Chang, Particle Swarm Optimization and Ant Colony Optimization, A ...

Computational Intelligence Optimization:

Particle Swarm Optimization and Ant Colony Optimization, A Gentle Introduction Department of Computer Science and Engineering Sogang University Hyeong Soo Chang System Modeling & Optimization Lab., Sogang University

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Intelligence

Still in debate for the definition

Dictionary definition: the ability to understand and profit from experience, having the capacity for thought and reason

In 1950, Turing believed that it would be possible for a computer with 109 bits of storage space to pass a 5-minute version of his test for computer intelligence with 0.7 probability by the year 2000. Has his belief come true?

Intelligent agent: a minimum decision making unit

System Modeling & Optimization Lab., Sogang University

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Computational Intelligence(CI)

The study of adaptive mechanisms to enable or facilitate “intelligent” (decision making) behavior in complex and changing environments based on mathematical models of biological systems.

A sub-branch of AI, Automatic Control, Operations Research, and Computational Biology


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CI Main Paradigms

Neural Networks (NN), Evolutionary Computing (EC), Swarm Intelligence (SI), Fuzzy Systems (FS), Hybrid of these, etc.

NN models biological neural systems.

EC models natural evolution (ex: genetic algorithm).

FS originated from studies of how organisms interact with their environment.

based on fuzzy sets and fuzzy logic (approximate reasoning)

In a sense, FS models common sense in human reasoning.


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Swarm Intelligence (SI)

Originated from the study of colonies, or swarms of social organisms

Social behavior increases the ability of an individual to adapt.

(Collective) Intelligence arises from interactions among individuals having simple behavioral intelligence.

Each individual in a swarm behaves in a distributed way with a certain information exchange protocol.

The Internet is, in a sense, a swarm intelligence!

A young (interdisciplinary) hot research field.

Much attention is being paid to SI for designing decentralized control systems (multi-agent systems).

Plenary talks given by Y. C. Ho at ACC/IEEE CDC 2004.


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SI-based Optimization

This talk introduces gently two recently proposed SIbased optimization techniques:

Particle Swarm Optimization (PSO)

An optimization technique designed for continuous optimization problem

Based on model of the social behavior of bird flocks

Ant Colony Optimization (ACO)

General purpose meta-heuristic for Combinatorial Optimization Problem (COP)

Based on model of ant colony’s social behavior


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Optimization Problem

Let S be the set of solutions for a given problem.

An objective (fitness) function f : S → ℜ is given.

The optimization problem is to find arg max f ( s ) or max f ( s ). s∈S

s∈S

Continuous Search Space: S=Rn

Discrete Search Space: S={0,1}n (binary string of the length n) (ex: 0-1 Knapsack, TSP, etc.)

Commonly called COP.

The given domain needs to be mapped to the search space by a mapping (general purpose meta-heuristic).


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Why SI for Optimization?

Continuous Optimization Problem

Nondifferentiable, highly nonlinear and complex, many local-maxima

Search speed

Discrete Optimization Problem

An algorithm is regarded as efficient or “good” if there exist a polynomial P(n) such that the time required for solving any problem instance of size n is bounded above by P(n).

NP-Complete problems: Nobody has found so far any good algorithm for any problem in this class.

It has been proved that if a good algorithm exists for some problem in this class, then a good algorithm exists for all NP-Complete problems.

Note: a continuous problem can be approximated by discretization.

What do we do then?


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Developed by Kennedy and Everhart

J. Kennedy, R.C. Everhart, Particle Swarm Optimization, Proc. of the IEEE Int. Conf. on Neural Networks, Vol. 4, pp. 1942-1948, 1995.

A population-based search algorithm based on the simulation of the social behavior of birds within a flock.

The initial intent of the particle swarm concept was to graphically simulate the graceful and unpredictable choreography of a bird flock, the aim of discovering patterns that govern the ability of birds to fly synchronously, and to suddenly change direction with a regrouping in an optimal formation.

From the initial objective, the concept evolved into a simple and efficient optimization algorithm.


