# A Simplified Recombinant PSO

Authors: Dan Bratton; Tim Blackwell
Journal: Journal of Artificial Evolution and Applications
 (2008)

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## Abstract

Simplified forms of the particle swarm algorithm are very
beneficial in contributing to understanding how a particle
swarm optimization (PSO) swarm functions. One of these forms,
PSO with discrete recombination, is extended and analyzed,
demonstrating not just improvements in performance relative to
a standard PSO algorithm, but also significantly different
behavior, namely, a reduction in bursting patterns due to the
removal of stochastic components from the update
equations.


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## Body

## 1. Introduction

Originally  conceived as a modification to the standard
PSO algorithm for use on self-reconfigurable adaptive systems used in on-chip
hardware processes, PSO with discrete recombination (PSO-DR) introduces several
appealing and effective modifications, resulting in a simpler variant of the
original [1]. It is one of the more interesting advances in PSO research over
the last few years because these simplifications apparently do not degrade
performance yet they remove various issues associated with the stochasticity of
the PSO acceleration parameters that hinder theoretical analysis of PSO.Physical creation of hardware-based optimizers is a
substantially more intricate undertaking than software implementations, so
fast, simple algorithms are desirable in order to minimize complexity. The
comparative straightforwardness of PSO to many other evolutionary optimization
algorithms makes it a good choice for this purpose, and further modifications
were applied by the authors of [1] in order to simplify it even further and to
introduce concepts from recombinant evolutionary techniques. The resulting
algorithm, which can be implemented using only addition and subtraction
operators and a simple 1-bit random number generator, is well suited for
dedicated hardware settings.Despite this rather specific original design
specification, PSO-DR has shown to be a robust optimizer in its own right,
equalling or surpassing a more common PSO implementation on a few tested
benchmarks [1]. In this paper we extend the original work of Peña et al. by
considering alternative topologies and parameter settings,  running comparisons
over a more comprehensive test suite, deriving simplified variants of the
algorithm, and subjecting the model to a burst analysis.The following section introduces PSO-DR (known here as
model 1) as originally defined by Peña et al. and summarizes the burst analysis
of [2]. Section 3 describes a series of simplifications to PSO-DR (models 2 and
3) which are introduced in this paper. The motivations for these
simplifications are explained. Section 4 presents the results of performance
experiments of models 1–3, and for comparative purposes, standard PSO.
Following this, the paper proceeds with an empirical investigation of bursting
patterns in recombinant PSO. The final section together draws together the
experimental results of this paper and advances some ideas for the immediate
future of PSO research.


