Tutorial rpo

52
Recent Advances in Real-Parameter Evolutionary Algorithms Presenters: P. N. Suganthan & Qin Kai School of Electrical and Electronic Engineering Nanyang Technological University, Singapore Some Software Resources Available from: http://www.ntu.edu.sg/home/epnsugan SEAL’06 October 2006 -2- RP-EAs & Related Issues Covered Benchmark Test Functions Methods Modeled After conventional GAs CMA-ES / Covariance Matrix Adapted Evolution Strategy PSO / Particle Swarm Optimization DE / Differential Evolution EDA / Estimation of Distribution Algorithms CEC’05, CEC’06 benchmarking Coverage is biased in favor of our research and results of the CEC-05/06 competitions while overlap with other SEAL tutorial presentations is reduced.

description

 

Transcript of Tutorial rpo

Page 1: Tutorial rpo

Recent Advances in Real-Parameter Evolutionary Algorithms

Presenters: P. N. Suganthan & Qin Kai School of Electrical and Electronic EngineeringNanyang Technological University, Singapore

Some Software Resources Available from:http://www.ntu.edu.sg/home/epnsugan

SEAL’06 October 2006

-2-

RP-EAs & Related Issues Covered

Benchmark Test Functions Methods Modeled After conventional GAsCMA-ES / Covariance Matrix Adapted Evolution StrategyPSO / Particle Swarm Optimization DE / Differential EvolutionEDA / Estimation of Distribution AlgorithmsCEC’05, CEC’06 benchmarking

Coverage is biased in favor of our research and results of the CEC-05/06 competitions while overlap with other SEAL tutorial presentations is reduced.

Page 2: Tutorial rpo

-3-

Benchmark Test Functions

Resources available fromhttp://www.ntu.edu.sg/home/epnsugan

(limited to our own work)

From Prof Xin Yao’s grouphttp://www.cs.bham.ac.uk/research/projects/ecb/

Includes diverse problems.

-4-

Why do we require benchmark problems?

Why we need test functions?To evaluate a novel optimization algorithm’s property on different types of landscapes

Compare different optimization algorithms

Types of benchmarks

Bound constrained problemsConstrained problems

Single / Multi-objective problems

Static / Dynamic optimization problems

Multimodal problems (niching, crowding, etc.)

Page 3: Tutorial rpo

-5-

Shortcomings in Bound constrained Benchmarks

Some properties of benchmark functions may make them unrealisticor may be exploited by some algorithms:

Global optimum having the same parameter values for different variables\dimensions Global optimum at the originGlobal optimum lying in the center of the search range Global optimum on the bound Local optima lying along the coordinate axes no linkage among the variables / dimensions or the same linkagesover the whole search rangeRepetitive landscape structure over the entire space

Do real-world problems possess these properties?Liang et. al 2006c (Natural Computation) has more

details.

-6-

How to Solve?

Shift the global optimum to a random position to make the global optimum to have different parameter values for different dimensions

Rotate the functions as below:

where R is an orthogonal rotation matrix

Use different kinds of benchmark functions, different rotation matrices to compose a single test problem.

Mix different properties of different basic test functions together to destroy repetitive structures and to construct novel Composition Functions

( ) ( * )F f=x R x

Page 4: Tutorial rpo

-7-

Novel Composition Test Functions

Compose the standard benchmark functions to construct a more challenging function with a randomly located global optimum and several randomly located deep local optima with different linkage properties over the search space.

Gaussian functions are used to combine these benchmark functions and to blur individual functions’ structures mainly around the transition regions.

More details in Liang, et al 2005.

-8-

Novel Composition Test Functions

is.

Page 5: Tutorial rpo

-9-

Novel Composition Test Functions

-10-

Novel Composition Test Functions

Page 6: Tutorial rpo

-11-

A couple of Examples

Many composition functions are available from our homepage

Composition Function 1 (F1):Made of Sphere Functions

Composition Function 2 (F2):

Made of Griewank’s Functions

Similar analysis is needed for other benchmarks too

-12-

Methods Modeled AfterConventional GAs

Some resources available fromhttp://www.iitk.ac.in/kangal/

Page 7: Tutorial rpo

-13-

Major Steps in Standard GAs

Population initialization

Fitness evaluation

Selection

Crossover (or recombination)

Mutation

Replacement

-14-

Mean-Centric Vs Parent-Centric Recombination

Individual recombination preserves the mean of parents (mean-centric) – works well if the population is randomly initialized within the search range and the global optimum is near the centre of the search range.Individual recombination biases offspring towards a particular parent, but with equal probability to each parent (parent-centric)

Page 8: Tutorial rpo

-15-

Mean-Centric Recombination [Deb-01]

SPX : Simplex crossover (Tsutsui et al. 1999)UNDX : Unimodal Normally distributed crossover (Ono et al. 1997, Kita 1998)BLX : Blend crossover (Eshelman et al 1993)Arithmetic crossover (Michalewicz et al 1991)Please refer to [Deb-01] for details.

SBX (Simulated Binary crossover) : variable-wiseVector-wise parent-centric recombination (PCX)

And many more …

Parent-Centric Recombination

-16-

Minimal Generation Gap (MGG) Model

1. Select μ parents randomly2. Generate λ offspring using a recombination scheme 3. Choose two parents at random from the population4. One is replaced with the best of λ offspring and other with a

solution by roulette-wheel selection from a combination of λoffspring and two parents

Steady-state and elitist GA Typical values to use: population N = 300, λ=200, μ=3 (UNDX) and μ=n+1 (SPX) (n: dimensions)

Satoh, H, Yamamura, M. and Kobayashi, S.: Minimal generation gap model for GAs considering both exploration and exploitation. Proc. of the IIZUKA 96, pp. 494-497 (1997).

Page 9: Tutorial rpo

-17-

Generalized Generation Gap (G3) Model

1. Select the best parent and μ-1 other parents randomly2. Generate λ offspring using a recombination scheme 3. Choose two parents at random from the population4. Form a combination of two parents and λ offspring, choose best

two solutions and replace the chosen two parentsParameters to tune: λ, μ and N

-18-

Mutation

In RP-EAs, the main difference between crossover and mutation may be the number of parents used for the operation. In mutation, just 1. In crossover, 2 or more (excluding DE’s terminology).

There are several mutation operators developed for real parameters

Random mutation : the same as re-initialization of a variable

Gaussian mutation: Xnew=Xold+N(0, )

Cauchy mutation with EP (Xin Yao et. al 1999)

Levy mutation / adaptive Levy mutation with EP (Lee, Xin Yao, 2004)

Making use of these RP-crossover (recombination), RP-mutation operators, RP-EAs can be developed in a manner similar to the Binary GA.

σ

Page 10: Tutorial rpo

-19-

Covariance Matrix Adapted Evolution Strategy CMAES

By Nikolaus Hansen & collaborators

Extensive Tutorials / Codes available fromhttp://www.bionik.tu-berlin.de/user/niko/cmaesintro.html

http://www.bionik.tu-berlin.de/user/niko/cmaes_inmatlab.html

-20-

Basic Evolution Strategy [BFM-02]

Randomly generate an initial population of M solutions. Compute the fitness values of these M solutions.

Use all vectors as parents to create nb offspring vectors by Gaussian mutation.

Calculate the fitness of nb vectors, and prune the population to Mfittest vectors.

Go to the next step if the stopping condition is satisfied. Otherwise, go to Step 2.

Choose the fittest one in the population in the last generation as the optimal solution.

Page 11: Tutorial rpo

-21-

Covariance Matrix Adapted Evolution Strategy CMAES

The basic ES generates solutions around the parents

CMAES keeps track of the evolution path.

It updates mean location of the population, covariance matrix and the standard deviation of the sampling normal distribution after every generation.

The solutions are ranked which is not a straight forward step in multi-objective / constrained problems.

Ranked solutions contribute according to their ranks to the updating of the mean position, covariance matrix, and the step-size parameter.

Covariance analysis reveals the interactions / couplings among the parameters.

-22-

CMAES Iterative Equations [Hansen 2006]

Page 12: Tutorial rpo

-23-

CMAES [HO-04]

Performs well on rotated / linked problems with bound constraints.

It learns global linkages between parameters.

The cumulation operation integrates a degree of forgetting.

The CMAES implementation is self-adaptive.

CMAES is not highly effective in solving constrained problems, multi-objective problems, etc.

CMAES is not as simple as PSO / DE.

-24-

Outline of PSO

PSO PSO variants CLPSO DMS-L-PSO Constrained Optimization Constraint-Handling Techniques DMS-C-PSO PSO for Multi-Objective Optimization

Page 13: Tutorial rpo

-25-

Particle Swarm Optimizer

Introduced by Kennedy and Eberhart in 1995(Eberhart & Kennedy,1995; Kennedy & Eberhart,1995)Emulates flocking behavior of birds to solve optimization problems Each solution in the landscape is a particleAll particles have fitness values and velocities

-26-

Particle Swarm Optimizer

Two versions of PSOGlobal version (May not be used alone to solve multimodal problems): Learning from the personal best (pbest) and the best position achieved by the whole population (gbest)

Local Version: Learning from the pbest and the best position achieved in the particle's neighborhood population (lbest)

The random numbers (rand1 & rand2) should be generated for each dimension of each particle in every iteration.

