Regrouping Particle Swarm
Optimization: A New Global
Optimization Algorithm with
Improved Performance Consistency
Across Benchmarks
George I. Evers
Advisor: Dr. Mounir Ben Ghalia
Electrical Engineering Department
The University of Texas – Pan American
1
Outline
I.
II.
From Physics to PSO
Visual Illustration of Stagnation
& the Regrouping Method
III. RegPSO Formulation
IV. Graph of Solution Quality
V.
Statistical Comparison with Basic PSO
VI. Summary
VII. Future Work
2
How PSO Derives from
Standard Physics Equations
I. From Physics to PSO
3
From Physics to PSO
Displacement Formula of Physics:
1 2
x  x0  v0t  at
2
assuming constant acceleration over the time
period
4
From Physics to PSO
Iterative Version:
1
x (k  1)  x (k )  v (k )  a (k )
2
Using 1 time unit between iterations:
• t = (k + 1) – k = 1 iteration per update
• t2 = 1 iteration2 per update
• For practical purposes, t drops out of the
equation.
5
From Physics to PSO
Subscript “i” Used for Particle Index:
1
x i (k  1)  x i (k )  v i (k )  a i (k ).
2
(All particles follow the same rule.)
6
From Physics to PSO
Particles are physical conceptualizations
accelerating according to social and
cognitive influences.
7
From Physics to PSO
Cognitive Acceleration
The cognitive acceleration is proportional to
(i) the distance,  p i (k )  x i (k )  , of a particle
from its personal best, and (ii) the cognitive
acceleration coefficient, c1.
8
From Physics to PSO
Social Acceleration
The social acceleration is proportional to
(i) the distance,  g (k )  x i (k )  , of a particle
from its global best, and (ii) the social
acceleration coefficient, c2.
9
From Physics to PSO
Total Acceleration
The overall acceleration can therefore be
written as a i (k )  c1  p i (k )  x i (k )   c2  g (k )  x i (k ) .
Substitution then leads from
1
x i (k  1)  x i (k )  v i (k )  a i (k )
2
to
1
1
x i (k  1)  x i (k )  v i (k )  c1  p i (k )  x i (k )   c 2  g (k )  x i (k ) .
2
2
10
From Physics to PSO
Total Acceleration
1
In place of constant 2 , a pseudo-random
1
number with an expected value of 2 is
generated per dimension to add an
element of stochasm to the algorithm.
In this manner
1
1
x i (k  1)  x i (k )  v i (k )  c1  p i (k )  x i (k )   c 2  g (k )  x i (k ) 
2
2
becomes
x i (k  1)  x i (k )  v i (k )  r1i c1  p i (k )  x i (k )   r2i c2  g (k )  x i (k ) .
11
From Physics to PSO
Simulating Friction
To prevent velocities from growing out of control, only
a fraction of the velocity is carried over to the next
iteration. This is accomplished by introducing an
inertia weight,  , which is set less than 1.
In this manner
1
1
x i (k  1)  x i (k )  v i (k )  c1  p i ( k )  x i ( k )   c 2  g ( k )  x i ( k ) 
2
2
becomes
x i (k  1)  x i (k )  v i (k )  r1i c1  p i (k )  x i (k )   r2i c2  g (k )  x i (k ) .
12
From Physics to PSO
Velocity and Position Updates
The previous equation is separated into two
more succinct equations, allowing velocities
and positions to be recorded and analyzed
separately.
Velocity UpdateEquation
v i (k  1)  v i (k )  c1r1i (k )   p i (k )  x i (k )   c 2r2i ( k )   g (k )  x i (k ) 
Position UpdateEquation
x i (k  1)  x i (k )  v i (k  1).
13
The Main Obstacle:
Premature Convergence/
Stagnation
II.
Visual Example of Stagnation
& The Regrouping Method
14
Rastrigin Benchmark
Used to Illustrate Stagnation
15
Swarm Initialization
Particles 1 and 3 are
selected to visually
illustrate how
velocities and
positions are
updated.
