pso
TRANSCRIPT
Particle Swarm Optimization (PSO) … and its applications…..
James [email protected]
Graduated in Psychology.
Not at all aprofessional researcher.
The Inventors (2)
PSO emulates the swarm behavior of insects, animals herding, birds flocking and fish schooling where these swarms search for food in a collaborative manner.
Each member in the swarm adapts its search patterns by learning from its own experience and other members’ experience.
Particle Swarm OptimizationParticle Swarm Optimization
Comparison with other Evolutionary Computation Techniques
Unlike in genetic algorithms, in PSO, there is no selection operation.
• All particles in PSO are kept as members of the populationthrough the course of the run
• PSO is the only algorithm that does not implement the survival ofthe fittest.
• No crossover operation in PSO.
• Velocity updating resembles mutation in EP.
• Balance between the global and local search is achieved by using an inertial weight factor (w) in velocity updatingequation.
• Each candidate solution is called PARTICLE and represents one individual of a population
• The population is set of vectors and is called SWARM
• The particles change their components and move (fly) in a space R2
• They can evaluate their actual position using the function to be optimized
• The function is called FITNESS FUNCTION
• Particles also compare themselves to their neighbors and imitate the best of that neighbors
Particle Swarm OptimizationParticle Swarm Optimization
Search space D-Dimensional
Xi = [xi1, …, xiD]T = ith particle of Swarm
Vi = [vi1, …, viD]T = Velocity of ith particle
Pi = [pi1, …, piD]T = Best previous position of the ith particle
Initialization
• Swarm of particles is flying through the parameter space and searching for optimum
• Each particle is characterized by• Position vector….. xi(t)
• Velocity vector…...vi(t)
xi(t)
vi(t)
Particle i-400
-200
0
200
400
-400
-200
0
200
400
-5
0
5
x 105
113
PSO Contd …PSO Contd …
Swarming :Swarming :
Bingo !
Swarming Contd …Swarming Contd …
Why Social insects ?Why Social insects ?
Problem solving benefits include:
– Flexible
– Robust
– Self-Organized
– Having no clear leader
– Can use past memory
– Swarming nature
– colonial life
• Each particle has – Individual knowledge pbest
– its own best-so-far position
xi(t)
vi(t)
Particle i. pbest
General PSO Algorithm:
• Each particle has – Individual knowledge pbest
– its own best-so-far position
– Social knowledge gbest – pbest of its best neighbor
xi(t)
vi(t)
Particle i. pbest .gbest
General PSO Algorithm Contd …
• Velocity update:
vi(t+1)= w×vi(t) +
+ c1×rand× (pbest(t) - xi(t)) +
+ c2×rand× (gbest(t) - xi(t))
Xi(t)
vi(t)
pbest_i(t)
gbest_i(t)
General PSO Algorithm Contd …
• Velocity update:
vi(t+1)=w×vi(t) +
+ c1×rand× (pbest(t) - xi(t)) +
+ c2×rand× (gbest(t) - xi(t))
Xi(t)
gbest_i(t)
vi(t)
pbest_i(t)
General PSO Algorithm Contd …
• Velocity update:
vi(t+1)=w×vi(t) +
+ c1×rand× (pbest(t) - xi(t)) +
+ c2×rand× (gbest(t) - xi(t))
Xi(t)
vi(t)
gbest_i(t)
pbest_i(t)
General PSO Algorithm Contd …
• Velocity update:
vi(t+1)=w vi(t) +
+c1*rand *(pbest(t) - xi(t)) +
+ c2*rand*(gbest(t) - xi(t))
• Position update:
xi(t+1)=xi(t) + vi(t+1) Xi(t)
vi(t)
gbest_i(t)
pbest_i(t)
General PSO Algorithm Contd …
• Velocity update:
vi(t+1)=w vi(t) +
+c1*rand*(pbest(t) - xi(t)) +
+c2*rand*(gbest(t) - xi(t))
• Position update:
xi(t+1)=xi(t) + vi(t+1)
Where
w > (1 / 2) (C1 + C2) – 1
0 < w < 1
Gbest(t)vi(t)
Xi(t) vi(t+1)
pbest_i(t)
General PSO Algorithm Contd …
Start
Initialize particles with random position and velocity vectors.
For each particle’s position (p) evaluate fitness
If fitness(p) better than fitness(pbest) then pbest= p
Loo
p u
nti
l all
p
arti
cles
exh
aust
Set best of pBests as gBest
Update particles velocity and position
Loop
unt
il m
ax it
er
Stop: giving gBest, optimal solution.
Flow chart depicting the General PSO Algorithm:
Maximum Velocity ?