1 2* 1 ( ) * 2 ( )← ∗ − + ∗ −

← +

d d d d d d di i i i i i

d d di i i

V c rand pbest X c rand gbest X

X X V

1 2* 1 ( ) * 2 ( )← ∗ − + ∗ −

← +

d d d d d d di i i i i k i

d d di i i

V c rand pbest X c rand lbest X

X X V

i – particle counter & d – dimension counter

Page 14: Tutorial rpo

-27-

Particle Swarm Optimizer

where and in the equation are the acceleration constants

are two random numbers in the range [0,1];

represents the position of the ith particle;

represents the rate of the position change (velocity) for particle i.

represents the best previous position (the position giving the best fitness value) of the ith particle;

represents the best previous position of the population;

represents the best previous position achieved by the neighborhood of the particle;

1c 2c

1 and 2d di irand rand

1 2( , ,..., )Di i i iX X X=X

1 2( , ,..., )Di i i ipbest pbest pbest=pbest

1 2( , ,..., )Dgbest gbest gbest=gbest

1 2( , ,..., )Di i i iV V V=V

1 2( , ,..., )Dk k k klbest lbest lbest=lbest

-28-

PSO variantsModifying the Parameters

Inertia weight ω (Shi & Eberhart, 1998; Shi & Eberhart, 2001; Eberhart& Shi, 2001, …)Constriction coefficient (Clerc,1999; Clerc & Kennedy, 2002)Time varying acceleration coefficients (Ratnaweera et al. 2004)Linearly decreasing Vmax (Fan & Shi, 2001)

… …

Using Topologies Extensive experimental studies (Kennedy, 1999; Kennedy & Mendes, 2002, …)Dynamic neighborhood (Suganthan,1999; Hu and Eberhart, 2002; Peram et al. 2003)Combine the global version and local version together (Parsopoulos and Vrahatis, 2004 ) named as the unified PSO.

Fully informed PSO or FIPS (Mendes & Kennedy 2004) and so on …

Page 15: Tutorial rpo

-29-

PSO variants and ApplicationsHybrid PSO Algorithms

PSO + selection operator (Angeline,1998)

PSO + crossover operator (Lovbjerg, 2001)

PSO + mutation operator (Lovbjerg & Krink, 2002; Blackwell & Bentley,2002; … …)

PSO + cooperative approach (Bergh & Engelbrecht, 2004)…

Various Optimization Scenarios & Applications

Binary Optimization (Kennedy & Eberhart, 1997; Agrafiotis et. al 2002; )

Constrained Optimization (Parsopoulos et al. 2002; Hu & Eberhart, 2002; … )

Multi-objective Optimization (Ray et. al 2002; Coello et al. 2002/04; … )

Dynamic Tracking (Eberhart & Shi 2001; … )

Yagi-Uda antenna (Baskar et al 2005b), Photonic FBG design (Baskar et al 2005a), FBG sensor network design (Liang et al June 2006)

-30-

Comprehensive Learning PSO

Comprehensive Learning Strategy :

where defines which particle’s pbest particle i should follow. can be the corresponding dimension of any particle’s pbest including its own pbest, and the decision depends on probability Pc, referred to as the learning probability, which can take different values for different particles.

For each dimension of particle i, we generate a random number. If this random number is larger than Pci, this dimension will learn from its own pbest, otherwise it will learn from another randomly chosen particle’s pbest.

( )* ( )d d d d di i i fi d iV w V c rand pbest X← ∗ + ∗ −

[ (1), (2),..., ( )]i i if f f D=if( )

dfi dpbest

Page 16: Tutorial rpo

-31-

Comprehensive Learning PSO

The tournament selection (2) procedure is employed to choose the exemplar for each dimension of each particle.

To ensure that a particle learns from good exemplars and to minimize the time wasted on poor directions, we allow the particle to learn from the exemplars until the particle ceases improving for a certain number of generations called the refreshing gap m (7 generations), after which we re-assign the dimensions of this particle.

-32-

Comprehensive Learning PSO

We observe three main differences between the CLPSO and the original PSO:

Instead of using particle’s own pbest and gbest as the exemplars, all particles’ pbests can potentially be used as the exemplars to guide a particle’s flying direction. Instead of learning from the same exemplar particle for all dimensions, each dimension of a particle in general can learn from different pbests for different dimensions for some generations. In other words, in any one iteration, each dimension of a particle may learn from the corresponding dimension of different particle’s pbest. Instead of learning from two exemplars (gbest and pbest) at the same time in every generation as in the original PSO, each dimension of a particle learns from just one exemplar for some generations.

Page 17: Tutorial rpo

-33-

Comprehensive Learning PSO

Experiments have been conducted on16 test functions which can be grouped into four classes. The results demonstrated good performance of the CLPSO in solving multimodal problems when compared with eight other recent variants of the PSO.

(The details can be found in Liang et al. 2006a. Matlab codes are also available for CLPSO and the 8 PSO variants.)

-34-

Dynamic Multi-Swarm PSO

Constructed based on the local version of PSO with a novel neighborhood topology.

This new neighborhood structure has two important characteristics:

Small Sized Swarms

Randomized Regrouping Schedule

Page 18: Tutorial rpo

-35-

Dynamic Multi-Swarm PSO

The population is divided into small swarms randomly.These sub-swarms use their own particles to search for better solutions and converge to good solutions. The whole population is regrouped into new sub-swarms periodically. New sub-swarms continue the search. This process is continued until a stop criterion is satisfied

Regroup

-36-

Dynamic Multi-Swarm PSO

Adaptive Self-Learning Strategy Each Particle has a corresponding Pci. Every R generations, keep_idi is decided by Pci. If the generated random number is larger than or equal to Pci, this dimension will be set at the value of its own pbest, keep_idi(d) is set to1. Otherwise keep_idi(d) is set to 0 and it will learn from the lbest and its own pbest as the PSO with constriction coefficient:

max max

If _ 0

0.729 1.49445 1 ( )

1.49445 2 ( )

min( , max( , ))

Otherwise

di

d d d d di i i i i

d d di k i

d d d di i

d d di i i

d di i

keep id

V V rand pbest X

rand lbest X

V V V V

X X V

X pbest

=

← ∗ + ∗ ∗ −

+ ∗ ∗ −

= −

← +

←Somewhat similar to DE & CPSO

Page 19: Tutorial rpo

-37-

Dynamic Multi-Swarm PSO

Adaptive Self-Learning StrategyAssume Pc normally distributed in a range with mean(Pc) and a standard deviation of 0.1.

Initially, mean(Pc) is set at 0.5 and different Pc values conforming to this normal distribution are generated for each individual in the current population.

During every generation, the Pc values associated with the particles which find new pbest are recorded. When the sub-swarms are regrouped, the mean of normal distribution of Pc is recalculated according to all the recorded Pc values corresponding to successful movements during this period. The record of the successful Pc values will be cleared once the normal distribution’s mean is recalculated.

As a result, the proper Pc value range for the current problem can be learned to suit the particular problem.

-38-

DMS-PSO with Local Search

Since we achieve a larger diversity using DMS-PSO, at the same time, the convergence rate is reduced. In order to alleviate this problem and to enhance local search capabilities, a local search is incorporated:

Every L generations, the pbests of five randomly chosen particles will be used as the starting points and the Quasi-Newton method (fminunc(…,…,…)) is employed to do the local search with the L_FEs as the maximum FEs.

In the end of the search, all particles form a single swarm to become a global PSO version. The best solution achieved so far is refined using Quasi-Newton method every L generations with the 5*L_FEs as the maximum FEs.

If local search results in improvements, the nearest pbest is replaced / updated.

Page 20: Tutorial rpo

-39-

Constrained Optimization

Optimization of constrained problems is an important area in the optimization field.

In general, the constrained problems can be transformed into the following form:

Minimize

subjected to:

q is the number of inequality constraints and m-q is the number of equality constraints.

1 2( ), [ , ,..., ]Df x x x=x x

( ) 0, 1,...,jh j q m= = +x

( ) 0, 1,...,ig i q≤ =x

-40-

Constrained Optimization

For convenience, the equality constraints can be transformed into inequality form:

where is the allowed tolerance.

Then, the constrained problems can be expressed asMinimize

subjected to

If we denote with the feasible region and the whole search space, if and all constraints are satisfied. In this case,x is a feasible solution.

| ( ) | 0jh ε− ≤x

ε

F∈x S∈xF S

1 2( ), [ , ,..., ]Df x x x=x x

1,..., 1,... 1,..., 1,...

