rand L(X 5 )) (3)

Particles that are not allowed to report, as they are tooclose to one or more of their promotion points, updatetheir velocity as:

VT+=(VT- GPC- LP) (4)where Pc is the unit vector towards the closest previous

promotion point and P1 is the unit vector in the direction

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of the smallest absolute component of VT. Without the

component Pt, exploration tends to be concentrated on ahyper plane that encompasses the positions of the first fewof the particle's promotion points.

Every particle is promoted when the velocitycomponent caused by the short-range force is more thanpromote factor times the vector sum of the other velocitycomponents. It either joins the highest current numberwave or, if it is already a member of the highest numberwave, starts a new wave with a wave number of onehigher. The position it was in when promoted is added tothe particle's list of promotion points. If this promotionleaves a wave with only one member, this is alsopromoted as part of the process of cleaning up old waves.This compulsory recruited particle does not record itsposition as a new promotion point.

4.1 The Computation Cost of the WoSP AlgorithmThe extra computational cost introduced to the basicswarm algorithm by the short range force and the WoSPalgorithm can be calculated by timing a series of repeatswith a fitness function that return a constant value. Noswarm coalescing takes place under these conditions andthe number of promotions is a function of the startingconditions. The average extra computation observed from100 repeats, compared to the basic swarm algorithm, wasabout 55% for the short-range force only and just over60% for the full WoSP algorithm. These values refer tothe basic algorithms excluding the time for fitnessevaluation. Since fitness assessment is often the dominantcomputational cost in real life problems, this means thatfor such problems the overhead introduced by the WoSPalgorithm would be very small relative to the basic swarm.

5 Experimental results

5.1 A Three Maximum ProblemThe surface shown in figure 2 has three maxima, labelledA, B and C. The swarm particles were initialised fromrandom positions within a restricted region as indicated bythe circle labelled 'Start' (at the front of the figure). Thispoint was chosen to be far from any maximum. Theclosest local maximum to this start region (maximum B) isthe poorest of the three. The second closest maximum(maximum A) is the highest. This test was designed notonly to see if particles would move from maximum B tomaximum A (a better maximum) but also to examine howwell they would explore beyond maximum A andinvestigate maximum C which was of intermediatefitness3. The exact fitness values and distances from thecentre of the particle start region are shown in Table 1.

3 Although, knowing the maxima in this problem, it is clear thatexploring from maximum A to maximum C is of no practicalpurpose, this information would not be available a priori for areal problem. Then the ability to leave a fit point and go on toexplore a point of lower fitness could be an important step onthe way to finding some other point of better fitness than anyyet explored.

This problem was designed so that the performance ofthe algorithm with just the short-range force was basicallythe same as for the basic PSO. This is because theprobability of a particle ejected from maximum Bencountering the limited region round maximum A thathas a higher fitness than maximum B before the globalcomponent returns it to maximum B is almost zero.

The key parameters used are listed in Table 2 and theresults for basic PSO and for WoSP are presented in Table3. Note that, as far as finding the first maximum isconcerned, the basic swarm and WoSP have the sameresults. However, the basic swarm does not explorefurther, with the result that just over 87% of the time itsettles for the poorest of the three maxima. The one casein which the basic swarm found two maxima appears to bea result of the swarm splitting into two and starting tocoalesce on both maxima. The particles converging onmaximum B met the criterion for 'found' just before theparticles converging on maximum A. The WoSPalgorithm, on the other hand, while again overwhelminglyidentifying maximum B first, continues to explore andfinds the global maximum A every time.

Figure 2 The three maximum problem surface.

Maximlum A B Cdlue 8.732 8.176 8.518

!average start 428.7position. 272.8 513.4

Table 1. Maximum values and distances from thecentre of the start circle for the problem surface

shown in Figure 2.

SRF coefficient 0.01 Search scale 50

L11 9SRFpowe J2 Promotion factor 10Table 2. The key SRF and wave parameter values

used.

Basic Swarm WoSPMaximum A B C A B CFound first 217 783 0 217 783 0

Found second 1 0 0 781 3 216Found third 0 0 0 2 21 765Total ound 218 783 0 1000 807 981

Table 3. The relative performance of the basic PSOand WoSP algorithms.

While it finds at least two maxima on every occasion, itonly found all three maxima 78.8% of the time before the

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run was terminated after 1000 iterations. The basic PSOtook just over 50 iterations (on average) to find the first(and, apart from one case, only) maximum. The WoSPalgorithm took almost the same time to find the firstmaximum, but took some 800 iterations on average to findall three maxima (when it did in fact find all three).