16
First Velocity Updates
17
First Position Updates
Particle 1 found a
new personal best,
but particle 3 did
not.
18
Second Velocity Updates
19
Second Position Updates
Particle 3 found a
new personal best,
while particle 1 did
not.
20
Swarm Snapshots
Having seen how particles iteratively update
their positions, the following slides show the
swarm state each 10 iterations to track the
progression from initialization to eventual
solution.
21
Swarm Initialization at Iteration 0
Particles are
randomly initialized
within the original
initialization space.
22
Swarm Collapsing at Iteration 10
Particles are
converging to a
local minimizer
near [2,0] via their
attraction to the
global best in that
vicinity.
23
Exploratory Momenta at Iteration 20
Momenta and
cognitive accelerations
keep particles
searching prior to
settling down.
24
Convergence in Progress
at Iteration 30
Personal bests move
closer to the global best
and momenta wane as no
better global best is
found. Particles continue
converging to the local
minimizer near [2,0].
25
Momenta Waning at Iteration 40
Momenta continue to
wane as particles are
repeatedly pulled toward
(a) the global best very
near [2,0] and (b) their
own personal bests in the
same vicinity.
26
Mostly Converged at Iteration 50
Most particles are
improving their
approximation of the
local minimizer found,
while two particles
still have some
momenta.
27
Momenta Waning at Iteration 60
The final two particles
are collapsing upon the
global best while the
remaining particles are
refining the solution.
28
Momenta Waning at Iteration 70
All particles are in
the same general
vicinity.
29
Cognitive Acceleration at Iteration 80
At least one particle
still has some
exploratory
momentum.
30
Premature Convergence Detected
at Iteration 102
All particles have converged to
within 0.011% of the diameter of
the initialization space. It is
important to allow particles to
refine each solution before
regrouping since they have no
prior knowledge of which
solution is the global minimizer.
31
Options for Dealing with Stagnation
• Terminate the search rather than wasting computations
while stagnated.
• Allow the search to continue and hope for solution
refinement.
• Restart particles from new positions and look for a better
solution.
• Somehow flag solutions already found so that each
restart finds new solutions, and continue restarting until
no better solutions are found.
• Reinvigorate the swarm with diversity to continue the
current search for the global minimizer.
32
“Regrouping” Definition
Regroup: “to reorganize (as after a setback)
for renewed activity”
– Merriam Webster’s online dictionary
33
Regrouping at Iteration 103
Regrouping is
more efficient than
restarting on the
original
initialization
space.
34
Exploration at Iteration 113
“Gbest” PSO continues
as usual within the
new regrouping space.
Particles move toward
the global best with
new momenta,
personal bests, and
positions/perspectives.
35
Swarm Migration at Iteration 123
The swarm is
migrating toward a
better region
discovered by an
exploring particle
near [1,0].
36
Differences of Opinion at Iteration 133
Some particles are
refining a local
minimizer near [1,0]
while others
continue exploring
in the vicinity.
37
Solution Comparison at Iteration 143
Cognition pulls some
particles back to the
local well containing
a local minimizer
near [1, 0].
38
Solution Comparison at Iteration 153
Cognition and
momenta keep
particles moving as
momenta wane.
39
Unconvinced of Optimality on
Horizontal Dimension
at Iteration 163
There is still some
uncertainty on the
horizontal
dimension.
40
New Well Agreed Upon
at Iteration 173
All particles agree
that the new well is
better than the
previous.
41
Waning Momenta at Iteration 183
Momenta wane.
42
Premature Convergence Detected
Again at Iteration 219
Regrouping improved
the function value
from approximately 4
to approximately 1,
and premature
convergence is
detected again.
43
Swarm Regrouped Again
at Iteration 220
The swarm is
regrouped a
second time.
44
Best Well Found
at Iteration 230
The well
containing the
global minimizer
is discovered.
45
Swarm Migration
at Iteration 240
The swarm
migrates to the
newly found
well.
46
Convergence at Iteration 250
Particles swarm to
the newly found
well due to its
higher quality
minimizer.
47
Cognition at Iteration 260
Momenta carry
particles beyond
the well.