( ) 0, 1,..., ,

( ) ( ), ( ) ( ) ε+ +

≤ =

= = −

j

q q q m q m

G j m

G g G h

x

x x x x

Page 21: Tutorial rpo

-41-

Constraint-Handling (CH) Techniques

Penalty Functions: Static Penalties (Homaifar et al.,1994;…)Dynamic Penalty (Joines & Houck,1994; Michalewicz& Attia,1994;…)Adaptive Penalty (Eiben et al. 1998; Coello, 1999; Tessema & Gary Yen 2006, …)…

Superiority of feasible solutionsStart with a population of feasible individuals (Michalewicz, 1992; Hu & Eberhart, 2002; …)Feasible favored comparing criterion (Ray, 2002; Takahama & Sakai, 2005; … )Specially designed operators (Michalewicz, 1992; …)…

-42-

Constraint-Handling (CH) Techniques

Separation of objective and constraintsStochastic Ranking (Runarsson & Yao, TEC, Sept 2000)

Co-evolution methods (Liang 2006e)

Multi-objective optimization techniques (Coello, 2000; Mezura-Montes & Coello, 2002;… )

Feasible solution search followed by optimization of objective (Venkatraman & Gary Yen, 2005)

While most CH techniques are modular (i.e. we can pick one CH technique and one search method independently), there are also CH techniques embedded as an integral part of the EA.

More details in the tutorial presented by Prof Z. Michalewicz

Page 22: Tutorial rpo

-43-

DMS-PSO for Constrained Optimization

Novel Constraint-Handling MechanismSuppose that there are m constraints, the population is divided into n sub-swarms with sn members in each sub-swarm and the population size is ps (ps=n*sn). n is a positive integer and ‘n=m’ is not required.

The objective and constraints are assigned to the sub-swarms adaptively according to the difficulties of the constraints.

By this way, it is expected to have population of feasible individuals with high fitness values.

-44-

DMS-PSO’s Constraint-Handling Mechanism

How to assign the objective and constraints to each sub-swarm?

Define

Thus 1 21 , [ , ,..., ]mfp p p p= − =p p

1

0

if a ba b

if a b

>⎧> = ⎨ ≤⎩

1

( ( ) 0)

, 1, 2,...,

ps

i jj

i

g

p i mps

=

>= =∑ x

1

( / ) 1m

ii

fp p m=

+ =∑

Page 23: Tutorial rpo

-45-

DMS-PSO’s Constraint-Handling Mechanism

For each sub-swarm,Using roulette selection according to fp and to assign the objective function or a single constraint as its target. If sub-swarm i is assigned to improve constraint j, set obj(i)=j and if sub-swarm i is assigned to improve the objective function, set obj(i)=0.

Assigning swarm member for this sub-swarm. Sort the unassigned particles according to obj(i), and assign the best and sn-1 worst particles to sub-swarm i.

/ip m

-46-

DMS-PSO’s Comparison Criteria

1. If obj(i) = obj(j) = k (particle i and j handling the same constraint k), particle i wins if

2. If obj(i) = obj(j) = 0 (particle i and j handling f(x) ) or obj(i) ≠ obj(j) (iand j handling different objectives), particle i wins if

( ) ( ) with ( ) 0

( ) ( ) & ( ), ( ) 0

( ) ( ) & ( ) ( )

< >

< ≤

< ==

k i k j k j

i j k i k j

i j i j

G G G

or V V G G

or f f V V

x x x

x x x x

x x x x

( ) ( )

( ) ( ) & ( ) ( )

<

< ==i j

i j i j

V V

or f f V V

x x

x x x x

1

( ) ( ( ) ( ( ) 0))=

= ⋅ ⋅ ≥∑m

i i ii

V weight G Gx x x

1

1/ max, 1, 2,...

(1/ max)=

= =

∑i

i m

ii

Gweight i m

G

Page 24: Tutorial rpo

-47-

DMS-PSO for Constrained Optimization

Step 1: Initialization -

Initialize ps particles (position X and velocity V), calculate f(X), Gj(X) (j=1,2...,m) for each particle.

Step 2: Divide the population into sub-swarms and assign obj for each sub-swarm using the novel constraint-handling mechanism, calculate the mean value of Pc (except in the first generation, mean(Pc)=0.5), calculate Pc for each particle. Then empty Pc.

Step 3: Update the particles according to their objectives; update pbest and gbest of each particle according to the same comparison criteria, record the Pc value if pbest is updated.

-48-

Step 5: Local Search-Every L generations, randomly choose 5 particles’ pbest and start local search with Sequential Quadratic Programming (SQP) method using these solutions as start points (fmincon(…,…,…) function in Matlab is employed). The maximum fitness evaluations for each local search is L_FEs.

Step 6: If FEs≤0.7*Max_FEs, go to Step 3. Otherwise go to Step 7.

Step 7: Merge the sub-swarms into one swarm and continue PSO (Global Single Swarm). Every L generations, start local search using gbest as start points using 5*L_FEs as the Max FEs. Stop search if FEs≥Max_FEs

DMS-PSO for Constrained Optimization

Page 25: Tutorial rpo

-49-

Multi-Objective Optimization

Mathematically, we can use the following formula to express the multi-objective optimization problems (MOP):

The objective of multi-objective optimization is to find a set of solutions which can express the Pareto-optimal set well, thus there are two goals for the optimization:

1) Convergence to the Pareto-optimal set

2) Diversity of solutions in the Pareto-optimal set

1 2Minimize ( ) ( ( ), ( ), , ( )) [ , ]

subject to ( ) 0, 1,...,

( ) 0, 1,...,

m

j

k

f f f f

g j z

h k q m

= = ∈

≤ =

= = +

y x x x x x Xmin Xmax

x

x

K

-50-

Extending PSO for MOP

Mainly used techniques:How to update pbest and gbest (or lbest)

Execute non-domination comparison with pbest or gbest

Execute non-domination comparison among all particles’pbests and their offspring in the entire population

How to chose gbest (or lbest)Choose gbest (or lbest) from the recorded non-dominated solutions

Choose good local guides

How to keep diversityCrowding distance sorting

Subpopulation

Page 26: Tutorial rpo

-51-

Extending CLPSO and DMS-PSO for MOP

Combine an external archive which is used to record the non-dominated solutions found so far.

Use Non-dominated Sorting and Crowding distance Sorting which have been used in NSGAII (Deb et al., 2002) to sort the members in the external archive.

Choose exemplars (CLPSO) or lbest (DMS-PSO) from the non-dominated solutions recorded in the external archive.

Experiments show that MO-CLPSO and DMS-MO-PSO are both capable of converging to the true Pareto optimal front and maintaining a good diversity along the Pareto front.

More details in Huang et. al Feb 2006 on MOCLPSO.Also please refer to SEAL-06 tutorials by Profs X. D. Li & G. Yen.

-52-

Outline

Motivation for Differential Evolution (DE)

Classic DE

DE Variants

Self-adaptive DE (SaDE)

Trends in DE Research

Real-parameter Estimation of Distribution Algorithms

(rEDA)

Page 27: Tutorial rpo

-53-

Motivation for DEDE, proposed by Price and Storn in 1995 [PS95], was motivated by the attempts

to use Genetic Annealing [P94] to solve the Chebychev polynomial fitting problem.

Genetic annealing is a population-based, combinatorial optimization algorithm that implements a thermodynamic annealing criterion via thresholds. Although successfully applied to solve many combinatorial tasks, genetic annealing cannot solve well the Chebychev problem.

Price modified genetic annealing by using floating-point encoding instead of bit-string one, arithmetic operations instead of logical ones, population-driven differential mutation instead of bit-inversion mutation and removed the annealing criterion. Storn suggested creating separate parent and children populations. Eventually, Chebychev problem can be solved effectively and efficiently.

DE is closely related to many other multi-point derivative free search methods [PSL05] such as evolutionary strategies, genetic algorithms, Nelder and Mead direct search and controlled random search.

-54-

DE at a glanceCharacteristics

Population-based stochastic direct searchSelf-referential mutationSimple but powerfulReliable, robust and efficientEasy parallelizationFloating-point encoding

Basic componentsInitializationTrial vector generation

MutationRecombination

ReplacementFeats

DE demonstrated promising performance in 1st and 2nd

ICEO [BD96][P97]

Page 28: Tutorial rpo

-55-

InitializationA population Px,0 of Np D-dimensional parameter vectors xi,0=[x1,i,0,…,xD,i,0], i=1,…,Np is randomly generated within the prescribed lower and upper bound bL= [b1,L,…,bD,L] and bU=[b1,U,…,bD,U]

Insight into classic DE (DE/rand/1/bin)

Trial vector generation

Example: the initial value (at generation g=0) of the jth parameter of the ith vector is generated by: xj,i,0 = randj[0,1] ·(bj,U-bj,L) + bj,L, j=1,…,D, i=1,…,Np

At the At the ggthth generation, a trial population generation, a trial population PPuu,,gg consisting of consisting of NpNp DD--dimensional trial dimensional trial vectors vectors vvi,gi,g=[=[vv1,1,i,gi,g,,……vvDD,,i,gi,g] is generated via mutation and recombination operations ] is generated via mutation and recombination operations applied to the current population applied to the current population PPxx,,gg

Differential mutation: with respect to each vector xi,g in the current population, called target vector, a mutant vector vi,g is generated by adding a scaled, randomly sampled, vector difference to a basis vector randomly selected from the current population