5.2 A Dual Cluster Problem.The dual cluster problem space was designed toinvestigate the effect of the search scale parameter on theperformance of the WoSP algorithm. The space consistsof two clusters of minima, each cluster consisting of sixlesser minima surrounding a central better minimum as

shown in Fi ure 3.

Figure 3. The 14 minima, two cluster problem space.

In this space the fitness of a particle at some position isthe minimum of the score values from that position to eachof the 14 defined points.

x y z FloorA- -200 -200 -200 0B- -150 -200 -200 5C- -250 -200 -200 5D- -200 -150 -200 5E- -200 -250 -200 5F- -200 -200 -150 5G- -200 -200 -250 5A+ 200 200 200 0B+ 150 200 200 5C+ 250 200 200 5D+ 200 150 200 5E+ 200 250 200 5F+ 200 200 150 5G+ 200 200 250 5

Table 4. The position and floor values of each of the 14minima

The score value of point X is the distance from theparticle to X plus the floor value of point X, the positionand floor values of the minima are given in table 4.

This produces two equal global minima with values ofzero at points A- and A+, together with twelve other local

minima (one at each of the other defined points) each witha value of five. Each of these local minima is a distance ofexactly 50 units from one of the global minima, pairs oflocal minima within a cluster are separated by either justunder 71 or by exactly 100 units. The two global maximawere approximately 700 units apart.

The results obtained from 100 independent runs at eachof five search scales are shown in table 5. The figures are

number of times this maximum (or group of maxima) were

found in 100 independent trials. The first search scale waschosen to be less than the closest spacing while the restwere chosen to be at approximately one of the spacingsbetween minima.

Table 5. The times each minimum, or selectedcombinations of minima, were found in 100 WoSP runs

5.3 A Problem with 830 Maxima.The results for a simple problem presented in table 5 donot illustrate the full potential of the WoSP algorithm.This becomes more obvious from consideration of theresults obtained seeking the maximum for Schwefel'sfunction in 30 dimensions (equation 5).

30

f= x sin( xj) where IxiI < 500. (5)i=l

This has 830 maxima, with one global maximum of12569.5 when xi=420.9687 for all values of i. Theconstraint that |x<500 for all values of i was hard

coded; any particle underwent a fully elastic reflection on

reaching this limit.Using the parameter values shown in Table 6, in a

series of 100 trials, each for 200,000 iterations (aniteration consists of all particles making one velocityupdate), the global maximum was found 41 times by theWoSP algorithm. On average this best position was one

of more than 100 explored during the run and was foundafter about 118,000 iterations. A second set of runs,identical except for the duration, which was set to

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Search scaleMinimum 30 50 70 100 700

A- 77 61 56 97 65B- 145 31 16 22 5C- 190 15 29 33 17D- 147 25 23 19 14E- 214 19 27 36 30F- 147 38 18 19 8G- 198 18 35 34 23A+ 66 50 61 95 49B+ 127 11 13 19 16C+ 163 38 33 21 35D+ 143 12 21 13 16E+ 194 36 23 27 34F+ 126 12 22 21 15G+ 182 33 17 42 34

Any other point 0 0 0 0 0A- or A+ found 100 100 98 100 99A- and A+ found 28 8 12 60 15A- found > once 8 0 1 12 0A+ found > once 5 2 3 13 0


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2,000,000 iterations, showed a slight performance change,but one that was insignificant when compared with theorder of magnitude increase in computing cost.

Parameter ValueNumber of particles 30

Maximum number of iterations 200,000Total number of evaluations 6,000,000

Momentum 0.95B global best factor 0.9

Normal L local best factor 0.5L if within search scale of a 20

promotion pointSearch scale 500

Promote factor 2Table 6. The parameter values used.

As each wave died (lost its last particle or had all itsmembers compulsorily promoted to a later wave that wasoutperforming it) a simple hill climbing local search agentwas used to find the local optimum in the vicinity of thebest position known to this wave. Many optimisationheuristics combine an algorithm with coarse global searchcapabilities together with a suitable local search heuristic.

Figure 4. A history of the number of particles ineach wave as a function of iteration for the first 8000iterations ofWoSP exploring Schwefel's function.This extra expenditure of computing resource is ideally

only warranted in the vicinity of an optimum, a conditionthat precludes its application to the basic particle swarm

algorithm. The high probability that each wave has by itsdeath investigated in the vicinity of an optimum makes itsuse in these few positions both highly rewarding andcomputationally reasonable

Figure 5. A history the best fitness discovered by eachwave as a function of iteration for the first 8000

iterations ofWoSP exploring Schwefel's function.

