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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

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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

System Modeling & Optimization Lab., Sogang University

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CI Main Paradigms „

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

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NN models biological neural systems.

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EC models natural evolution (ex: genetic algorithm).

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

based on fuzzy sets and fuzzy logic (approximate reasoning)

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

System Modeling & Optimization Lab., Sogang University

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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.

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

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

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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).

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

System Modeling & Optimization Lab., Sogang University

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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

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Based on model of the social behavior of bird flocks

Ant Colony Optimization (ACO) „

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

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Based on model of ant colony’s social behavior

System Modeling & Optimization Lab., Sogang University

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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

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Continuous Search Space: S=Rn

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

Commonly called COP.

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

System Modeling & Optimization Lab., Sogang University

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Why SI for Optimization? „

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Continuous Optimization Problem „

Nondifferentiable, highly nonlinear and complex, many local-maxima

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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).

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

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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.

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Note: a continuous problem can be approximated by discretization.

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What do we do then?

System Modeling & Optimization Lab., Sogang University

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Particle Swarm Optimization (PSO) „

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.

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

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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.

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

System Modeling & Optimization Lab., Sogang University

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Particle Swarm Optimization (PSO) „

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.

System Modeling & Optimization Lab., Sogang University

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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

System Modeling & Optimization Lab., Sogang University

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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System Modeling & Optimization Lab., Sogang University

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 )

System Modeling & Optimization Lab., Sogang University

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

0 15

System Modeling & Optimization Lab., Sogang University

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 )

System Modeling & Optimization Lab., Sogang University

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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System Modeling & Optimization Lab., Sogang University

Particle Swarm Optimization (PSO) 5. Repeat until convergence or a stopping cond. is met t=2

gbest

P1

P3

P2

r r x1 (t ) x3 (t )

x* x2 (t )

System Modeling & Optimization Lab., Sogang University

r

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Particle Swarm Optimization (PSO) 5. Repeat until convergence or a stopping cond. is met t=3

gbest

P1

P3

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

P2

r

x* x2 (t )

x3 (t )

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Particle Swarm Optimization (PSO) 5. Repeat until convergence or a stopping cond. is met t=4

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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Particle Swarm Optimization (PSO) „

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.

System Modeling & Optimization Lab., Sogang University

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Particle Swarm Optimization (PSO) „

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.

System Modeling & Optimization Lab., Sogang University

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Particle Swarm Optimization (PSO) „

Other details cont. „

Usually set c1 and c2 to 2.

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

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The convergence of PSO has not been proved yet.

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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.

System Modeling & Optimization Lab., Sogang University

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Particle Swarm Optimization (PSO) „

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)

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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

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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

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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

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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

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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

System Modeling & Optimization Lab., Sogang University

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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.

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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.

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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!

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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