## 2. PSO with Discrete Recombination

The velocity update for particlei in standard PSO
(SPSO) in the inertia weight formalism is
(1)IW:vidt+1=wvidt+ϕ2u1(pid−xidt)+ϕ2u2(pnd−xidt), where d labels
components of the position and velocity vectors, d=1,2,…,D, p→i is the personal
best position achieved by i, p→n is the best
position of informers in i's  social
neighborhood and u1,2∼U(0,1) [3]. After
velocity update, the particle position is adjusted:
(2)xidt+1=vidt+1+xidt.Peña et al. introduced a recombinant version of PSO by
replacing either the personal best or the neighborhood best position by the
recombinant position [1]. We focus here on the former for reasons of improved
performance and the more interesting social aspect. A recombinant position
vector r→ is defined by
(3)rid=ηdpld+(1−ηd)prd, where ηd=U{0,1} and p→l,r are immediate
left and right neighbors of i in a ring
topology. While separate random numbers ηd are used for
separate dimensions d, a single value is generated for each single
dimension and used for both occurrences of ηd in that
dimension. This places r→i at a corner of
the smallest D-dimensional
box which has pl and pr at its corners.The authors of [1], in a search for a very efficient
implementation, argued for the removal of the random numbers u1,2 from
(1) and parameter settings ϕ=2 and w=0.5. The velocity update for the original form of PSO-DR
is
(4)DR:vidt+1=wvidt+ϕ2(rid−xidt)+ϕ2(pnd−xidt).The choice ofϕ was based on
the observation that ϕ≈4.0 in standard
PSO, but, since u1,2 are uniform in [0,1], the expectation value of ϕu1,2 is 2.0.
Furthermore, the multiplication by w=0.5 can be
implemented in hardware by a right shift operation. While optimal efficiency is
desirable for hardware implementations, this issue does not concern us to the
same degree in this study of (4) and it is one aim of this paper to study
PSO-DR for arbitrary parameter values.Although (4) contains a random element in the
recombinant position, the acceleration parameters are constant. In other words,
the update rule has additive rather than multiplicative stochasticity [2]. This
has two ramifications; first, a stability condition can be computed based on
the theory of second order, fixed parameter, difference equations and second,
recombinant PSO is predicted not to exhibit particle velocity bursts. The
details of these results are to be found in [2]. The stability condition is
(5)|w|<1,0<ϕ<2(1+w).It is known that PSO at stagnation, that is, when no
improvements to personal bests are occurring, and the particles effectively
decouple, exhibits bursts of outliers [4]. These are temporary excursions of
the particle to large distances from the attractors. A burst will typically
grow to a maximum and then return through a number of damped oscillations to
the region of the attractors. The origin of bursts, and of the concomitant
fattening of the tails of the position distribution at stagnation, can be
traced to the second-order stochastic difference equation
(6)x(t+1)+a(t)x(t)+bx(t−1)=c(t) which is equivalent to SPSO with the identification a(t)=(ϕ/2)(u1+u2)−w−1, b=w, and c(t)=(ϕ/2)(u1p1+u2p2) for fixed
attractors p1,2. Since max(|a|)>0, amplification of x(t) can occur
through repeated multiplication of x(t) by a despite the
second order reduction by multiplication by the constant b. Interestingly, the distribution tail of |x|, by virtue of the bursts that become increasingly
less probable for increasing size, is fattened compared to an exponential
falloff as provided by, for example, a Gaussian. A theoretical justification of
these power laws and some empirical tests can be found in [2].PSO bursts differ from the random outliers generated
by PSO models which replace velocity by sampling from a distribution with fat
tails such as a Richer and Blackwell [5]. In contradistinction to the outliers of these “bare
bones" formulations [6], the outliers from bursts occur in sequence, and they
are one dimensional. Bursting will therefore produce periods of rectilinear
motion where the particle will have a large velocity parallel to a coordinate
axis. Furthermore, large bursts may take the particle outside the search space.
Although this will not incur any penalty in lost function evaluations if
particles that exit the feasible bounds of the problem are not evaluated, as is
the common approach to this situation, they are not contributing to the search
while in outer space. PSO-DR, which is predicted not to have bursts [2],
therefore provides a salient comparison.


