PARTICLE SWARM OPTIMISATION (PSO)

Perry BrownAlexander Mathews

Image: http://www.cs264.org/2009/projects/web/Ding_Yiyang/ding-robb/pso.jpg

Introduction• Concept first introduced by Kennedy and Eberhart (1995)• Original idea was to develop a realistic visual simulation of

bird flock behaviour• Simulation was then modified to include a point in the

environment that attracted the virtual bird agents• Potential for optimisation applications then became

apparent

The natural metaphor• A flock of birds (or school of fish, etc.) searching for food• Their objective is to efficiently find the best source of food• Nature-based theory underlying PSO:

The advantage of sharing information within a group outweighs the disadvantage of having to share the reward

Image: http://www.nerjarob.com/nature/wp-content/uploads/Flock-of-pigeons.jpg

Terminology• The “particles” in PSO have no mass or volume

(essentially they are just points in space), but they do have acceleration and velocity

• Behaviour of groups in the developed algorithm ended up looking more like a swarm than a flock

• Hence the name Particle Swarm Optimisation

Swarm intelligence

• Millonas’ five basic principles of swarm intelligence: Proximity: agents can perform basic space and time calculations Quality: agents can respond to environmental conditions Diverse response: population can exhibit a wide range of

behaviour Stability: behaviour of population does not necessarily change

every time the environment does Adaptability: behaviour of population must be able to change when

necessary to adapt to environment

• A PSO swarm satisfies all of the above conditions

Population and environment• Multidimensional search space• Each point in the search space has some value

associated with it, goal is to find the “best” value• Numerous agent particles navigating the search space• Each agent has the following properties:

a current position within the search space a velocity vector

• Additionally, each agent knows the following information: the best value it has found so far (pbest) and its location the best value any member of the population has found so far

(gbest) and its location

Kennedy and Eberhart’s (1995) refined algorithm• Some number of agent particles are initialised with individual positions

and velocities (often just done randomly)• The following steps are then performed iteratively:

The position of each agent is updated according to its current velocity:new position = old position + velocity

The value at each agent’s new position is checked, with pbest and gbest information updated if necessary

Each component of each agent’s velocity vector is then adjusted as a function of the differences between its current location and both the pbest and gbest locations, each weighted by a random variable:

new velocity = old velocity + 2 * rand1 * (pbest location - current location)+ 2 * rand2 * (gbest location - current location)

where rand1 and rand2 are random numbers between 0 and 1.(Multiplying by the constant 2 causes particles to “overshoot” their target about half of the time, resulting in further exploration.)

A (partial) example in two dimensions0 1 2 3 4 5 6 7 8 9

0 3 2 1

1 3 1 1

2 2 1 2 2 1

3 1 3 4 3 2

4 2 4 5 3 1 1 1

5 1 3 2 3 2 1 2 1

6 1 3 1 1 1 3 2

7 1 3 2 1 1 2 1

8 1 4 2 1 1 1

9 1 2 1

pbestsBlue: 0Green: 0Red: 0

gbest: 0

(dots indicate agents, yellow star indicates the global optimum)

Begin with random velocities0 1 2 3 4 5 6 7 8 9

0 3 2 1

1 3 1 1

2 2 1 2 2 1

3 1 3 4 3 2

4 2 4 5 3 1 1 1

5 1 3 2 3 2 1 2 1

6 1 3 1 1 1 3 2

7 1 3 2 1 1 2 1

8 1 4 2 1 1 1

9 1 2 1

pbestsBlue: 0Green: 0Red: 0

gbest: 0

Update particle positions0 1 2 3 4 5 6 7 8 9

0 3 2 1

1 3 1 1

2 2 1 2 2 1

3 1 3 4 3 2

4 2 4 5 3 1 1 1

5 1 3 2 3 2 1 2 1

6 1 3 1 1 1 3 2

7 1 3 2 1 1 2 1

8 1 4 2 1 1 1

9 1 2 1

pbestsBlue: 1 at (6, 2)Green: 0Red: 2 at (8, 7)

gbest: 2 at (8, 7)

Update particle velocities0 1 2 3 4 5 6 7 8 9

0 3 2 1

1 3 1 1

2 2 1 2 2 1

3 1 3 4 3 2

4 2 4 5 3 1 1 1

5 1 3 2 3 2 1 2 1

6 1 3 1 1 1 3 2

7 1 3 2 1 1 2 1

8 1 4 2 1 1 1

9 1 2 1

pbestsBlue: 1 at (6, 2)Green: 0Red: 2 at (8, 7)

gbest: 2 at (8, 7)

For example, Blue’s velocity in the horizontal dimension calculated by:

velocity = 1 +2 * rand() * (6 – 6) +2 * rand() * (8 – 6)

Update particle positions again and repeat…0 1 2 3 4 5 6 7 8 9

0 3 2 1

1 3 1 1

2 2 1 2 2 1

3 1 3 4 3 2

4 2 4 5 3 1 1 1

5 1 3 2 3 2 1 2 1

6 1 3 1 1 1 3 2

7 1 3 2 1 1 2 1

8 1 4 2 1 1 1

9 1 2 1

pbestsBlue: 3 at (8, 6)Green: 1 at (4, 1)Red: 2 at (8, 7)

gbest: 3 at (8, 6)

Algorithm termination

• The solution to the optimisation problem is (obviously) derived from gbest

• Possible termination conditions that might be used: Solution exceeds some quality threshold Average velocity of agents falls below some threshold (agents may

never become completely stationary) A certain number of iterations is completed

An example visualisation

• http://www.youtube.com/watch?v=_bzRHqmpwvo• Velocities represented by trailing lines• After some individual exploration, particles all converge on

global optimum• Particles can be seen oscillating about the optimum

Reference

Kennedy, J.; Eberhart, R.; , "Particle swarm optimization," Neural Networks, 1995. Proceedings., IEEE International Conference on , vol.4, no., pp.1942-1948 vol.4, Nov/Dec 1995URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=488968&isnumber=10434