Intelligent Systems II · TEE · read

Read the PSO equation and work three individual updates

4. Particle swarm optimization

First understand position, velocity and memory; then solve complete two-iteration minimisation examples.

First read the update in words

New velocity = retained old velocity + pull toward personal best + pull toward the group best. After finding that velocity, add it to the old position. You are calculating a movement, then applying it.

A negative difference pulls toward a best position on the left.

Example: current x=4 and personal best=1 gives the difference 1−4=−3. A coefficient of .5 gives personal pull .5×(−3)=−1.5. The negative sign says “move left”; it does not mean a negative probability. If current x=−2 and best=1, the difference is 1−(−2)=1+2=3, which pulls right.

In v_i,j^t, i chooses the particle, j chooses its coordinate, and t chooses the recorded iteration. For particle B, coordinate 2, iteration 0, look at row B and the second velocity entry in the initial table. The letters identify a stored number; they are not extra multiplication factors.

The velocity equation is three contributions added together

v_{i,j}^{t+1}=w v_{i,j}^{t}+c_1r_1(pbest_{i,j}^{t}-x_{i,j}^{t})+c_2r_2(gbest_j^{t}-x_{i,j}^{t})

x_{i,j}^{t+1}=x_{i,j}^{t}+v_{i,j}^{t+1}

Symbol	Meaning
i	Which particle
j	Which coordinate of that particle
t / t+1	Before / after this iteration; these are labels, not powers
w	Inertia weight: how much old velocity is kept
c₁r₁	Combined strength of the pull to personal best
c₂r₂	Combined strength of the pull to global best
r₁,r₂	Random factors, normally in [0,1]; use the values supplied in an exam

A best position to the left gives a negative difference and pulls left; a best position to the right gives a positive difference and pulls right. The multiplication sets the strength of each pull.

For one coordinate, x=4, v=0, pbest=4, gbest=2. With w=0.5, c₁=c₂=1 and r₁=r₂=0.5, find the next velocity and position. [3 marks · Practice]

Given

x=4, v=0, pbest=4, gbest=2; w=.5; c₁r₁=.5; c₂r₂=.5.

To find

New velocity followed by new position.

Formula and why it applies

Use the three-term velocity equation, then add the new velocity to x.

v_{new}=0.5v+0.5(pbest-x)+0.5(gbest-x)

Solution

Step 1. Inertia

0.5\times0=0

Step 2. Personal pull

0.5(4-(4))=0

Step 3. Social pull

0.5(2-(4))=-1

Step 4. New velocity

v_{new}=0+0+(-1)=-1

Step 5. New position

x_{new}=4+(-1)=3

Final answer. vnew=-1; xnew=3.

For one coordinate, x=4, v=-2, pbest=3, gbest=1. With w=0.5, c₁=c₂=1 and r₁=r₂=0.5, find the next velocity and position. [3 marks · Practice]