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Solution space : n-dimension Euclidian space Fitness function f : ℜ → ℜ n

Swarm : a set of particles denoted by P Particle : a potential solution Pi (i=1,…,m)

Particle Pi consists of the following components:

r Position xi (t ) = ( x1 , x2 ,..., xn ), x j ∈ ℜ at step t

r Velocity vi (t ) = (v1 , v2 ,..., vn ), v j ∈ ℜ

gbest is xi (t ) such that f ( xi (t )) ≥ f ( x j (t )) for all x j (t ), Pj ∈ P * * n x is a position such that f ( x ) ≥ f ( y ), y ∈ ℜ We call x* an optimal solution position.


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Particle Swarm Optimization (PSO) fitness

Basic Concepts of PSO Maximum fitness

fitness of r position xi

r velocity v1 of

particle P1 on r position x1

particle P1 gbest

r position xi

r x1

x*

r x2

Solution space 11


Particle Swarm Optimization (PSO) 1. Initialize the swarm in solution space

r t = 0, Pi ∈ P (t ) is random position xi (t ) within the solution space r Velocity vi (t ) = (0,...,0)

P1

r x1 (t ) System Modeling & Optimization Lab., Sogang University

P3

r x3 (t )

P2

x*

r x2 (t ) 12

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Particle Swarm Optimization (PSO) 2. Evaluate fitness of individual particles r Fitness of Pi = f ( xi (t ))

fitness of P3

fitness of P1

P1

fitness of P2

P3

r x1 (t )

r x3 (t )

P2

r x2 (t )

x*

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Particle Swarm Optimization (PSO) 3. Modify gbest, pbest and velocity

r r (2) If f ( xi (t )) > gbest , (1) If f ( xi (t )) > pbesti , r r r r r r (a) gbest = f ( xi (t )), (b) x gbest = xi (t ) (a) pbesti = f ( xi (t )), (b) x pbesti = xi (t ) r r r r r r (3) vi (t ) = vi (t − 1) + ρ1 ( x pbesti − xi (t )) + ρ 2 ( x gbest − xi (t )) ( ρ1 , ρ 2 are random variables) gbest

fitness of P2

P1

r x1 (t )

r v1 (t )


r v2 (t )

P3

r x3 (t )

x*

P2

r x2 (t )

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Particle Swarm Optimization (PSO) r r r r r r vi (t ) = vi (t − 1) + ρ1 ( x pbesti − xi (t )) + ρ 2 ( x gbest − xi (t )) ( ρ1 , ρ 2 are random variables) 2-dimensonal solution space

fitness

x r x pbest1 pbest1 r r ( xrpbest − xr1 (t )) ρ1 ( x pbest11 − x1 (t ))

r v1 (t )

P1

r v1 (t − 1)

r x1 (t )

r

gbest

r x gbest

r

r

r

ρ1 ( x pbest − x1 (t )) + ρ 2 ( xgbest − x1 (t ))

r r ρ 2((xrx gbest −− xrx1((tt))))

1

1

x gbest

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Particle Swarm Optimization (PSO) 4. Move each particle to a new position r r r xi (t ) = xi (t − 1) + vi (t )

t ← t +1 gbest

> pbest1

r v1 (t )

r x pbest1

>

P1

r x1 (t )


P2 vr (t ) 2

P3

r r x3 (t ), x gbest

x*

r x2 (t ) 16

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Particle Swarm Optimization (PSO) 5. Repeat until convergence or a stopping cond. is met t=1

gbest

P1

P3

r x1 (t )

r x3 (t )

P2

r x2 (t )

x*

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gbest

P1

P3

P2

r r x1 (t ) x3 (t )

x* x2 (t )


r

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gbest

P1

P3

r x1 (t r) System Modeling & Optimization Lab., Sogang University

P2

r

x* x2 (t )

x3 (t )

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gbest

P2

r

x* = x2 (t ) System Modeling & Optimization Lab., Sogang University

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Particle Swarm Optimization (PSO) 5. Repeat until convergence or a stopping cond. is met t=*

gbest

P1=P2=P3

r

r

r

x* = x1 (t ) = x2 (t ) = x3 (t ) System Modeling & Optimization Lab., Sogang University

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Basic flow of PSO 1. Initialize the swarm from the solution space. 2. Evaluate fitness of individual particles. 3. Modify gbest, pbest and velocity. 4. Move each particle to a new position. 5. Go to step 2, and repeat until convergence or a stopping condition is satisfied.