-56-

Insight into classic DE (DE/rand/1/bin)

Replacement

Example: at the gth generation, the ith mutant vector vi,g with respect to ith target vector xi,g in the current population is generated by vi,g = xr0,g + F·(xr1,g-xr2,g), i≠r0≠r1≠r2, mutation scale factor F∈(0,1+)

Discrete recombination: with respect to each target vector xi,g in the current population, a trial vector ui,g is generated by crossing the target vector xi,g with the corresponding mutant vector vi,g under a pre-specified crossover rate Cr∈[0,1]

Example: at the gth generation, the ith trial vector ui,g with respect to ith target vector xi,g in the current population is generated by:

vj,i,g if randj[0,1]≤Cr or j=jrand

xj,i,g otherwiseuj,i,g=

If the trial vector ui,g has the better objective function value than that of its corresponding target vector xi,g, it replaces the target vector in the (g+1)th

generation; otherwise the target vector remains in the (g+1)th generation

Page 29: Tutorial rpo

-57-

Illustration of classic DE

x2

x1

Illustration of classic DE

-58-

xi,g

xr1,g

xr2,g

xr0,g

x2

x1

Target vector

Base vector Two randomly selected vectors

Illustration of classic DE

Illustration of classic DE

Page 30: Tutorial rpo

-59-

xi,g

xr1,g

xr2,g

xr0,g

x2

x1

Four operating vectors in 2D continuous space

Illustration of classic DE

-60-

xi,g

xr1,g

xr2,g

xr0,g

F·(xr1,g-xr2,g)

vi,g

x2

x1

Mutation

Illustration of classic DE

Page 31: Tutorial rpo

-61-

xi,g

xr1,g

xr2,g

xr0,g

F·(xr1,g-xr2,g)

vi,g ui,g

x2

x1

Crossover

Illustration of classic DE

-62-

Replacement

xi,g

xr1,g

xr2,g

xr0,g

xi,g+1

x2

x1

Illustration of classic DE

Page 32: Tutorial rpo

-63-

Differential vector distribution

Most important characteristics of DE: self-referential mutation!

ES: fixed probability distribution function with adaptive step-size

DE: adaptive distribution of difference vectors with fixed step-size

A population of 5 vectors 20 generated difference vectors

-64-

DE variants

Modification of different components of DE can result in many DEModification of different components of DE can result in many DEvariantsvariants:

Initialization

Uniform distribution and Gaussian distribution

Trial vector generation

Base vector selection

Random selection without replacement: r0=ceil(randi[0,1]·Np)

Permutation selection: r0=permute[i]

Random offset selection: r0=(i+rg)%Np (e.g. rg=2)

Biased selection: global best, local best and better

Page 33: Tutorial rpo

-65-

DE variants

Differential mutation

One difference vector: F·(xr1- xr2)

Two difference vector: F·(xr1- xr2)+F·(xr3- xr4)

Mutation scale factor F

Crucial role: balance exploration and exploitation

Dimension dependence: jitter (rotation variant) and dither (rotation invariant)

Randomization: different distributions of F

DE/rand/1:

DE/best/1:

DE/current-to-best/1:

DE/rand/2:

DE/best/2:

( )GrGrGrGi F ,,,, 321XXXV −⋅+=

( )GrGrGbestGi F ,,,, 21XXXV −⋅+=

( ) ( )GrGrGiGbestGiGi FF ,,,,,, 21XXXXXV −⋅+−⋅+=

( )GrGrGrGrGrGi F ,,,,,, 54321XXXXXV −+−⋅+=

( )GrGrGrGrGbestGi F ,,,,,, 4321XXXXXV −+−⋅+=

-66-

DE variantsRecombination

Discrete recombination (crossover) (rotation variant)

One point and multi-point

Exponential

Binominal (uniform)

Arithmetic recombination

Line recombination (rotation invariant)

Intermediate recombination (rotation variant)

Extended intermediate recombination (rotation variant)

x1

x2 xb

xa

line

discrete

discrete

intermediate

x 1x 2

Page 34: Tutorial rpo

-67-

DE variants

Replacement

Degenerate cases in the trial vector generation

For example, in classic DE, r1=r2, r0=r1, r0=r2, i=r0, i=r1, i=r2

Better to generate mutually exclusive indices for target vector, base vector and vectors constituting the difference vector

One-to-one replacement

Neighborhood replacement

Crossover rate CR∈[0,1]

Decomposable (small CR) and indecomposable functions (large CR)

-68-

Motivation for self-adaptation in DE

The performance of DE on different problems depends on:

Population size

Strategy and the associated parameter setting to generate trial vectors

Replacement scheme

It is hard to choose a unique combination to successfully solve any problem at hand

Population size usually depends on the problem scale and complexity

During evolution, different strategies coupled with specific parameter settings

may favor different search stages

Replacement schemes influence the population diversity

Trial and error scheme may be a waste of computational time & resources

Automatically adapt the configuration in DE so as to generate efAutomatically adapt the configuration in DE so as to generate effective fective trial vectors during evolutiontrial vectors during evolution

Page 35: Tutorial rpo

-69-

Related worksPractical guideline [SP95], [SP97], [CDG99], [BO04], [PSL05],[GMK02]: for example, Np∈[5D,10D]; Initial choice of F=0.5 and CR=0.1/0.9; Increase NP and/or F if premature convergence happens. Conflicting conclusions with respect to different test functionsConflicting conclusions with respect to different test functions..

Fuzzy adaptive DE [LL02]: use fuzzy logical controllers whose inputs incorporate the relative function values and individuals of successive generations to adapt the mutation and crossover parameters.

Self-adaptive Pareto DE [A02]: encode crossover rate in each individual, which is simultaneously evolved with other parameters. Mutation scale factor is generated for each variable according to Gaussian distribution N(0,1).

Zaharie [Z02]: theoretically study the DE behavior so as to adapt the control parameters of DE according to the evolution of population diversity.

Self-adaptive DE (1) [OSE05]: encode mutation scale factor in each individual, which is simultaneously evolved with other parameters. Crossover rate is generated for each variable according to Gaussian distribution N(0.5,0.15).

DE with self-adaptive population [T06]: population size, mutation scale factor and crossover rate are all encoded into each individual.

Self-adaptive DE (2) [BGBMZ06]: encode mutation scale factor and crossover rate in each individual, which are reinitialized according to two new probability variables.

-70-

Steps:Initialize selection probability pi=1/num_st, i=1,…,num_st for each strategyAccording to the current probabilities, we employ stochastic universal selection to assign one strategy to each target vector in the current population For each strategy, define vectors nsi and nfi, i=1,…num_st to store the number of trial vectors successfully entering the next generation or discarded by applying such strategy, respectively, within a specified number of generations, called “learning period (LP)”Once the current number of generations is over LP, the first element of nsi and nfi with respect to the earliest generation will be removed and the behavior in current generation will update nsi and nfi

Self-adaptive DE (SaDE)

Strategy adaptation: select one strategy from a pool of candidate strategies with the probability proportional to its previous successful rate to generate effective trial vectors during a certain learning period

DE with strategy and parameter self-adaptation (QS05, HQS06)

Page 36: Tutorial rpo

-71-

Self-adaptive DE (SaDE)

Parameter adaptation

Mutation scale factor (F): for each target vector in the current population, we randomly generate F value according to a normal distribution N(0.5,0.3). Therefore, 99% F values fall within the range of [–0.4,1.4]

Crossover rate (CRj): when applying strategy j with respect to a target vector, the corresponding CRj value is generated according to an assumed distribution, and those CRj values that have generated trial vectors successfully entering the next generation are recorded and updated every LP generations so as to update the parameters of the CRj distribution. We hereby assume that each CRj, j=1,…,num_st is normally distributed with its mean and standard deviation initialized to 0.5 and 0.1, respectively

The selection probability pi is updated by:

nsnum_st(L)…ns1(L)

nsnum_st(1)…ns1(1)

nsnum_st(L+1)…ns1(L+1)

nsnum_st(2)…ns1(2)

nsnum_st(L+2)…ns1(L+2)

nsnum_st(3)…ns1(3)

( )∑+∑∑ iii nfnsns . Go to 2nd step

-72-

Instantiations

In CEC’05, we use 2 strategies:

In CEC’06, we employ 4 strategies:

( )GrGrGrGi F ,,,, 321XXXV −⋅+=

DE/rand/1/bin:

DE/rand/2/bin:

DE/current-to-best/2/bin:

DE/current-to-rand/1:

( )GrGrGrGi F,321 ,,, XXXV −⋅+=

( ) ( )GrGrGrGrGiGbestGiGi FF ,,,,,,,, 4321XXXXXXXV −+−⋅+−⋅+=

( )GrGrGrGrGrGi F,54,321 ,,,, XXXXXV −+−⋅+=

DE/rand/1/bin:

DE/current-to-best/2/bin: ( ) ( )GrGrGrGrGiGbestGiGi FF ,,,,,,,, 4321XXXXXXXV −+−⋅+−⋅+=

