swarm intelligence 虞台文. content overview swarm particle optimization (pso) – example ant...

Swarm Intelligence

虞台文

Content Overview Swarm Particle Optimization (PSO)

– Example Ant Colony Optimization (ACO)

Swarm Intelligence

Overview

Swarm Intelligence

Collective system capable of accomplishing difficult tasks in dynamic and varied environments without any external guidance or control and with no central coordination

Achieving a collective performance which could not normally be achieved by an individual acting alone

Constituting a natural model particularly suited to distributed problem solving

Swarm Intelligence

http://www.scs.carleton.ca/~arpwhite/courses/95590Y/notes/SI%20Lecture%203.pdf

Swarm Intelligence

Swarm Intelligence

Swarm Intelligence

Swarm Intelligence

Swarm Intelligence Particle Swarm Optimization (PSO)

Basic Concept

The Inventors

Russell Eberhart James Kennedysocial-psychologistelectrical engineer

Particle Swarm Optimization (PSO)

PSO is a robust stochastic

optimization technique based

on the movement and

intelligence of swarms.

PSO applies the concept of

social interaction to problem

solving.

Developed in 1995 by James Kennedy and Russell Eberhart.

PSO Search Scheme

It uses a number of agents, i.e., particles, that constitute a swarm moving around in the search space looking for the best solution.

Each particle is treated as a point in a N-dimensional space which adjusts its “flying” according to its own flying experience as well as the flying experience of other particles.

Particle Flying Model

pbest the best solution achieved so far by that particle. gbest the best value obtained so far by any particle in the ne

ighborhood of that particle.

The basic concept of PSO lies in accelerating each particle toward its pbest and the gbest locations, with a random weighted acceleration at each time.

Particle Flying Model

kskpbest

kgbest

kv1kv

1ks

kpbestd

kgbestd

1 2

k kpbestk gbestd dv w w 11 ()c rw and 22 ()c rw and

kv

Particle Flying Model

Each particle tries to modify its position using the following information:– the current positions, – the current velocities,– the distance between the curre

nt position and pbest,– the distance between the curre

nt position and the gbest.

kskpbest

kgbest

kvkv1kv 1kv

1ks

kpbestd

kgbestd

1 2

k kpbestk gbestd dv w w 11 ()c rw and 22 ()c rw and

kv kv

Particle Flying Model

1 1k k ki i is s v

1k k ki i iv v v

1 2() ( ) () ( )k k k k ki i i iv c rand pbest s c rand gbest s

kskpbest

kgbest

kvkv1kv 1kv

1ks

kpbestd

kgbestd

1 2

k kpbestk gbestd dv w w 11 ()c rw and 22 ()c rw and

kv kv

PSO Algorithm

For each particle Initialize particleEND

Do For each particle Calculate fitness value If the fitness value is better than the best fitness value (pbest) in history set current value as the new pbest End

Choose the particle with the best fitness value of all the particles as the gbest For each particle Calculate particle velocity according equation (*) Update particle position according equation (**) End While maximum iterations or minimum error criteria is not attained

For each particle Initialize particleEND

Do For each particle Calculate fitness value If the fitness value is better than the best fitness value (pbest) in history set current value as the new pbest End

Choose the particle with the best fitness value of all the particles as the gbest For each particle Calculate particle velocity according equation (*) Update particle position according equation (**) End While maximum iterations or minimum error criteria is not attained

1k k ki i iv v v

1 2() ( ) () ( )k k k k ki i i iv c rand pbest s c rand gbest s

1k k ki i is s v

*

**

Swarm Intelligence Particle Swarm Optimization (PSO)

Examples

Schwefel's Function

1

( ) ( ) sin( )

with 500 500

n

i ii

i

f x x x

x

( ) 418.9829;

Global

420.9687, 1

minimum

: i

f x n

x i n

Simulation Initialization

Simulation After 5 Generations

Simulation After 10 Generations

Simulation After 15 Generations

Simulation After 20 Generations

Simulation After 25 Generations

Simulation After 100 Generations

Simulation After 500 Generations

Summary

Iterations gBest

0 416.245599

5 515.748796

10 759.404006

15 793.732019

20 834.813763

100 837.911535

5000 837.965771

Optimun 837.9658

400

450

500

550

600

650

700

750

800

850

1 4 16 64 256 1024 4096

"sample.dat"

Exercises Compare PSO with GA

Can we use PSO to train neural networks? How?

Particle Swarm Optimization (PSO)

Ant Colony

Optimization

(ACO)

Facts

Many discrete optimization problems are difficult to solve, e.g., NP-Hard

Soft computing techniques to cope with these problems:– Simulated Annealing (SA)

Based on physical systems– Genetic algorithm (GA)

based on natural selection and genetics– Ant Colony Optimization (ACO)

modeling ant colony behavior

Ant Colony Optimization

Background

Introduced by Marco Dorigo (Milan, Italy), and others in early 1990s.

A probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs.

They are inspired by the behaviour of ants in finding paths from the colony to food.

Typical Applications

TSP Traveling Salesman Problem Quadratic assignment problems Scheduling problems Dynamic routing problems in networks

Natural Behavior of Ant

ACO Concept

Ants (blind) navigate from nest to food source

Shortest path is discovered via pheromone

trails

– each ant moves at random, probabilistically

– pheromone is deposited on path

– ants detect lead ant’s path, inclined to follow, i.e.,

more pheromone on path increases probability of

path being followed

ACO System

Virtual “trail” accumulated on path segments Starting node selected at random Path selection philosophy

– based on amount of “trail” present on possible paths from starting node

– higher probability for paths with more “trail”

Ant reaches next node, selects next path Continues until goal, e.g., starting node for TSP, reached Finished “tour” is a solution

ACO System A completed tour is analyzed for optimality “Trail” amount adjusted to favor better

solutions– better solutions receive more trail– worse solutions receive less trail

higher probability of ant selecting path that is part of a better-performing tour

New cycle is performed Repeated until most ants select the same tour

on every cycle (convergence to solution)

, cont.

Ant Algorithm for TSP

Randomly position m ants on n cities

Loop

for step = 1 to n

for k = 1 to m

Choose the next city to move by applying

a probabilistic state transition rule (to be described)

end for

end for

Update pheromone trails

Until End_condition

Randomly position m ants on n cities

Loop

for step = 1 to n

for k = 1 to m

Choose the next city to move by applying

a probabilistic state transition rule (to be described)

end for

end for

Update pheromone trails

Until End_condition

Pheromone Intensity

( )ij t(0) , ,ij C i j

Ant Transition Rule

( ) .

( )( ) .

ki

ij ijkij

il ill J

tp t

t

Probability of ant k going from city i to j:

the set of nodes applicableto ant k at city i

kiJ :

, 0

1/ij ijd

visibility

Ant Transition Rule

= 0 : a greedy approach = 0 : a rapid selection of tours that may not be optimal. Thus, a tradeoff is necessary.

( ) .

( )( ) .

ki

ij ijkij

il ill J

tp t

t

Probability of ant k going from city i to j:

, 0

1/ij ijd

Pheromone Update

/ ( ) ( , ) ( )( )

0

k kkij

Q L t i j T tt

otherwise

( ) (1 ) ( ) ( )ij ij ijt t t

Q: a constantTk(t): the tour of ant k at time tLk(t): the tour length for ant k at time t

1

( ) ( )m

kij ij

k

t t

Demonstration