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Other details

ρ1 = r1c1 , ρ 2 = r2 c2 , with r1 , r2 ~ U (0,1),

and c1 and c2 are positive acceleration constants. c1 + c2 ≤ 4

If

J. Kennedy, The Behavior of Particles, in V.W. Porto, N Saravanan, D. Waagen (eds), Proc. of the 7th Int. Conf. on Evolutionary Programming, 1998, pp 581-589.

c1 + c2 > 4, velocities and positions tend to explode toward infinity.

lbest (in the local best version of PSO)

While lbest is slower in convergence than gbest, lbest results in much better solutions. R.C. Everhart, R.W. Dobbins and P. Simpson, Computational Intelligence PC Tools, Academic Press, 1996.


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Other details cont.

Usually set c1 and c2 to 2.

Usually, an upper limit is placed on the velocity in all dimensions and the limit depends on the range of the domain.

The convergence of PSO has not been proved yet.

Binary-PSO for discrete optimization problem has been developed:

J. Kennedy, R.C. Everhart, A Discrete Binary Version of the Particle Swarm Algorithm, Proc. of the Conf. on Systems, Man, and Cybernetics, 1997, pp. 4104-4109.


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Some applications

Y. Shi, R.C. Everhart, Empirical Study of Particle Swarm Optimization, Proceedings of the IEEE Congress on Evolutionary Computation, Vol 4, 1999, pp 1945-1950. (study on benchmark functions)

A multiagent-based particle swarm optimization approach for optimal reactive power dispatch Zhao, B.; Guo, C.X.; Cao, Y.J.; Power Systems, IEEE Transactions on Volume 20, Issue 2, May 2005 Page(s):1070 - 1078

A particle swarm optimization approach for optimum design of PID controller in AVR system Zwe-Lee Gaing; Energy Conversion, IEEE Transactions on Volume 19, Issue 2, June 2004 Page(s):384 - 391

Use of intelligent-particle swarm optimization in electromagnetics Ciuprina, G.; Ioan, D.; Munteanu, I.; Magnetics, IEEE Transactions on Volume 38, Issue 2, Part 1, March 2002 Page(s):1037 - 1040

Inversion of ocean color observations using particle swarm optimization Slade, W.H.; Ressom, H.W.; Musavi, M.T.; Miller, R.L.; Geoscience and Remote Sensing, IEEE Transactions on Volume 42, Issue 9, Sept. 2004 Page(s):1915 - 1923

Efficiency-Constrained Particle Swarm Optimization of a Modified Bernstein Polynomial for Conformal Array Excitation Amplitude Synthesis Boeringer, D.W.; Werner, D.H.; Antennas and Propagation, IEEE Transactions on Volume 53, Issue 8, Part 2, Aug. 2005 Page(s):2662 - 2673


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Ant Colony Optimization (ACO)

PSO is based on model of the choreography of relatively small swarms, where all individuals have the same behavior and characteristics.

The next technique considers swarms that consist of large numbers of individuals, where individuals typically have different morphological structures and tasks – but all contributing to a common goal.

Such swarms model distributed systems where components of the system are capable of distributed operation.

Roughly, 108 living organisms in the earth; Only 2% of those are social insects: Only 2% of all insects live in swarms where social interaction is the most important aspect to ensure survival; Of these social insects, 50% are ants!

Ant colonies consist of from 30 to millions of individuals.


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Real Ant Behavior A single ant has a limited capability. But an ant colony is highly efficient, capable to find shortest paths between foods and the nest (real experiment). Communication through a chemical substances pheromone, which is accumulative and also evaporative.


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Real Ant Behavior Initial route (t=0)


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Real Ant Behavior Change the environment (t=1) ACB