Recent Advances in Real-Parameter Evolutionary Algorithms

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

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.

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Benchmark Test Functions

Resources available from
http://www.ntu.edu.sg/home/epnsugan
(limited to our own work)

From Prof Xin Yao’s group
http://www.cs.bham.ac.uk/research/projects/ecb/
Includes diverse problems.

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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 problems
Constrained problems
Single / Multi-objective problems
Static / Dynamic optimization problems
Multimodal problems (niching, crowding, etc.)

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Shortcomings in Bound constrained Benchmarks
Some properties of benchmark functions may make them unrealistic
or may be exploited by some algorithms:
Global optimum having the same parameter values for different
variablesdimensions
Global optimum at the origin
Global 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 linkages
over the whole search range
Repetitive landscape structure over the entire space

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

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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:
F ( x) = f ( R * x)
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

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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.

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is.

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A couple of Examples
Many composition functions are available from our homepage

Composition Function 1 (F1): Composition Function 2 (F2):
Made of Sphere Functions Made of Griewank’s Functions

Similar analysis is needed for other benchmarks too
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Methods Modeled After
Conventional GAs
Some resources available from
http://www.iitk.ac.in/kangal/

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Major Steps in Standard GAs

Population initialization
Fitness evaluation
Selection
Crossover (or recombination)
Mutation
Replacement

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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)

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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.
Parent-Centric Recombination
SBX (Simulated Binary crossover) : variable-wise
Vector-wise parent-centric recombination (PCX)

And many more …

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Minimal Generation Gap (MGG) Model
1. Select μ parents randomly
2. Generate λ offspring using a recombination scheme
3. Choose two parents at random from the population
4. One is replaced with the best of λ offspring and other with a
solution by roulette-wheel selection from a combination of λ
offspring 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).

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Generalized Generation Gap (G3) Model
1. Select the best parent and μ-1 other parents randomly
2. Generate λ offspring using a recombination scheme
3. Choose two parents at random from the population
4. Form a combination of two parents and λ offspring, choose best
two solutions and replace the chosen two parents
Parameters to tune: λ, μ and N

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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, σ )
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.
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Covariance Matrix Adapted Evolution
Strategy CMAES

By Nikolaus Hansen & collaborators

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

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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 M
fittest 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.

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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.

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CMAES Iterative Equations [Hansen 2006]

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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.

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

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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 particle
All particles have fitness values and velocities

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Two versions of PSO
Global 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)
Vi d ← c1 * rand1i d ∗ ( pbesti d − X i d ) + c2 * rand 2i d ∗ ( gbest d − X i d )
X i d ← X i d + Vi d i – particle counter & d – dimension counter
Local Version: Learning from the pbest and the best position
achieved in the particle's neighborhood population (lbest)
Vi d ← c1 * rand1i d ∗ ( pbesti d − X i d ) + c2 * rand 2i d ∗ (lbestk d − X i d )
X i d ← X i d + Vi d

The random numbers (rand1 & rand2) should be
generated for each dimension of each particle in every
iteration.
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where c1 and c2 in the equation are the acceleration constants
rand 1i d and rand 2i d are two random numbers in the range [0,1];
Xi = ( X i1 , X i 2 ,..., X i D ) represents the position of the ith particle;
Vi = (Vi1 ,Vi 2 ,...,Vi D ) represents the rate of the position change
(velocity) for particle i.
pbest i = ( pbesti1 , pbesti 2 ,..., pbesti D ) represents the best
previous position (the position giving the best fitness value) of
the ith particle;
gbest = ( gbest1 , gbest 2 ,..., gbest D ) represents the best previous
position of the population;
lbest k = (lbestk 1 , lbestk 2 ,..., lbestk D ) represents the best previous
position achieved by the neighborhood of the particle;

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PSO variants
Modifying 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 …

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PSO variants and Applications
Hybrid 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)

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Comprehensive Learning PSO
Comprehensive Learning Strategy :

Vi d ← w ∗ Vi d + c * rand i d ∗ ( pbest d ( d ) − X i d )
fi

where f i = [ fi (1), fi (2),..., fi ( D)] defines which particle’s pbest particle
d
i should follow. pbest fi ( d ) 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.

