Regrouping Particle Swarm Optimization: A New Global Optimization Algorithm with

Improved Performance Consistency Across Benchmarks

George I. EversAdvisor: Dr. Mounir Ben Ghalia

Electrical Engineering DepartmentThe University of Texas – Pan American

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OutlineI. From Physics to PSOII. Visual Illustration of Stagnation

& the Regrouping MethodIII. RegPSO FormulationIV. Graph of Solution QualityV. Statistical Comparison with Basic PSOVI. SummaryVII. Future Work

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How PSO Derives from Standard Physics Equations

I. From Physics to PSO

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From Physics to PSO

Displacement Formula of Physics:

20 0

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x x v t at

assuming constant acceleration over the time period

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From Physics to PSO

Iterative Version:

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 theequation.

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1( 1) ( ) ( ) ( )2

x k x k v k a k

From Physics to PSO

Subscript “i” Used for Particle Index:

(All particles follow the same rule.)

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1( 1) ( ) ( ) ( ).2i i i ix k x k v k a k

From Physics to PSO

Particles are physical conceptualizations accelerating according to social andcognitive influences.

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From Physics to PSO

Cognitive AccelerationThe cognitive acceleration is proportional to(i) the distance, , of a particle from its personal best, and (ii) the cognitive acceleration coefficient, .

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( ) ( )i ip k x k

1c

From Physics to PSO

Social AccelerationThe social acceleration is proportional to(i) the distance, , of a particle from its global best, and (ii) the social acceleration coefficient, .

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( ) ( )ig k x k

2c

From Physics to PSO

Total AccelerationThe overall acceleration can therefore be

written as

Substitution then leads from

to

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1 2( ) c ( ) ( ) c ( ) ( ) .i i i ia k p k x k g k x k

1 21 1( 1) ( ) ( ) c ( ) ( ) c ( ) ( ) .2 2i i i i i ix k x k v k p k x k g k x k

1( 1) ( ) ( ) ( )2i i i ix k x k v k a k

From Physics to PSO

Total AccelerationIn place of constant , a pseudo-random

number with an expected value of is generated per dimension to add anelement of stochasm to the algorithm.

In this manner

becomes

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1 1 2 2( 1) ( ) ( ) c ( ) ( ) c ( ) ( ) .i i i i i i i ix k x k v k r p k x k r g k x k

1 21 1( 1) ( ) ( ) c ( ) ( ) c ( ) ( )2 2i i i i i ix k x k v k p k x k g k x k

12 1

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From Physics to PSO

Simulating FrictionTo 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

becomes

12 1 1 2 2( 1) ( ) ( ) c ( ) ( ) c ( ) ( ) .i i i i i i i ix k x k v k r p k x k r g k x k

1 21 1( 1) ( ) ( ) c ( ) ( ) c ( ) ( )2 2i i i i i ix k x k v k p k x k g k x k

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.

1 1 2 2( 1) ( ) c ( ) ( ) ( ) c ( ) ( ) ( )

( 1) ( ) ( 1).

i ii i i i i

i i i

v k v k r k p k x k r k g k x k

x k x k v k

Velocity UpdateEquation

PositionUpdateEquation

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The Main Obstacle: Premature Convergence/

Stagnation

II. Visual Example of Stagnation& The Regrouping Method

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Rastrigin BenchmarkUsed to Illustrate Stagnation

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Swarm Initialization

Particles 1 and 3 are selected to visually

illustrate how velocities and positions are

updated.

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First Velocity Updates

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First Position Updates

Particle 1 found a new personal best, but particle 3 did

not.

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Second Velocity Updates

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Second Position Updates

Particle 3 found a new personal best, while particle 1 did

not.

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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.

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Swarm Initialization at Iteration 0

Particles are randomly initialized within the original initialization space.

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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.

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Exploratory Momenta at Iteration 20

Momenta and cognitive accelerations keep particles searching prior to settling down.

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Convergence in Progressat 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].

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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.