## 3. Simplifying Recombinant PSO

This section details the two new recombinant models
that are being proposed in this paper. To begin, an investigation into PSO-DR
reveals more interesting properties of the formulation. Performance plots for a
sweep through parameter space to find an optimal balance between the inertia
weight coefficientw and the ϕ coefficients
show that while the optimal region is spread across the parameter space, it
also intersects the axis for the w term (see Figure 1
for results on selected functions from Table 1). This demonstrates that the
system is able to obtain good optimal results even at w=0.0 and there is no
inertia term in the velocity update equations.Table 1
Benchmark
function equations.
Equationf1=∑i=1Dxi2f2=∑i=1D(∑j=1ixj)2f3=∑i=1D−1{100(xi+1−xi2)2+(xi−1)2}f4=−∑i=1Dxisin(xi)f5=∑i=1D{xi2−10cos(2πxi)+10}f6=−20exp{−0.21D∑i=1Dxi2}−exp{1D∑i=1Dcos(2πxi)}+20+ef7=14000∑i=1Dxi2−∏i=1Dcos(xii)+1f8=πD{10sin2(πyi)+∑i=1D−1(yi−1)2{1+10sin2(πyi+1)}+(yD−1)2}+∑i=1Dμ(xi,10,100,4)yi=1+14(xi+1)μ(xi,a,k,m)={k(xi−a)mxi>a0−a≤xi≤ak(−xi−a)mxi<−af9=0.1{sin2(3πxi)+∑i=1D−1(xi−1)2{1+sin2(3πxi+1)}+(xD−1)2×{1+sin2(2πxD)}}+∑i=1Dμ(xi,5,100,4)f10=4x12−2.1x14+13x16+x1x2−4x22+4x24f11={1+(x1+x2+1)2(19−14x1+3x12−14x2+6x1x2+3x22)}×{30+(2x1−3x2)2(18−32x1+12x12+48x2−36x1x2+27x22)}f12=−∑i=15{∑j=14(xj−aij)2+ci}−1f13=−∑i=17{∑j=14(xj−aij)2+ci}−1f14=−∑i=110{∑j=14(xj−aij)2+ci}−1Optimal regions forw versus ϕ in PSO-DR model
1, found empirically through 30 runs for each combination of parameters w=0.0,…,1.0,ϕ=0.0,…,5.0 at a granularity
of 0.1. Each contour line represents a 10% improvement in performance with the
region within the innermost line representing the best performing 10% of
possible combinations of w and ϕ.
(a)
f1(b)
f3(c)
f5(d)
f12Model 2 PSO-DR setsw=0, with a velocity update,
(7)DR2:vidt+1=ϕ2(rid−xidt)+ϕ2(pnd−xidt).Velocity now serves as a dummy variable in the update
equations (1) and (2) and model 2 can be represented as a single, velocity-free
rule
(8)DR2:xidt+1=xidt+ϕ2(rid−xidt)+ϕ2(pnd−xidt).At this point, the twoϕ terms were
detached and another sweep through parameter space to find an optimal
combination of the recombinant component via its coefficient ϕ1 and the
neighborhood best component via its coefficient ϕ2 was performed.
Surprisingly, results again showed that the optimal region intersects an axis,
this time for the neighborhood term (pgd−xidt) (see Figure 2 for
selected  results).Performance plots
forϕ1 and ϕ2 in PSO-DR model
2, found empirically through 30 runs for each combination of parameters ϕ1=0.0,…,4.5,ϕ2=0.0,…,4.5 at granularity
0.1.
(a)
f1(b)
f5This allows a further simplification to the update
equation (4), down to PSO-DR model 3:
(9)DR3:xidt+1=xidt+ϕ(rid−xidt) which is clearly a substantial reduction of the
original PSO-DR equation. This PSO variant, if it proves to be viable, would
raise a couple of interesting questions. To what extent is velocity a necessary
component, or is it a relic of the biological origins of PSO [6]? Secondly, how
important is the neighborhood component drawn from the single best neighbor?
The optimization process of Model 3 is entirely driven by the recombinant
component; this idea is reminiscent of fully informed particle swarms (FIPS)
[7], where the entire neighborhood influences particle behavior.
However, whereas FIPS allows every neighbor to influence a particle's behavior in every dimension, Model 3 allows only a single randomly chosen neighbor to fully influence the particle in each dimension. This gives the
particle an updated position that is a combination of the best positions of all
of its neighbors throughout all dimensions.The following section presents evidence that PSO-DR3
is a viable alternative to standard PSO by reporting on performance results for
all three models of PSO-DR over a number of commonly used test functions.