( ) ( )GrGrGiGrGiGi FF ,,,,,, 321XXXXXV −⋅+−⋅+=

LP = 50

LP = 50

Page 37: Tutorial rpo

-73-

Local search enhancement

To improve convergence speed,To improve convergence speed, we apply a local search we apply a local search procedure every 500 generations:procedure every 500 generations:

To apply local search, we choose n = 0.05·Np individuals, which include the individual having the best objective function value and the n-1 individuals randomly selected from the top 50% individuals in the current population

We perform the local search by applying the Quasi-Newton method to the selected n individuals

-74-

SaDE for constrained optimization

Modification of the replacement procedure

Each trial vector ui,g is compared with its corresponding target vector xi,g in the current population in terms of both the objective function value and the degree of constraint violation

uui,gi,g will replace will replace xxi,gi,g if any of the following conditions is satisfied:if any of the following conditions is satisfied:

ui,g is feasible while xi,g is not

ui,g and xi,g are both feasible, and ui,g has a better or equal objective function value than that of xi,g

ui,g and xi,g are both infeasible, but ui,g has a smaller degree of overall constraint violation

Page 38: Tutorial rpo

-75-

The overall constraint violation is evaluated by a weighted average constraint violation as follows:

max1

i

iw

G=

maxiG is the maximum violation of the ith constraint obtained so far

is a weighting factor

SaDE for constrained optimization

( ) ( )∑

∑==

=mi i

mi ii

w

Gwv

1

1 xx

-76-

Overview of DE research trends

Number of publications per year in ISI web of knowledge database

Page 39: Tutorial rpo

-77-

Overview of DE research trends

Digital Filter Design

Multiprocessor synthesis

Neural network learning

Diffraction grating design

Crystallographic characterization

Beam weight optimization in radiotherapy

Heat transfer parameter estimation in a trickle bed reactor

Electricity market simulation

Scenario-Integrated Optimization of Dynamic Systems

Optimal Design of Shell-and-Tube Heat Exchangers

Optimization of an Alkylation's Reaction

Optimization of Thermal Cracker Operation

Optimization of Non-Linear Chemical Processes

Optimum planning of cropping patterns

Optimization of Water Pumping System

Optimal Design of Gas Transmission Network

Differential Evolution for Multi-Objective Optimization

Bioinformatics

DE Applications

-78-

DE Resources

http://www.icsi.berkeley.edu/~storn/code.htm

Books

Website

Blog

1999 [CDG99] 2004 [BO04] 2005 [PSL05]

http://dealgorithm.blogspot.com/

Matlab code of SaDE http://www.ntu.edu.sg/home/epnsugan

Page 40: Tutorial rpo

-79-

Estimation of distribution algorithm (EDA)

New approach to evolutionary computation, which was proposed in New approach to evolutionary computation, which was proposed in 1996 [1996 [MP96MP96] and described in details in [LL01, ] and described in details in [LL01, LLIB06LLIB06]]

Characteristics:

Population-based

No crossover and mutation operators

In each generation, the underlying probability distribution of the selected individuals is estimated by certain machine learning model, which can capture the dependence among parameters

New population of individuals is sampled from the estimated probability distribution model

Repeat the two previous steps until a stopping criterion is met

-80-

Real-parameter EDA (rEDA)Without parameter dependencies

Stochastic Hill-Climbing with Learning by Vectors of Normal Distribution (SHCLVND) [RK96]

Population Based Incremental Learning – Continuous (PBILc) [SD98]

Univariate Marginal Distribution Algorithm - Continuous (UMDAc) [LELP00]

Bivariate parameter dependencies

Mutual Information Maximization for Input Clustering - Continuous - Gaussian (MIMICG

c) [LELP00]

Multiple parameter dependencies

Estimation of Multivariate Normal Algorithm: EMNAglobal, EMNAa, EMNAi[LLB01, LL01]

Estimation of Gaussian Network Algorithm: EMNAee, EMNABGe, EMNABIC[LLB01, LL01]

Iterated Density Evolutionary Algorithm (IDEA) [BT00]

Real-coded Bayesian Optimization Algorithm (rBOA) [ARG04]

Page 41: Tutorial rpo

-81-

rEDA resources

http://medal.cs.umsl.edu/

http://www.iba.k.u-tokyo.ac.jp/english/EDA.htm

http://www.evolution.re.kr/

Books

Websites

2001 [LL01] 2006 [LLIB06]

-82-

Benchmarking Evolutionary Algorithms

CEC05 comparison results (Single obj. + bound const.)

CEC06 comparison results (Single obj + general const.)

Page 42: Tutorial rpo

-83-

CEC’05 Comparison ResultsAlgorithms involved in the comparison:

BLX-GL50 (Garcia-Martinez & Lozano, 2005 ): Hybrid Real-Coded Genetic Algorithms with Female and Male DifferentiationBLX-MA (Molina et al., 2005): Adaptive Local Search Parameters for Real-Coded Memetic AlgorithmsCoEVO (Posik, 2005): Mutation Step Co-evolutionDE (Ronkkonen et al.,2005):Differential EvolutionDMS-L-PSO: Dynamic Multi-Swarm Particle Swarm Optimizer with Local SearchEDA (Yuan & Gallagher, 2005): Estimation of Distribution AlgorithmG-CMA-ES (Auger & Hansen, 2005): A restart Covariance Matrix Adaptation Evolution Strategy with increasing population sizeK-PCX (Sinha et al., 2005): A Population-based, Steady-State real-parameter optimization algorithm with parent-centric recombination operator, a polynomial mutation operator and a niched -selection operation.L-CMA-ES (Auger & Hansen, 2005): A restart local search Covariance Matrix Adaptation Evolution StrategyL-SaDE (Qin & Suganthan, 2005): Self-adaptive Differential Evolution algorithm with Local SearchSPC-PNX (Ballester et al.,2005): A steady-state real-parameter GA with PNX crossover operator

-84-

CEC’05 Comparison Results

Problems: 25 minimization problems (Suganthan et al. 2005)Dimensions: D=10, 30Runs / problem: 25 Max_FES: 10000*D (Max_FES_10D= 100000; for 30D=300000; for 50D=500000)Initialization: Uniform random initialization within the search space, except for problems 7 and 25, for which initialization ranges are specified. The same initializations are used for the comparison pairs (problems 1, 2, 3 & 4, problems 9 & 10, problems 15, 16 & 17, problems 18, 19 & 20, problems 21, 22 & 23, problems 24 & 25). Global Optimum: All problems, except 7 and 25, have the global optimum within the given bounds and there is no need to perform search outside of the given bounds for these problems. 7 & 25 are exceptions without a search range and with the global optimum outside of the specified initialization ranges.

Page 43: Tutorial rpo

-85-

CEC’05 Comparison Results

Termination: Terminate before reaching Max_FES if the error in the function value is 10-8 or less.

Ter_Err: 10-8 (termination error value)

Successful Run: A run during which the algorithm achieves the fixed accuracy level within the Max_FES for the particular dimension.

Success Rate= (# of successful runs) / total runs

Success Performance = mean (FEs for successful runs)*(# of total runs) / (# of successful runs)

-86-

CEC’05 Comparison Results

Success Rates of the 11 algorithms for 10-D

*In the comparison, only the problems in which at least one algorithm

succeeded once are considered.

Page 44: Tutorial rpo

-87-

CEC’05 Comparison Results

Empirical distribution over all successful functions for 10-D (SP here means the Success Performance for each problem. SP=mean (FEs for successful runs)*(# of total runs) / (# of successful runs). SPbest is the minimal FES of all algorithms for each problem.)

First three algorithms:

1. G-CMA-ES

2. DE

3. DMS-L-PSO

-88-

CEC’05 Comparison Results

Success Rates of the 11 algorithms for 30-D

*In the comparison, only the problems which at least one algorithm

succeeded once are considered.

Page 45: Tutorial rpo

-89-

CEC’05 Comparison Results

Empirical distribution over all successful functions for 30-D (SP here means the Success Performance for each problem. SP=mean (FEs for successful runs)*(# of total runs) / (# of successful runs). SPbest is the minimal FES of all algorithms for each problem.)