Figures 4 and 5 show the number of particles in eachwave (Figure 4) and the best fitness yet found by eachwave (Figure 5) for each of the first 8000 iterations of a

run in which the WoSP algorithm was finding maxima ofSchwefel's function in 30 dimensions. Note that waves 8to 10 have not yet died by the time the plots finish. Notehow later waves (e.g. wave 8) absorb the particles fromearlier waves whose fitness they exceed. Also note thatwaves exploring poorer regions tend persist for shortertimes than waves exploring better regions.

6 Discussion

6.1 The Dual Cluster Problem Space.Examination of these result for the dual cluster problemspace reveals the following:

* No wave ever converges at a point that is not in thenear vicinity of a minimum.

* Although a particle was prohibited from reportingwhen within search scale of any of it's promotionpoints, minima can be re-explored by different waves.

This can occur since the particles in a particular wavemay have reached that wave with different wave

membership histories. This occurs as a promotedparticle joins the highest number wave existing at thetime of promotion, creating this wave if necessary.Once a particle that has not previously explored in theregion of some optimum A reaches this region it can

report to the wave, other particles that have alreadyexplored this region are none the less now attracted toit and will again converge on it and be promoted fromit. This is an important feature of the algorithm,

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without it once an optimum has been explored it couldonly be used as a base for further exploration once.

* With a search scale less than the smallest inter-minimum spacing, there is substantial re-exploration,especially of the 'lesser' minima. As the search scaleincreases, this decreases but does not totally cease.

* The algorithm is effective at finding at least one or thetwo equal global minima for all values of search scale,but the probability of finding both increases as thesearch scale is itself increased as this prevents so

much effort being spent on examining the 'lesser'minima. However, once the search scale approachesthe separation between the two global minima, findingone will preclude finding the second unless this is thevery next minimum found after the first global one.

* Note how the two global minima A- and A+ may befound more than once: if the search scale is smallthere is much exploration of all minima and so theparticles of a particular wave may have many differentwave membership histories. This, of course, can allowre-exploration of 'lesser' minima too, but as the searchscale is increased the number of lesser minimaavailable to be found is decreased.

6.2 The Problem Space with 830 Maxima.Figure 6 shows the best fitness achieved by each wave as

it died (or by 200,000 iterations when the run was

terminated). In each case, at death or termination, a localsearch agent used directed random search to explore thelocal optimum. It is the fitness found by this local searchagent that assigned to the wave and shown above. Aswaves are numbered sequentially as they are created, thewave number is an approximation to time. In thisparticular run 98 different maxima were explored. One ofthese, the global maximum, was explored three times, nineothers were explored twice and two were explored threetimes. All other 86 maxima were only explored once.

The absorption by a later wave of the particles of any

earlier wave that it is outperforming is the reason for thefairly steady improvement leading up to the firstexploration of the global maximum by wave twenty-eight.This absorption process is effectively a survival of thefittest with the absorbed particles moving to join theparticles already in the absorbing wave. Once there theycan explore the immediate vicinity but they, like allparticles there, may also be promoted again thus using thisas a springboard for further exploration seeking even

better maxima. Thus an evolution process is taking placewith these waves, which provides the early sharpimprovement in the fitness of maxima explored. Once theglobal maximum has been found evolution can onlyoperate beneficially on waves that are exploring sub-global maxima.

Particles are actively rejected if they again approach a

position from which they have been promoted and are

prohibited from providing feedback to their wave if withinsearch scale of such a position. The particles of wave 28are, in time, promoted into later waves but cannot notifytheir wave of any fitness in the vicinity of the global

maximum. The limited re-exploration that occurs must beinitiated by some particle that has not been promoted fromthe global maximum and is thus able to report: the otherparticles in its wave can still respond to the best position itreports even though they themselves cannot report.

Fitness v Wave number

n(A

CD

U-

13000

12000

11000

10000

9000

8000

7000

60uu00 20 40 60 80 100 120 140

Wave number

Figure 6. The best fitness found by each waveduring a typical run.

Re-exploration could be totally prohibited if eachparticle, instead of maintaining a list of its own promotionpoints, had access to a global list of all promotion points.However, this would likely be of disadvantage as it wouldmake the progress of waves in the problem space a seriesof linear steps from maximum to maximum without (asnow) allowing for forking to occur and one maximum tobe used as a base for exploring two or more other maxima.