## 4. Performance Experiments

Algorithms were tested over a series of 14 benchmark
functions chosen for their variety, shown in Tables1 and 2. Functions f1−f3 are unimodal
functions with a single minimum, f4−f9 are complex
high-dimensional multimodal problems, each containing many local minima and a
single global optimum, and f10−f14 are
lower-dimensional multimodal problems with few local minima and a single global
optimum apart from f10, which is symmetric about the origin with two global
optima.Table 2
Benchmark
function details.
FunctionNameDFeasible boundsf1Sphere/parabola30(−100,100)Df2Schwefel
1.230(−100,100)Df3Generalized
Rosenbrock30(−30,30)Df4Generalized
Schwefel 2.630(−500,500)Df5Generalized
Rastrigin30(−5.12,5.12)Df6Ackley30(−32,32)Df7Generalized
Griewank30(−600,600)Df8Penalized
function P830(−50,50)Df9Penalized
function P1630(−50,50)Df10Six-hump
camel-back2(−5,5)Df11Goldstein-price2(−2,2)Df12Shekel
54(0,10)Df13Shekel
74(0,10)Df14Shekel
104(0,10)DParticles were initialized using theregion scaling technique where initialization takes place in an area of the search space known
not to contain the global optimum [8]. To avoid initializing the entire swarm
directly within a local minimum, as could be possible with f12−f14 if
initialization takes place in the bottom quarter of the search space in each
dimension (as is common), an area of initialization composed of the randomly
chosen top or bottom quarter of each dimension was defined, into which all
particles were placed with uniform distribution. This method ensures that the
swarm will not be initialized within the same area for every optimization run,
but will still be confined to an area at most 0.25D of the search
space, making the chance of initialization directly on or near the global
optimum extremely unlikely. In instances where the global optimum was located
at the center of the search space (i.e., f1,f2,f5−f7), the function
was shifted by a random vector with maximum magnitude of a tenth of the
size of the search space in each dimension for each run to remove any chance of
a centrist bias [9].This investigation tested PSO-DR model 1 using both
global (as used in the originally proposed algorithm) and local ring topologies
for selecting the neighborhood operatorpn. The parameter settings were Pena's, giving a
velocity update with the form
(10)vidt+1=0.5vidt+(rid−xidt)+(pnd−xidt).Results shown for PSO-DR  model 2 use the valueϕ≈ 1.6, while
those for PSO-DR model 3 use ϕ≈1.2. These
values were empirically determined to be optimal for these algorithms; an
analytical determination is the subject of current research. Results for both
models 2 and 3 are shown for runs using a ring topology, which showed superior
performance in testing.For comparison, results are presented for a standard
PSO algorithm (SPSO), which  operates using the constricted  velocity update
equation(11)vt+1=χ(vt+ϕ2u1(pi−xt)+ϕ2u2(pg−xt)) with ϕ=4.1,χ=0.72984 and with 50
particles [3]. All PSO-DR model tests were carried out using 50 particles as
well. Algorithm performance was measured as the minimum error |f(x)−f(x⋆)| found over the
trial where f(x⋆) is the fitness
at the global optimum for the problem. Results were averaged over 30
independent trials, and are displayed, with standard error, in Table 3. Values
less than 10−15 have been
rounded to 0.0.Table 3
Mean error
after 30 trials of 300 000 evaluations. Necessary function evaluations are
shown where 0.0 error was attained.
SPSOSPSOPSO-DRPSO-DRPSO-DRPSO-DRRingGlobalM1 RingM1 GlobalM2M3f10.0±0.00.0±0.00.0±0.00.0±0.00.0±0.00.0±0.097063±37746897±42159322±12533290±17060063±4175810±322f20.12±0.010.0±0.00.01±0.0020.0±0.03.7E-7±7.5E-85.14±1.27—297800±928—168852±1205——f36.18±1.078.37±2.2616.79±0.490.80±0.2934.57±5.4618.64±4.45——————f43385±403522±322697±363754±482418±271830±46——————f5163.50±5.64140.16±5.8744.64±2.71115.51±7.0335.21±2.139.88±0.86——————f618.28±0.8512.93±1.590.68±0.6718.51±0.900.0±0.00.0±0.0————287220±2105248160±1945f70.0±0.00.019±0.0040.0±0.00.008±0.0020.0±0.00.0±0.0110616±3320—101526±9227—81226±656070348±2954f80.004±0.0030.15±0.050.0±0.00.05±0.020.0±0.00.0±0.0——61370±249—85101±58195810±655f90.0±0.00.003±0.0010.0±0.00.002±0.00070.0±0.00.0±0.0106163±537—61793±221—86031±37792416±437f100.0±0.00.0±0.00.0±0.00.0±0.00.0±0.00.0±0.09348±19011808±44544577±760840015±44836103±1045918±75f110.0±0.00.0±0.00.0±0.00.0±0.00.0±0.00.0±0.08258±1047080±1084772±473968±276720±666846±87f120.59±0.334.61±0.540.17±0.174.34±0.590.0±0.00.0±0.0————59958±4316229±855f131.09±0.454.40±0.600.0±0.02.55±0.628.1E-11±1.8E-110.0±0.0——13433±249——14012±764f140.96±0.453.24±0.660.0±0.03.13±0.666.6E-11±1.7E-110.0±0.0——12760±1386——121031004±Performance results in Table3 for all models of
