📘 PSO Notes — Simple Explanation

What is PSO?

Imagine birds flying over mountains trying to find the highest peak. Each bird remembers its own best spot and knows the group's best spot. They adjust their flight speed to move toward these best spots. That's PSO!

GA = population of chromosomes evolving via crossover & mutation

PSO = swarm of particles (birds) flying via velocity updates — much simpler, just 2 formulas!

The Algorithm (5 Steps)

Step	What Happens	GA Equivalent
1	Drop N birds at random positions	Create initial random population
2	Check f(x) at each bird's position	Evaluate fitness of chromosomes
3	Set all bird speeds to 0	— (no equivalent)
4a	Track P_best (personal best) & G_best (global best)	Track best chromosome
4b	Update velocity using the formula	Selection + Crossover
4c	Move bird: x_new = x_old + v_new	New offspring generation
5	Repeat until all birds converge	Repeat until convergence

Step

What Happens

GA Equivalent

Drop N birds at random positions

Create initial random population

Check f(x) at each bird's position

Evaluate fitness of chromosomes

Set all bird speeds to 0

— (no equivalent)

Track P_best (personal best) & G_best (global best)

Track best chromosome

Update velocity using the formula

Selection + Crossover

Move bird: x_new = x_old + v_new

New offspring generation

Repeat until all birds converge

Repeat until convergence

The Velocity Formula

v_new = v_old + c₁·r₁·(P_best − x) + c₂·r₂·(G_best − x) ↑ ↑ ↑ momentum "go toward MY best" "go toward GROUP best"

All Terms Defined

Term	Symbol	Meaning	Example (Book 13.4)
Lower Bound	x^(l)	Minimum allowed value of x. Bird can't fly below this.	−2
Upper Bound	x^(u)	Maximum allowed value of x. Bird can't fly above this.	2
Particle	x_j	One bird's current position (a candidate solution)	x₁=−1.5, x₂=0.0, etc.
Swarm Size	N	Total number of birds	4
Velocity	v_j	Speed & direction of bird j. Positive = right, negative = left.	v₁(1) = 2.6241
P_best	P_best,j	Personal best — best position bird j has found so far	Bird 1's own best spot
G_best	G_best	Global best — best position found by ANY bird	G_best = 1.25
c₁ (Cognitive)	c₁	How much bird trusts its own memory. Higher = more self-trust.	1
c₂ (Social)	c₂	How much bird trusts the group. Higher = follows crowd more.	1
r₁, r₂	r₁, r₂	Random numbers (0 to 1). Adds randomness like mutation in GA.	0.3294, 0.9542
Iteration	i	One complete round of velocity + position update for all birds	i = 1, 2, 3...
Inertia Weight	θ	Controls previous speed memory. High θ = explore. Low θ = fine-tune.	0.9 → 0.4
θ_max	θ_max	Starting inertia (bird flies fast initially to explore)	0.9
θ_min	θ_min	Final inertia (bird slows down to settle at optimum)	0.4
Convergence	—	When all birds reach the same position, answer is found	All x ≈ 1.0
f(x)	f(x)	Objective function — the "height" at position x. Birds want max.	−x²+2x+11

Term

Symbol

Meaning

Example (Book 13.4)

Lower Bound

x^(l)

Minimum allowed value of x. Bird can't fly below this.

−2

Upper Bound

x^(u)

Maximum allowed value of x. Bird can't fly above this.

Particle

x_j

One bird's current position (a candidate solution)

x₁=−1.5, x₂=0.0, etc.

Swarm Size

Total number of birds

Velocity

v_j

Speed & direction of bird j. Positive = right, negative = left.

v₁(1) = 2.6241

P_best

P_best,j

Personal best — best position bird j has found so far

Bird 1's own best spot

G_best

G_best

Global best — best position found by ANY bird

G_best = 1.25

c₁ (Cognitive)

c₁

How much bird trusts its own memory. Higher = more self-trust.

c₂ (Social)

c₂

How much bird trusts the group. Higher = follows crowd more.

r₁, r₂

r₁, r₂

Random numbers (0 to 1). Adds randomness like mutation in GA.

0.3294, 0.9542

Iteration

One complete round of velocity + position update for all birds

i = 1, 2, 3...

Inertia Weight

Controls previous speed memory. High θ = explore. Low θ = fine-tune.

0.9 → 0.4

θ_max

θ_max

Starting inertia (bird flies fast initially to explore)

0.9

θ_min

θ_min

Final inertia (bird slows down to settle at optimum)

0.4

Convergence

—

When all birds reach the same position, answer is found

All x ≈ 1.0

f(x)

f(x)

Objective function — the "height" at position x. Birds want max.

−x²+2x+11

How c₁ and c₂ Work

Bird's current position (x) ↓ ────────●────────────────────────── ←── c₁·r₁ pulls toward P_best (own memory) ←──────── c₂·r₂ pulls toward G_best (group best) If c₁ > c₂ → bird trusts itself more If c₂ > c₁ → bird follows the group more Usually c₁ = c₂ = 2 (book example uses 1)

GA vs PSO Comparison

	GA	PSO
Representation	Binary strings (chromosomes)	Real numbers (positions)
Evolution	Crossover + mutation	Velocity update
Memory	No memory of past	Each particle remembers its best
Operators	Selection, crossover, mutation	Just velocity + position update
Simplicity	More complex	Much simpler (just 2 formulas)

PSO

Representation

Binary strings (chromosomes)

Real numbers (positions)

Evolution

Crossover + mutation

Velocity update

Memory

No memory of past

Each particle remembers its best

Operators

Selection, crossover, mutation

Just velocity + position update

Simplicity

More complex

Much simpler (just 2 formulas)

Why Inertia θ Decreases?

Early iterations (θ = 0.9): Birds fly fast → explore large area → don't miss the peak

Late iterations (θ = 0.4): Birds slow down → fine-tune → settle exactly on the peak

Think of it like: first search the whole city for the restaurant, then slowly narrow down to the exact address.

Book Example 13.4 in Plain Words

Find maximum of f(x) = −x² + 2x + 11 (answer: x=1, f=12)

4 birds start at: −1.5, 0.0, 0.5, 1.25

Bird 4 at x=1.25 has highest f = 11.9375, so G_best = 1.25

All birds get pulled toward 1.25. After iteration 1, they're at: 1.12, 1.19, 1.22, 1.25 (all near x=1!)

They keep adjusting until all converge to x ≈ 1.0

PSO Dashboard

Ready

Best Solution Found

Run the optimization to see results.

Convergence Log

Iter	G_best Value	G_best x	Max \|Δx\|

All Iteration Details (Step-by-Step)

Run the optimization to see detailed iteration computations for every particle.