First three algorithms:

1. G-CMA-ES

2. DMS-L-PSO

3. L-CMA-ES

-90-

CEC’06 Comparison Results

Involved AlgorithmsDE (Zielinski & Laur, 2006): Differential Evolution

DMS-C-PSO (Liang & Suganthan, 2006): Dynamic Multi-Swarm Particle Swarm Optimizer with the New Constraint-Handling Mechanism

ε- DE [TS06] Constrained Differential Evolution with Gradient-Based Mutation and Feasible Elites

GDE (Kukkonen & Lampinen, 2006) : Generalized Differential Evolution

jDE-2 (Brest & Zumer, 2006 ): Self-adaptive Differential Evolution

MDE (Mezura-Montes, et al. 2006): Modified Differential Evolution

MPDE (Tasgetiren & Suganthan,2006): Multi-Populated DE Algorithm

PCX (Ankur Sinha, et al, 2006): A Population-Based, Parent Centric Procedure

PESO+ (Munoz-Žavala et al, 2006): Particle Evolutionary Swarm Optimization Plus

SaDE (Huang et al, 2006 ): Self-adaptive Differential Evolution Algorithm

Page 46: Tutorial rpo

-91-

CEC’06 Comparison Results

Problems: 24 minimization problems with constraints (Liang, 2006b)Runs / problem: 25 (total runs)Max_FES: 500,000 Feasible Rate = (# of feasible runs) / total runsSuccess Rate = (# of successful runs) / total runsSuccess Performance = mean (FEs for successful runs)*(# of total runs) / (# of successful runs)

The above three quantities are computed for each problem separately. Feasible Run: A run during which at least one feasible solution is found in Max_FES.Successful Run: A run during which the algorithm finds a feasible solution x satisfying ( ) ( *) 0.0001f f− ≤x x

-92-

CEC’06 Comparison Results

ωNP, LP, LS_gapSaDE

, c1, c2 , n, not sensitive to , c1, c2PESO+

N , λ, r (a different N is used for g02), PCX

F, CR, np1, np2MPDE

µ, CR, Max_Gen, λ, Fα, Fβ,MDE

NP, F, CR, k, ljDE-2

NP, F, CRGDE

N, F, CR, Tc, Tmax, cp, Pg, Rg, Neε_DE

, c1, c2 , Vmax, n, ns, R, L, L_FESDMS-PSO

NP, F, CRDE

ω

ω

Algorithms’ Parameters

Page 47: Tutorial rpo

-93-

CEC’06 Comparison Results

Success Rate for Problems 1-11

-94-

CEC’06 Comparison Results

Success Rate for Problems 12-19,21,23,24

Page 48: Tutorial rpo

-95-

CEC’06 Comparison Results

Empirical distribution over all functions( SP here means the Success Performancefor each problem. SP=mean (FEs for successful runs)*(# of total runs) / (# of successful runs). SPbest is the minimal FES of all algorithms for each problem.)

GDE, PESO+9th

jDE-28th

DE7th

MPDE6th

SaDE5th

MDE, PCX3rd

DMS-PSO2nd

ε_DE1st

-96-

Acknowledgement

Thanks to the SEAL’06 organizers or giving us this opportunity

Our research into real parameter evolutionary algorithms is financially supported by an A*Star (Agency for Science, Technology and Research), Singapore

Thanks to Jane Liang Jing for her contributions in PSO, and CEC05/06 benchmarking studies.

Some slides on G3-PCX were from Prof Kalyanmoy Deb

Page 49: Tutorial rpo

References [A02] H. A. Abbass, "Self-adaptive pareto differential evolution," Proc. of 2000 IEEE Congress on Evolutionary

Computation, pp. 831-836, Honolulu, USA, 2002. Agrafiotis, D. K.and Cedeno, W. (2002), "Feature selection for structure-activity correlation using binary particle

swarms,” Journal of Medicinal Chemistry, 2002, 45, 1098-1107. [ARG04] C. W. Ahn, R. S. Ramakrishna, and D. E. Goldberg, “Real-coded Bayesian optimization algorithm:

bringing the strength of BOA into the continuous world,” LNCS (GECCO'04), 3102: 840-851, 2004. Angeline, P. J. (1998) "Using selection to improve particle swarm optimization," Proc. of the IEEE Congress on

Evolutionary Computation (CEC 1998), Anchorage, Alaska, USA. Auger, A. and Hansen, N. (2005), "A restart CMA evolution strategy with increasing population size," the 2005

IEEE Congress on Evolutionary Computation, Vol.2, pp.1769 - 1776, 2005 Auger, A. and Hansen, N. (2005), "Performance evaluation of an advanced local search evolutionary algorithm,"

2005 IEEE Congress on Evolutionary Computation, Vol.2, pp.1777 - 1784, 2005 [BO04] B. V. Babu, G. Onwubolu (eds.), New Optimization Techniques in Engineering, Springer, Berlin, 2004. [BFM-02] T. Back, D. B. Fogel and Z. Michalewicz, Handbook of Evolutionary Computation, 1st edition, IOP

Publishing Ltd., UK, 2002. Ballester, P. J., Stephenson, J., Carter, J. N. and Gallagher, K. (2005), "Real-Parameter Optimization Performance

Study on the CEC-2005 benchmark with SPC-PNX," 2005 IEEE Congress on Evolutionary Computation, Vol.1, pp.498-505, 2005

S. Baskar, R. T. Zheng, A. Alphones, N. Q. Ngo, P. N. Suganthan (2005a), “Particle Swarm Optimization for the Design of Low-Dispersion Fiber Bragg Gratings,” IEEE Photonics Technology Letters, 17(3):615-617, 2005.

S. Baskar, A. Alphones, P. N. Suganthan, J. J. Liang (2005b), “Design of Yagi-Uda Antennas Using Particle Swarm Optimization with new learning strategy”, IEE Proc. on Antenna and Propagation 152(5):340-346 OCT. 2005

Bergh van den F. and Engelbrecht, A. P. (2004) "A cooperative approach to particle swarm optimization," IEEE Transactions on Evolutionary Computation; 8 (3):225~239

[BD96] H. Bersini, M. Dorigo, L. Gambarella, S. Langerman and G. Seront, "Results of the first international contest on evolutionary optimization," P. IEEE Int. Conf. on Evolutionary Computation, Nagoya, Japan, 1996.

Blackwell, T. M. and Bentley, P. J. (2002 ) "Don't push me! Collision-avoiding swarms," Proc. of the IEEE Congress on Evolutionary Computation (CEC 2002), Honolulu, Hawaii USA.

Brest, J., Zumer, V. and Maucec, M. S. (2006) "Self-Adaptive Differential Evolution Algorithm in Constrained Real-Parameter Optimization," in Proc. of the sixth IEEE Conference on Evolutionary Computation, 2006.

[BGBMZ06] J. Brest, S. Greiner, B. Boškovic, M. Mernik, and V. Zumer, "Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems," IEEE Transactions on Evolutionary Computation, 2006

[BT00] P. A. N. Bosman and D. Thierens, “Continuous iterated density estimation evolutionary algorithms within the IDEA framework,” in M. Pelikan and A. O. Rodriguez (eds.), P. of the Optimization by Building and Using Probabilistic Models OBUPM Workshop at GECCO’00, pp.197-200, San Francisco, CA, Morgan Kauffmann.

Clerc, M. (1999), "The swarm and the queen: towards a deterministic and adaptive particle swarm optimization," Proc of the IEEE Congress on Evolutionary Computation (CEC 1999), pp. 1951-1957, 1999.

Clerc M. and Kennedy, J. (2002), The particle swarm-explosion, stability, and convergence in a multidimensional complex space IEEE Transactions on Evolutionary Computation, vol. 6, pp. 58-73, 2002.

Coello Coello A. C., Constraint-handling using an evolutionary multiobjective optimization technique. Civil Engineering and Environmental Systems, 17:319-346, 2000.

Coello Coello A. C. and Lechuga, M. S. 2002 "MOPSO: a proposal for multiple objective particle swarm optimization," Proc. of the IEEE Congress on Evolutionary Computation (CEC 2002), Honolulu, Hawaii USA.

[CDG99] D. Corne, M. Dorigo, and F. Glover (eds.), New Ideas in Optimization, McGraw-Hill, 1999. [DK05] Das, S., and Konar, A.,(2005) "An improved differential evolution scheme for noisy optimization

problems."Lecture Notes Comput SC 3776: 417-421 2005. Deb, K.,Pratap, A., Agarwal, S. and Meyarivan (2002), T. "A fast and elitist multi-objective genetic algorithms:

NSGA-II," IEEE Trans. on Evolutionary Computation. 6(2): 182-197, 2002 [Deb-01] K. Deb. Multi-Objective Optimization using Evolutionary Algorithms, Wiley, Chichester, 2001

(http://www.iitk.ac.in/kangal/papers). Eberhart R. C. and Shi, Y. 2001 "Tracking and optimizing dynamic systems with particle swarms," Proceedings of

the IEEE Congress on Evolutionary Computation (CEC 2001), Seoul, Korea. pp. 94-97. Eberhart, R. C. and Shi, Y. (2001), "Particle swarm optimization: developments, applications and resources,"

Proc. of the IEEE Congress on Evolutionary Computation (CEC 2001), Seoul, Korea. Eberhart, R.C and Kennedy, J. (1995), "A new optimizer using particle swarm theory," Proc. of the Sixth

International Symposium on Micromachine and Human Science, Nagoya, Japan, pp. 39-43.

Page 50: Tutorial rpo

Eiben, A. E., Hauw van der J. K. and Hermert van J. I. (1998), "Adaptive penalties for evolutionary gragh coloring," in Artificial Evolution' 97, pp. 95-106, Berlin,.

Coello Coello A. Carlos (1999), "Self-Adaptation Penalties for GA-based Optimization," in Proceedings of the 1999 Congress on Evolutionary Computation, Washington, D.C., Vol. 1, pp. 573-580.