PSO-DR versus SPSO clearly indicate that it is a competitive variant,
especially on highly complex problems such as f5 (Rastrigin).
Statistical tests were performed on these results to determine the significance
of the performance differences between the two algorithms. To avoid the problem
of the probabilistic nature of t-tests potentially affecting results when
conducting multiple significance tests, a modified Bonferroni procedure was
applied to values of α for successive
tests [10]. This procedure involves inversely ranking observations by ascending
values of p, then setting
(12)α'=αinverserank.Results for these statistical tests on PSO-DR model 3
and SPSO are shown in Table4 and confirm that the performance is significantly
improved on 3 of the 14 tested functions, equivalent for 10 functions, and
worsened for 1 function for PSO-DR model 3 versus SPSO with ring topology.
Perhaps the most impressive improvement comes for f5 (Rastrigin), a
notoriously difficult multimodal problem that PSO algorithms  perform
poorly on some problems in high dimensionality.Table 4
Significance
for SPSO versus PSO-DR model 3 with ring topologies.
Functionp-valueInverse rankα'Significantf40140.003 571Yesf50130.003 846Yesf60120.004 167Yesf22.11e-11100.005Yesf30.0086110.004 545Nof130.0290.005 556Nof140.0480.00 625Nof120.266360.08Nof80.321570.007 143Nof1150.01Nof7140.0125Nof9130.016 667Nof10120.025Nof11110.05NoDue to the high number of function evaluations that
were performed to obtain these results  relative to previous work (where only
30 k–60 k function evaluations might be performed), selected convergence plots
are shown in Figure3. These show that the standard PSO obtains superior
results at the very start of the optimization process, up to 5000 function evaluations for the highest
observed value (Figure 3(b)). After the point at which
this occurs, PSO-DR model 3 surpasses the standard algorithm in performance, and maintains this advantage to the end of the
300 k function evaluations on 7 of the 14 tested problems (f4−f6,f8,f12−f14). On problems
for which both algorithms attained equal error levels of 0.0 (f1,f7,f9−f11), the point at
which this occurs, that is, when SPSO “catches up" to PSO-DR model 3,
can be observed in Table 3. On average, SPSO took 25% more function evaluations to attain the
optimum than PSO-DR model 3 on these problems. Finally, for the two problems on
which SPSO outperformed PSO-DR model 3 (f2,f3), the same
early performance is seen with PSO-DR model 3 surpassing SPSO in performance
early in the optimization process; in these cases, SPSO eventually repasses the
other algorithm by 50 k function evaluations.Convergence plots for SPSO and PSO-DR model 3 early in the
optimization process.
(a)
f1(b)
f5A potential explanation for this behavior lies in the
diversity of the swarms at this point in the optimization process. Figure4
shows the mean Euclidean distance between particles for the corresponding
convergence plots of Figure 3. It should be noted that uniform initialization
was used in the trials used to generate these plots; relative performance
between the algorithms was unaffected, and initializing particle positions
uniformly throughout the search space removes an unrelated phenomenon in
subspace initialization wherein the swarm expands greatly beyond the relatively
small initialization region at the start of the optimization process to explore
the search space. Expansion is common in the first few iterations using uniform
initialization as well, but this is inherent to the swarm behavior and
influenced only by the size of the entire search space.Diversity plots for SPSO and PSO-DR model 3 early in the
optimization process.
(a)
f1(b)
f5(c)
f5(zoomed)As can be seen in the plots of Figure4, neither swarm
type begins converging immediately following initialization but rather they
maintain their diversity or expand slightly. On a comparative basis, the
standard PSO swarm expands substantially more than the PSO-DR model 3 swarm;
for example Figure 4(c) shows that after the first 100 function evaluations, the mean distance
between particles in the standard PSO swarm increases from 23 to 31.5, while
the PSO-DR swarm diversity increases only from 23 to 24.5. Similar disparities
were observed for all other tested problems.It is reasonable to gather from these results that the
higher swarm diversity for the standard PSO algorithm early in the optimization
process demonstrates a wider spread of particle dispersion, and hence an
improved probability of finding and starting to explore the basin of attraction
for global or good local optima. PSO-DR model 3 expands very little, if at all,
early in the optimization process, resulting in delayed acquisition of optimal
regions of the search space.