Maximal velocity
• Velocity must be limited
• Prevention of swarm explosion
• Vmax – if velocity of particle is greater than Vmax or less than
–Vmax, its set to Vmax
• Vmax is saturation point of velocity
Comments on the Inertial weight factor: A large inertia weight (w) facilitates a global search while a small inertia weight facilitates a local search. By linearly decreasing the
inertia weight from a relatively large value to a small value
through the course of the PSO run gives the best PSO performance compared with fixed inertia weight settings.
Larger w ----------- greater global search abilitySmaller w ------------ greater local search ability.
PSO with linearly decreasing inertia Weight, w
I
kwwwkw
*)(*)( 10
0
4.0
9.0
..max
1
0
w
w
iterationsofnoI
numbersearchk
The linearly decreasing inertia weight, w is given by
where
Proposed by Shi and Eberhart.
PSO with constriction factor
Introduced by Clerc anf Kennedy.
The expression for velocity has been modified as
))()((*)(2*))()((*)(1*)(*)( 21 dxdpdrandcdxdpdrandcdVKdV igiiiiii
,42
22
K ,21 cc 4
is set to 4.1which gives and = 0.729. ,05.221 cc K
As a result each of the (pi-xi) terms is calculated by multiplying 0.729 * 2.05= 1.49445.
Constriction factor guarantees the convergence and improves the convergence velocity.
zwRcxywRbxxRavwv tkkk ...)ˆ.(..)ˆ.(.. 3211
Repulsive PSO
RPSO is a global optimization algorithm, belongs to the classof stochastic evolutionary global optimizers, a variant ofparticle swarm optimization (PSO).
Where R1, R2, R3 : random numbers between 0 and 1ω : inertia weight between 0.01 and 0.7 :best position of a particleŷ : best position of a randomly chosen other particle from within the swarmz : a random velocity vectora, b, c : constants
x̂
Comprehensive Learning PSO
Overcomes premature convergenceConventional PSO– Learns simultaneously from pbest and gbest.– Possibility of trapping in local minima.CLPSO– Learns from gbest of swarm, pbest of particle andpbest of all other particles at different dimension.
– If D is the total number of Dimension of optimizingproblem then, m- dimensions are randomly chosen tolearn from the ‘gbest’, out of the remaining D- mdimensions some learn from a randomly selectedparticle’s ‘pbest’ and the remaining dimensions learnfrom its own ‘pbest’.
)(. ,,, dibestdindi XgrandVwV
)(. ,,,, didbestfdindi XprandVwV
Initialization of all the particle velocity and position randomlySet the initial position as pbest (pi,d) and Find gbest (pg,d);For n= 1 to the max. Number of generations,For i=1 to the population size,For d=1 to the problem dimensionality,Update velocity according following to equation;If ai (d) ==1
Else if bi (d) ==1
Else)(. ,,,, didbestdindi XprandVwV
Where ai (d) and bi (d) are the deciders of randomly chosen dimension for learning for random particles.And gbest is global best value, pbest,d is personal best of ith particle in dth dimension and pbestf,d is the personal best value of fth particle in dth dimensionEnd ifLimit VelocityUpdate Position according to equationXi,d=Xi,d + Vi,d
End- of-d;Compute fitness of the particle;Update historical information regarding pbest i and gbest i if needed ;End-of-i;Terminate if meet the requirements;End-of-n;
The best features of PSO and Clonal Selection Principle of Artificial Immune System are combined.
The local and global best positions of particles are not directly used in deciding their new positions. Rather velocity of each particle is updated from the knowledge of the previous velocity and the variable inertia weights.
The location of each particle is then changed in the next search using its updated velocity information. The fitness values of all particles are then evaluated and the overall best location is selected. In the next search, each particle following clonal selection principle occupies the best position achieved so far and then they diversify .As the search process continues all particles proceed towards the best possible location which then represents the desired solution.
Clonal PSO
)(*)()1( kvkHkv idid
I
kHHHkH
*)(*)( 10
0
The velocity update equation of CPSO becomes
4.0
9.0
..max
1
0
H
H
iterationsofnoI
munbersearchk
Position update: x(k+1)=xi(k) + vid(k+1)
Where
Comparison of CPU time used during training operation
Nonlinear system PSO (in second)
GA(in second)
Example-1 trained at -30dB 0.3200 13.5090
Example-1 trained at -10dB 0.3300 12.4080
Example-2 trained at -30dB 0.3500 12.7990
Example-2 trained at -10dB 0.3700 13.8390
Conclusion
The PSO based nonlinear system identification involves :
(1)Low computational complexity
(2)Faster convergence during training operation compared to GA based approach
(3)The present method avoids local minima during training of weights (derivative free training)