Fan, H. Y. and Shi, Y. (2001), "Study on Vmax of particle swarm optimization," Proc. of the Workshop on Particle Swarm Optimization 2001, Indianapolis, IN.

[FJ06] V. Feoktistov and S. Janaqi, “New energetic selection principle in differential evolution” ENTERPRISE INFORMATION SYSTEMS VI :151-157, 2006

[GMK02] R. Gaemperle, S. D. Mueller and P. Koumoutsakos, “A Parameter Study for Differential Evolution”, A. Grmela, N. E. Mastorakis, editors, Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation, WSEAS Press, pp. 293-298, 2002.

Garcia-Martinez C. and Lozano M. (2005), "Hybrid real-coded genetic algorithms with female and male differentiation," the 2005 IEEE Congress on Evolutionary Computation, Vol.1, pp. 896 - 903, 2005.

N. Hansen (2006), ‘Covariance Matrix Adaptation (CMA) Evolution Strategy,’ Tutorial at PPSN-06, Reykjavik, Iceland, http://www.bionik.tu-berlin.de/user/niko/cmaesintro.html

[HO-04] N. Hansen, A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies”, Evolutionary Computation, 9(2), pp. 159-195, 2004.

Homaifar, A., Lai, S. H. Y. and Qi X., (1994)"Constrained Optimization via Genetic Algorithms," Simulation, 62(4):242-254.

Hu X. and Eberhart, R. C. (2002) "Multiobjective optimization using dynamic neighborhood particle swarm optimization," Proc. IEEE Congress on Evolutionary Computation (CEC 2002), Hawaii, pp. 1677-1681, 2002.

Hu X. and Eberhart, R. C. (2002)" Solving constrained nonlinear optimization problems with particle swarm optimization". Proc. of World Multiconf. on Systemics, Cybernetics and Informatics (SCI 2002), Orlando, USA.

[HQS06] Huang, V. L., Qin, A. K. and Suganthan, P. N. (2006) "Self-adaptive Differential Evolution Algorithm for Constrained Real-Parameter Optimiation," Proc. of IEEE Congress on Evolutionary Computation, 2006

V. L. Huang, P. N. Suganthan and J. J. Liang, “Comprehensive Learning Particle Swarm Optimizer for Solving Multiobjective Optimization Problems”, Int. J of Intelligent Systems, Vol. 21, No. 2, pp. 209-226, Feb. 2006.

[IKL03] J. Ilonen, J.-K. Kamarainen and J. Lampinen, “Differential Evolution Training Algorithm for Feed-Forward Neural Networks,” In: Neural Processing Letters vol. 7, no. 1, pp. 93-105. 2003.

[IL04] A. Iorio and X. Li. “Solving Rotated Multi-objective Optimization Problems Using Differential Evolution,” Australian Conference on Artificial Intelligence 2004: 861-872

Joines J. and Houck, C. (1994) "On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GAs," P. IEEE Conf. on Evolutionary Computation, pp.579-584, Orlando, Florida.

Kennedy, J. and Mendes, R. (2002), "Population structure and particle swarm performance," Proc. of the IEEE Congress on Evolutionary Computation (CEC 2002), Honolulu, Hawaii USA.

Kennedy, J. (1999) "Small worlds and mega-minds: effects of neighborhood topology on particle swarm performance," Proc. IEEE Congress on Evolutionary Computation (CEC 1999), pp. 1931-1938.

Kennedy, J. and Eberhart, R. C. 1997 "A discrete binary version of the particle swarm algorithm," Proc. of the World Multiconference on Systemics,Cybernetics and Informatics, Piscataway, NJ, pp. 4104-4109, 1997.

Kennedy, J. and Eberhart, R. C. (1995), "Particle swarm optimization," Proc. of IEEE International Conference on Neural Networks, Piscataway, NJ, pp. 1942-1948.

Kukkonen S. and Lampinen, J. (2006) "Constrained Real-Parameter Optimization with Generalized Differential Evolution," in Proceedings of the sixth IEEE Conference on Evolutionary Computation, 2006.

[LELP00] P. Larrañaga, R. Etxeberria, J. A. Lozano, and J. M. Peña, “Optimization in continuous domains by learning and simulation of Gaussian networks,” in Proc. of the Workshop in Optimization by Building and using Probabilistic Models. A Workshop within the GECCO 2000, pp. 201-204, Las Vegas, Nevada, USA.

[LLB01] P. Larrañaga, J. A. Lozano and E. Bengoetxea, “Estimation of distribution algorithms based on multivariate normal and Gaussian networks,” Technical Report KZZA-IK-1-01, Department of Computer Science and Artificial Intelligence, University of the Basque Country, 2001.

[LL01] P. Larrañaga and J. A. Lozano. Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation, Kluwer Academic Publishers, 2001.

Laskari, E. C., Parsopoulos, K. E. and Vrahatis, M. N. 2002, "Particle swarm optimization for minimax problems," Proc. of the IEEE Congress on Evolutionary Computation (CEC 2002), Honolulu, Hawaii USA.

Chang-Yong Lee, Xin Yao, “Evolutionary programming using mutations based on the Levy probability distribution’, IEEE Transactions on Evolutionary Computation, Volume 8, Issue 1, Feb. 2004 Page(s):1 - 13

[L05] Li, X. (2005), "Efficient Differential Evolution using Speciation for Multimodal Function Optimization", Proc. of GECCO'05, eds. Beyer, Hans-Georg et al., Washington DC, USA, 25-29 June, p.873-880.

Page 51: Tutorial rpo

Liang, J. J., Suganthan, P. N., Qin, A. K. and Baskar, S (2006a). "Comprehensive Learning Particle Swarm Optimizer for Global Optimization of Multimodal Functions," IEEE Transactions on Evolutionary Computation, June 2006.

Liang, J. J. Runarsson Thomas Philip, Mezura-Montes Efren, Clerc, Maurice, Suganthan, P. N., Coello Coello A. Carlos and Deb, K. (2006b) "Problem Definitions and Evaluation Criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization," TR, Nanyang Technological University, Singapore, March 2006.

J. J. Liang, S. Baskar, P. N. Suganthan and A. K. Qin (2006c), “Performance Evaluation of Multiagent Genetic Algorithm”, Volume 5, Number 1, pp. 83 – 96, Natural Computing, March 2006.

J. J. Liang, P. N. Suganthan, C. C. Chan, V. L. Huang (2006d), “Wavelength detection in FBG sensor network using tree search dynamic multi-swarm particle swarm optimizer”, IEEE Photonics Tech. Lets, 18(12):1305-1307.

J. J. Liang & P. N. Suganthan (2006e), "Dynamic Multi-Swarm Particle Swarm Optimizer with a Novel Constrain-Handling Mechanism," in Proc. of IEEE Congress on Evolutionary Computation, pp. 9-16, 2006.

J. J. Liang, P. N. Suganthan and K. Deb, "Novel composition test functions for numerical global optimization," Proc. of IEEE International Swarm Intelligence Symposium, pp. 68-75, 2005.

[LL02] J. Liu and J. Lampinen. “Adaptive Parameter Control of Differential Evolution,” In Proceedings of the 8th International Conference on Soft Computing (MENDEL 2002), pages 19–26, 2002.

Lovbjerg, M., Rasmussen, T. K. and Krink, T. (2001) "Hybrid particle swarm optimiser with breeding and subpopulations," Proc. of the Genetic and Evolutionary Computation Conference 2001 (GECCO 2001).

Lovbjerg M. and Krink, T. (2002)"Extending particle swarm optimisers with self-organized criticality," Proc. of the IEEE Congress on Evolutionary Computation (CEC 2002), Honolulu, Hawaii USA.

[LLIB06] J. A. Lozano, P. Larrañaga, I. Inza, and E. Bengoetxea (eds.), Towards a New Evolutionary Computation: Advances on Estimation of Distribution Algorithms Series: Studies in Fuzzy & Soft Computing, Springer, 2006.

Mezura-Montes, E. , Velazquez-Reyes J. and Coello Coello A. Carlos (2006) "Modified Differential Evolution for Constrained Optimization," in Proceedings of the sixth IEEE Conference on Evolutionary Computation, 2006.

E. Mezura-Montes and C. A. C. Coello, "A Numerical Comparison of some Multiobjective-Based Techniques to Handle Constraints in Genetic Algorithms", Technical Report EVOCINV-0302002, Evolutionary Computation Group at CINVESTAV, México, 2002.

Michalewicz, Z. and Attia, N. (1994), "Evolutionary Optimization of Constrained Problems," in Proceedings of the third Annual Conference on Evolutionary Programming, pp. 98-108.