48
Convergence at Iteration 270
Solution
refinement of the
global minimizer
is in progress.
49
Regrouping PSO (RegPSO)
Formulation
III. RegPSO Formula
50
Regrouping PSO (RegPSO)
Detection of Premature Convergence
Range of theSearch Space
range   r    range1   r  , range2   r  ,..., ranged   r  
r represents
Diameter of theSearch Space
the search space for
r
r
diam     range   
regrouping index r.
Maximum Euclidean Distance from Global Best
 (k )  max xi (k )  g (k )
i1, , s
Terminate When Maximum Distance from Global Best is Less Than
a User - Specified Percentage of the Diameter of the Search Space
 norm 
 (k )
diam( )
r

51
Regrouping PSO (RegPSO)
Regrouping the Swarm
Uncertainty per Dimension
 j  max xi , j  k   g j  k 
i1, , s
Range of New Search Space
range j ( r )  min  range j ( 0 ),  j 
range   r    range1   r  , range2   r  ,..., ranged   r  
New Search Space Centered at Global Best
1
xi  k  1  g  k   ri  range( )   range( r )
2
where ri   r1i , r2i ,..., rdi 
r
with each rji  U (0,1) randomly selected.
52
Regrouping PSO (RegPSO)
High-Level Pseudo Code
Do
Run Gbest PSO until premature convergence.
Regroup the swarm.
Re-calculate the velocity clamping value based on
the range of the new initialization space.
Re-initialize velocities.
Re-initialize personal bests.
Remember the global best.
Until Search Termination
53
Effectiveness of RegPSO
Demonstrated Graphically
IV. Graphical Comparison of Mean
Function Values
54
Mean Behavior on 30D Rastrigin
A swarm size of 20 suffices for RegPSO to approximate the global minimizer of the
30-D Rastrigin and reduce the cost function to approximately true minimum across
all 50 trials.
55
Effectiveness of RegPSO
Demonstrated Statistically
V.
Statistical Comparison
56
Regrouping PSO (RegPSO)
Compared to Gbest, Lbest PSO
RegPSO Compared to Gbest PSO & Lbest PSO of neighborhood size 2
s = 20, c1 = c2 = 1.49618, 50 trial sets, 800,000 function evaluations
RegPSO used   1.1104 ;   1.2 1; 100,000 evaluations max per grouping.
Benchmark
d
30
Gbest PSO
  0.5,
  0.72984
Mean: 3.6524
Gbest PSO
  0.15,
  0.9 to 0.4
1.1191e-014
Lbest PSO
  0.5,
  0.72984
0.046206
Ackley
Lbest PSO
RegPSO
  0.5,
  0.15,
  0.9 to 0.4   0.72984
1.0623e-014 5.2345e-007
Griewangk
30
Mean: 0.055008
0.022023
9.1051e-003 0.012538
Quadric
30
Mean: 4.1822e-75
2.3189e-014
3.4340e-012 5.9577e-022 3.1351e-010
Quartic
with noise
30
Mean: 0.0039438
0.0015241
1.2630e-002 0.0025417
0.00064366
Rastrigin
30
Mean: 71.63686
25.252
52.812
31.2746
2.6824e-011
Rosenbrock
30
Mean: 2.06915
18.859
2.6106
1.0713
0.0039351
Schaffer’s f6
2
Mean: 0.0033034
0
1.2025e-003 0
Sphere
30
Mean: 2.4703e-323
1.0834e-094
2.0146e-160 2.1967e-215 9.2696e-015
Weighted Sphere 30
Mean: 1.0869e-321
4.4182e-093
6.5519e-158 1.2102e-225 9.8177e-014
0.013861
0
57
Summary
By regrouping the swarm within an efficiently
sized regrouping space when premature
convergence is detected, RegPSO considerably
improves performance consistency, as
demonstrated with a suite of popular
benchmarks.
58
Future Work
Theoretical Improvements
• Give the algorithm the ability to progress from
regrouping to a solution refinement phase.
Testing
• NP hard problems
• Applications to real-world problems
59