7 Comparison to Other ApproachesA number of variations to the basic PSO algorithm havebeen proposed that are intended to either promoteconvergence to a number of sub-swarms so that a range ofoptima may be explored in parallel and / or permit thetracking of optima of a problem domain with a temporalcomponent. [7] and [8] discuss two such methods andcontain a review of a number of other approaches.

These PSO versions use techniques already proven ingenetic algorithms, such as niching and speciation. Ingeneral particles will remain in the optima found, althoughis some cases the fitness function is modified so that itbecomes unattractive to particles which therefore leave toexplore further. In the author's experience, along with thedesirable result of forcing continued exploration, suchfunction modification is prone to producing false optima.

In its current form the WoSP algorithm does not lenditself to tracking optima. While at any time several optimamay be being explored in parallel, the main emphasis is onthe sequential exploration of optima. The fitness functiondoes not require any modification after an optimum islocated; the maintenance of a tabu list by each particleappears to achieve the same end without producing extra(false) optima, but at the cost of introducing an extraparameter, the search scale.

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8 Concluding RemarksThis paper introduces a version of the particle swarm

optimisation algorithm intended to be efficient whenexploring problem spaces with multiple optima. As noadditional fitness evaluations are required for the WoSPalgorithm compared to basic PSO, the additionalcomputational cost of the behaviour modification is likelyto be relatively small when compared to the overallcomputation involved, especially for problems withcomplex fitness functions.

The results obtained on a simple contrived three-maximum problem clearly show that the WoSP algorithmis able to escape from local sub-optima and continue tosearch for other optima.

The dual cluster problem space results are instructive inshowing both the effect of the choice of search scale onthis simple problem but also how re-exploration ofpreviously explored optima is important in preventing thealgorithm from being too greedy by allowing more thanone wave of exploration to be initiated from one place.

Results obtained from the more challenging Schwefel'sfunction are most instructive. When one considers that6x106 evaluations were done during the 200,000 iterationsfor a 41% chance of finding the best of approximately1.2x1027 maxima, the performance of this technique onthis problem is quite remarkable and a testament to thepower of combining swarm exploration of each maximumwith an evolutionary driven search for further maxima.The reports on the locations and fitness's of other maximaobtained are a bonus that may be of considerable use inpractical problems, such as scheduling.

The regular spacing of the maxima in this particularproblem may have made it particularly suited to the WoSPalgorithm, but the results are sufficiently encouraging toaugur well for other problem domains.

Proceedings of the 2000 Congress on EvolutionaryComputation, pp. 84-88. (2000)

5 Hendtlass T. "A Combined Swarm Differential EvolutionAlgorithm for Optimization Problems". Lecture Notes inArtificial Intelligence, Vol 2070, pp. 374-382, Springer,Berlin. (2001)

6 Hendtlass T. and Rodgers T. "Discrete Evaluation and theParticle Swarm Algorithm". Proceedings of Complex04,Cairns, Australia, pp. 14-22 (2004)

7 Daniel Parrot and Xiaodong Li. "A Particle SwarmModel for Tracking Multiple Peaks in a DynamicEnvironment using Speciation " Proceeding of the 2004Congress on Evolutionary Computation (CEC'04), p.98- 103 (2004)

8 R Brits, A.P Engelbrecht, F van den Bergh. "A NichingParticle Swarm Optimizer" Proceedings of the 4thAsia-Pacific Conference on Simulated Evolution andLearning 2002 (SEAL 2002), Singapore. pp. 692-696,(2002)

Acknowledgements.

The author would like to express his thanks to the othermembers of CISCP for informative and stimulatingdiscussion. He would also like to thank the anonymousreferees for their helpful comments on the first version ofthis paper.

Bibliography1 Kennedy J., and Eberhart R.C. "Particle Swarm

Optimization", Proc. IEEE International Conference onNeural Networks, Perth Australia, IEEE Service Centre,Piscataway NJ USA IV: 1942-1948. (1995)

2 Eberhart R. C., Dobbins P. and Simpson P. ComputationalIntelligence PC Tools, Academic Press, Boston. (1996)

3 Kennedy J., and Eberhart R.C., The Particle Swarm: SocialAdaptation in Information-Processing Systems. Chapter 25 inNew Ideas in Optimization. Come D., Dorigo M., and GloverF. (Editors) McGraw-Hill Publishing Company, England,ISBN 007 709506 5. (1999)

4 Eberhart R. and Shi Y. "Comparing Inertia Weights andConstriction Factors in Particle Swarm Optimisation",

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