Michalewicz, Z.(1992), "Genetic Algorithms + Data Structures = Evolution Programs," Springer-Verlag. Molina, D., Herrera, F. and Lozano, M. (2005)"Adaptive Local Search Parameters for Real-Coded Memetic

Algorithms," the 2005 IEEE Congress on Evolutionary Computation, Vol.1, pp.888 - 895, 2005 [MP96] H. Mühlenbein and G. Paaß, “From recombination of genes to the estimation of distributions I. Binary

parameters,” in Proc. of Parallel Problem Solving from Nature - PPSN IV. LNCS 1411, pp. 178-187, 1996. Munoz-Zavala A. E., Hernandez-Aguirre A., Villa-Diharce E. R. and Botello-Rionda, S. (2006) "PESO+ for

Constrained optimization," in Proceedings of the sixth IEEE Conference on Evolutionary Computation, 2006. [OSE05] G. H. M. Omran, A. Salman, and A. P. Engelbrecht, "Self-adaptive differential evolution," International

Conference on Computational Intelligence and Security (CIS'05), Xi'an, China, December 2005. Parsopoulos, K. E. and Vrahatis, M. N. (2004), "UPSO- A Unified Particle Swarm Optimization Scheme," Lecture

Series on Computational Sciences, pp. 868-873. Parsopoulos, K. E. and Vrahatis, M. N. 2002 "Particle swarm optimization method for constrained optimization

problems," Proc. of the Euro-International Symposium on Computational Intelligence. Peram, T. , Veeramachaneni, K. , and Mohan, C. K. , (2003 )"Fitness-distance-ratio based particle swarm

optimization," Proc. IEEE Swarm Intelligence System, Indianapolis, Indiana, USA., pp. 174-181. Posik, P. (2005), "Real-Parameter Optimization Using the Mutation Step Co-evolution," the 2005 IEEE Congress on

Evolutionary Computation, Vol.1, pp.872 - 879, 2005 [PSL05] K. V. Price, R. Storn, J. Lampinen, Differential Evolution - A Practical Approach to Global Optimization,

Springer, Berlin, 2005. [P99] K. V. Price, “An Introduction to Differential Evolution”. In: Corne, D., Dorigo, M.,and Glover, F. (eds.): New

Ideas in Optimization. McGraw-Hill, London (UK):79-108,1999. [P97] K. V. Price, "Differential evolution vs. the functions of the 2nd ICEO," in Proc. of 1997 IEEE International

Conference on Evolutionary Computation (ICEC’97), pp. 153-157, Indianapolis, USA, 1997. [P94] K. V. Price, "Genetic annealing," Dr. Dobb's Journal, Miller Freeman, San Mateo, Ca., pp. 127-132, 1994. [QS05] Qin, A. K. and Suganthan, P. N. (2005), "Self-adaptive differential evolution algorithm for numerical

optimization," the 2005 IEEE Congress on Evolutionary Computation, Vol.2, pp.1785-1791, 2005 Ratnaweera, A., Halgamuge, S. and Watson, H. (2002) ,"Particle swarm optimization with self-adaptive acceleration

coefficients," Proc. of Int. Conf. on Fuzzy Systems and Knowledge Discovery 2002 (FSKD 2002), pp. 264-268,

Page 52: Tutorial rpo

Ratnaweera, A. ,Halgamuge, S. and Watson, H. (2004), "Self-organizing hierachical particle swarm optimizer with time varying accelerating coefficients," IEEE Transactions on Evolutionary Computation, 8(3): 240-255

Ray T. and Liew, K. M. 2002, A swarm metaphor for multiobjective design optimization Engineering Optimization, vol. 34, pp. 141-153, 2002.

Ray, T. (2002) "Constraint Robust Optimal Design using a Multiobjective Evolutionary Algorithm," in Proceedings of Congress on Evolutionary Computation 2002 (CEC'2002), vol.1, pp. 419-424,

[RKP05] Ronkkonen, J. , Kukkonen, S. and Price, K.V. , (2005), "Real-Parameter Optimization with Differential Evolution," the 2005 IEEE Congress on Evolutionary Computation, Vol.1, pp.506 - 513, 2005

[RK96] S. Rudlof and M. Köppen, “Stochastic hill climbing by vectors of normal distributions. in Proc. of the First Online Workshop on Soft Computing (WSC1), Nagoya, Japan, 1996.

Runarsson, T.P., Xin Yao, “Stochastic ranking for constrained evolutionary optimization”, IEEE Transactions on Evolutionary Computation, Vol. 4, Issue 3, Sept. 2000 Page(s):284 - 294

[SD98] M. Sebag and A. Ducoulombier, “Extending population-based incremental learning to continuous search spaces,” Proc. Conference on Parallel Problem Solving from Nature - PPSN V, pp. 418-427, Berlin, 1998.

Shi, Y. and Eberhart, R. C. , (1998), "A modified particle swarm optimizer," Proc. of the IEEE Congress on Evolutionary Computation (CEC 1998), Piscataway, NJ, pp. 69-73.

Shi, Y. and Eberhart, R. C., (2001) "Particle swarm optimization with fuzzy adaptive inerita weight," Proc. of the Workshop on Particle Swarm Optimization, Indianapolis, IN.

Shi, Y. and Krohling, R. A. 2002 "Co-evolutionary particle swarm optimization to solve Min-max problems," Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2002), Honolulu, Hawaii USA.

Sinha, A., Srinivasan A. and Deb, K. (2006) "A Population-Based, Parent Centric Procedure for Constrained Real-Parameter Optimization," in Proceedings of the sixth IEEE Conference on Evolutionary Computation, 2006.

Sinha, S. Tiwari and Deb, K. (2005), "A Population-Based, Steady-State Procedure for Real-Parameter Optimization," the 2005 IEEE Congress on Evolutionary Computation, Vol.1, pp.514 - 521, 2005

[SP95] R. Storn and K. V. Price, “Differential Evolution - a simple and efficient adaptive scheme for global optimization over continuous spaces,” TR-95-012, ICSI, http://http.icsi.berkeley.edu/~storn/litera.html

[SP97] R. Storn and K. V. Price, “Differential evolution-A simple and Efficient Heuristic for Global Optimization over Continuous Spaces,” Journal of Global Optimization, 11:341-359,1997.

Suganthan P. N., Hansen, N. Liang, J. J. Deb, K. Chen, Y.-P. Auger, A. and Tiwari, S. (2005), "Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on Real-Parameter Optimization," Technical Report, Nanyang Technological University, Singapore, May 2005 and KanGAL Report #2005005, IIT Kanpur, India.

Suganthan, P. N.,(1999)"Particle swarm optimizer with neighborhood operator," Proc. of the IEEE Congress on Evolutionary Computation (CEC 1999), Piscataway, NJ, pp. 1958-1962.

[TS06] Takahama T. and Sakai, S. (2006) "Constrained Optimization by the Constrained Differential Evolution with Gradient-Based Mutation and Feasible Elites," Proc. IEEE Congress on Evolutionary Computation, 2006.

Takahama T. and S. Sakai, (2005) "Constrained optimization by constrained particle swarm optimizer with -level control," Proc. IEEE Int. Workshop on Soft Computing as Transdisciplinary Science and Technology (WSTST’05), Muroran, Japan, May 2005, pp. 1019–1029.

Tasgetiren, M. Fatih and Suganthan, P. N. (2006) "A Multi-Populated Differential Evolution Algorithm for Solving Constrained Optimization Problem," Proc. IEEE Congress on Evolutionary Computation, 2006.

[TPV06] Tasoulis, DK., Plagianakos, VP., and Vrahatis, MN.(2006),“Differential evolution algorithms for finding predictive gene subsets in microarray data” Int Fed Info Proc 204: 484-491 2006.

[T06] J. Teo, "Exploring dynamic self-adaptive populations in differential evolution," Soft Computing - A Fusion of Foundations, Methodologies and Applications, 10(8): 673-686, 2006.

Tessema, B. Yen, G.G., “A Self Adaptive Penalty Function Based Algorithm for Constrained Optimization”, IEEE Congress on Evolutionary Computation, CEC 2006 16-21 July 2006 Page(s):246 – 253.

Venkatraman, S. Yen, G.G., “A Generic Framework for Constrained Optimization Using Genetic Algorithms”, IEEE Transactions on Evolutionary Computation, Volume 9, Issue 4, Aug. 2005 Page(s):424 – 435.

X. Yao, Y. Liu, G. M. Lin, “Evolutionary programming made faster”, IEEE Transactions on Evolutionary Computation, Volume 3, Issue 2, July 1999 Page(s):82 - 102

Yuan, B. and Gallagher, M. (2005), "Experimental results for the special session on real-parameter optimization at CEC 2005: a simple, continuous EDA," IEEE Congress on Evolutionary Computation, pp.1792 - 1799, 2005

[Z02] D. Zaharie, “Critical values for the control parameters of differential evolution algorithms,” in in Proc. of the 8th Int.l Conference on Soft Computing (MENDEL 2002), pp. 62-67, Brno,Czech Republic, June 2002.

[ZX03] W. J. Zhang and X. F. Xie. “DEPSO: hybrid particle swarm with differential evolution operator.” IEEE International Conference on Systems, Man & Cybernetics (SMCC), Washington D C, USA, 2003: 3816-3821.

Zielinski K. and Laur, R., (2006) "Constrained Singel-Objective Optimization Using Differential Evolution", in Proceedings of the sixth IEEE Conference on Evolutionary Computation, 2006.