## 5. Examination of Bursting

Bursts in the velocities of particles are commonly
observed using the standard PSO algorithm. These are generated by means of the
multiplicative stochasticity of the algorithm [2]. In order to investigate
bursting behavior in PSO-DR and SPSO an empirical measure was devised.This bursting measure was implemented to highlight
when a particle had a velocity in a single dimension that was considerably
higher than the next highest dimensional velocity. Bursting patterns of
behavior were detected by reporting that every time particle velocity in a
single dimension was a set amountλ times higher
than velocity in the next highest dimension. Bursting behavior is demonstrated
in Figure 6, where the velocity of a single particle in a 10-dimensional
problem is shown. On the plot of the multidimensional velocity of the SPSO
particle, it can be seen that velocity in a single dimension increases suddenly
and dramatically while remaining relatively level and low in all other
dimensions. This is an example of a velocity burst. While the figure shows
velocity for a single particle on a single run, examination of velocity plots
for hundreds of particles over dozens of runs confirmed this to be
representative of general particle behavior.Velocity for a PSO-DR particle is also shown in Figure6, and demonstrates the absence of bursts. Similarly to the SPSO plot,
examination of a large number of plots confirmed this to be representative of
general behavior for PSO-DR.Examination of these empirical analyses show that
PSO-DR clearly does not contain bursting behavior on the scale of SPSO while
demonstrating equal or superior performance on 13 of the 14 benchmark
functions, leading to the hypothesis that bursts are not, in fact, integral to
the successful operation of particle swarm algorithms. The fact that a very few
bursts do occur with PSO-DR indicates that it is a highly improbable feature of
DR dynamics.Analysis performed on statistics of several functions
shows that particle updates involving bursts are far less effective than more
common nonbursting updates. For example, results showed that for SPSO onf5 with λ=100, on average 20.1% of all particle, updates involve an improvement to the particle's best found position pi, whereas only 1.8% of updates
involving bursts result in an improvement to pi. Likewise, on average 0.9% of all
particle, updates improve the best found swarm position g, as opposed to only 0.01% for bursting
particles. Burst frequencies for values of λ from 10 to 150
are shown in Figure 5.Figure 5
Frequency of updates showing burst
behavior for values ofλ.Representative particle velocities for SPSO and PSO-DR on
10D Rastrigin.
(a)
SPSO(b)
PSO-DRIt is also interesting to note that far fewertotal updates result in an improved pi or g for PSO-DR when
compared to SPSO, for example, results showed that 20.1% of all updates improve pi for SPSO
compared with 0.64% for PSO-DR, and 0.91% improve g for SPSO
compared with 0.02% for PSO-DR on f5 for λ=100.


## 6. Conclusions

Simplification of the standard PSO algorithm is an
important step toward understanding how and why it is an effective optimizer.
By removing components of the algorithm and seeing how this affects
performance, we are granted insight into what those components contribute to
overall particle and swarm behaviors.In particular, this paper has proposed a very simple
PSO(13)DR3:xidt+1=xidt+ϕ(rid−xidt) which offers competitive performance to standard PSO,
but removes multiplicative randomness, inertia, and the personal memory term pi from the
position update.There is still much to be done before questions
concerning PSO behavior can be completely answered, and it is expected that the
next decade of PSO research will be focused on understanding the basic
algorithm that powers both the standard implementation and its variants.In that light, the PSO-DR variant is important not
only because of its improved performance on several benchmark functions, but
also because its simplified state allows us to examine what happens to the
standard algorithm when pieces are modified or removed. Based on the results
presented here, it can be argued that large bursts are not generally beneficial
or integral to PSO performance, and may possibly be detrimental. Although the
presence of particle outliers is demonstrably important for swarm optimization
(as demonstrated in bare bones analysis, [6]), bursts, which are sequences of
extreme particle positions, occurring along an axis and reaching outside the
search space, remain a special feature of velocity-based swarms. This work,
which compares standard PSO to a burst-free but comparable optimizer suggests
that bursts are disadvantageous in general. (However, in the coincidence that
the objective function has a rectangular symmetry aligned with the axes, then
bursting may actually be fortuitous.)Further, the replacement of the direct personal
influence operatorpi from SPSO with
the recombinant term ri derived from
its neighborhood in PSO-DR strengthens the case for PSO being mostly reliant on
social interaction as opposed to personal experience. This is further supported
by the effectiveness of PSO-DR model 3, which lacks a cognitive term
altogether. The social behavior occurring inside of a swarm is still a
wide-open area in the field, and will hopefully constitute a great deal of the
future research devoted to the development of a better understanding of this
deceptively simple optimizer.Another property of PSO-DR resides in attractor
jiggling that takes place even at stagnation (no updates to anypi) since ri is never fixed.
This jiggling will work against convergence and could propel the swarm onwards.
This, and other matters concerning the nature of recombination within PSO, will
be the subject of further study.



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*Source: 654184-2008-02-19.xml*