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Mostly Converged at Iteration 50

Most particles are improving their approximation of the local minimizer found, while two particles still have some momenta.

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Momenta Waning at Iteration 60

The final two particles are collapsing upon the global best while the remaining particles are refining the solution.

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Momenta Waning at Iteration 70

All particles are in the same general vicinity.

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Cognitive Acceleration at Iteration 80

At least one particle still has some exploratory momentum.

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Premature Convergence Detectedat 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.

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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.

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“Regrouping” Definition

Regroup: “to reorganize (as after a setback)for renewed activity”

– Merriam Webster’s online dictionary

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Regrouping at Iteration 103

Regrouping is more efficient than restarting on the original initialization space.

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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.

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Swarm Migration at Iteration 123

The swarm is migrating toward a better region discovered by an exploring particle near [1,0].

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Differences of Opinion at Iteration 133

Some particles are refining a local minimizer near [1,0] while others continue exploring in the vicinity.

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Solution Comparison at Iteration 143

Cognition pulls some particles back to the local well containing a local minimizer near [1, 0].

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Solution Comparison at Iteration 153

Cognition and momenta keep particles moving as momenta wane.

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Unconvinced of Optimality on Horizontal Dimension

at Iteration 163

There is still some uncertainty on the horizontal dimension.

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New Well Agreed Uponat Iteration 173

All particles agree that the new well is better than the previous.

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Waning Momenta at Iteration 183

Momenta wane.

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Premature Convergence Detected Again at Iteration 219

Regrouping improved the function value from approximately 4 to approximately 1, and premature convergence is detected again.

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Swarm Regrouped Againat Iteration 220

The swarm is regrouped a second time.

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Best Well Foundat Iteration 230

The well containing the global minimizer is discovered.

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Swarm Migrationat Iteration 240

The swarm migrates to the newly found well.

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Convergence at Iteration 250

Particles swarm to the newly found well due to its higher quality minimizer.

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Cognition at Iteration 260

Momenta carry particles beyond the well.

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Convergence at Iteration 270

Solution refinement of the global minimizer is in progress.

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Regrouping PSO (RegPSO) Formulation

III. RegPSO Formula

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Regrouping PSO (RegPSO)Detection of Premature Convergence

1 2

1, ,

, ,...,

( ) max ( ) ( )

r r r rd

r r

ii s

range range range range

diam range

k x k g k

Range of theSearch Space

Diameter of theSearch Space

Maximum Euclidean Distance from Global Best

TerminateWh

norm( )( )r

kdiam

en Maximum Distance from Global Best is Less Thana User - Specified Percentage of the Diameter of the Search Space

representsthe search space forregrouping index .

r

r

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Regrouping PSO (RegPSO)Regrouping the Swarm

,1, ,

0

1 2

max

( ) min ( ),

, ,...,

1

j i j ji s

rj j j

r r r rd

i i

x k g k

range range

range range range range

x k g k r ra

Uncertainty per Dimension

Range of New Search Space

New Search Space Centered at Global Best

1 2

1( ) ( )2

where , ,...,

with each (0,1) randomly selected.i i i

i

r r

i d

j

nge range

r r r r

r U

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Regrouping PSO (RegPSO)High-Level Pseudo Code

DoRun 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

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Effectiveness of RegPSO Demonstrated Graphically

IV. Graphical Comparison of Mean Function Values

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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.

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Effectiveness of RegPSO Demonstrated Statistically

V. Statistical Comparison

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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 4 11.1 10 ; 1.2 ; 100,000 evaluations max per grouping. Benchmark d Gbest PSO

0.5,0.72984

Gbest PSO 0.15,0.9 to 0.4

Lbest PSO 0.5,0.72984

Lbest PSO 0.15,0.9 to 0.4

RegPSO 0.5,0.72984

Ackley 30 Mean:

3.6524 1.1191e-014

0.046206 1.0623e-014 5.2345e-007

Griewangk 30 Mean:

0.055008 0.022023 9.1051e-003 0.012538 0.013861

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 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

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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.

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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

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