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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.

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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.

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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.)

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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

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Regroup

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

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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:
If keep _ idi d = 0
Vi d ← 0.729 ∗ Vi d + 1.49445 ∗ rand1i d ∗ ( pbesti d − X i d )
+1.49445 ∗ rand 2i d ∗ (lbestk d − X i d )
Vi d = min(Vmax , max(−Vmax , Vi d ))
d d

X i d ← X i d + Vi d
Otherwise Somewhat similar to
X i ← pbesti
d d
DE & CPSO
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Adaptive Self-Learning Strategy
Assume 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.

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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.

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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 f (x), x = [ x1 , x2 ,..., xD ]
subjected to:
gi (x) ≤ 0, i = 1,..., q
h j (x) = 0, j = q + 1,..., m

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

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For convenience, the equality constraints can be transformed into
inequality form:
| h j ( x) | −ε ≤ 0

where ε is the allowed tolerance.

Then, the constrained problems can be expressed as
Minimize f (x), x = [ x1 , x2 ,..., xD ]

subjected to G j (x) ≤ 0, j = 1,..., m,
G1,..., q (x) = g1,...q (x), Gq +1,..., m (x) = hq +1,...m (x) − ε

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

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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 solutions
Start 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; …)
…

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Constraint-Handling (CH) Techniques
Separation of objective and constraints
Stochastic 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

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DMS-PSO for Constrained Optimization

Novel Constraint-Handling Mechanism
Suppose 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.

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DMS-PSO’s Constraint-Handling Mechanism
How to assign the objective and constraints to each sub-swarm?

Define
⎧1 if a>b
a>b=⎨
⎩0 if a≤b
ps

∑ ( g (x
j =1
i j ) > 0)
pi = , i = 1, 2,..., m
ps

Thus fp = 1 − p, p = [ p1 , p2 ,..., pm ]
m
fp + ∑ ( pi / m) = 1
i =1

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DMS-PSO’s Constraint-Handling Mechanism
For each sub-swarm,
Using roulette selection according to fp and pi / m 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.

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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
Gk ( x i ) < Gk ( x j ) with Gk ( x j ) > 0
or V ( x i ) < V ( x j ) & Gk ( x i ), Gk (x j ) ≤ 0
or f ( xi ) < f ( x j ) & V (x i ) == V ( x j )

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

V (xi ) < V (x j )
or f ( xi ) < f ( x j ) & V ( x i ) == V ( x j )
m
V (x) = ∑ ( weighti ⋅ Gi (x) ⋅ (Gi (x) ≥ 0))
i =1

1/ Gi max
weighti = m
, i = 1, 2,...m
∑ (1/ Gi max)
i =1

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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.

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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

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Multi-Objective Optimization
Mathematically, we can use the following formula to
express the multi-objective optimization problems (MOP):
Minimize y = f (x) = ( f1 (x), f 2 (x),K , f m (x)) x ∈ [ Xmin, Xmax]
subject to g j (x) ≤ 0, j = 1,..., z
hk (x) = 0, k = q + 1,..., m

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

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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 diversity
Crowding distance sorting
Subpopulation

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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.

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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)

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Motivation for DE
DE, 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.
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DE at a glance
Characteristics
Population-based stochastic direct search
Self-referential mutation
Simple but powerful
Reliable, robust and efficient
Easy parallelization
Floating-point encoding
Basic components
Initialization
Trial vector generation
Mutation
Recombination
Replacement
Feats
DE demonstrated promising performance in 1st and 2nd
ICEO [BD96][P97]

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Insight into classic DE (DE/rand/1/bin)
Initialization
A 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]
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 gth generation, a trial population Pu,g consisting of Np D-dimensional trial
vectors vi,g=[v1,i,g,…vD,i,g] is generated via mutation and recombination operations
=[v1,i,g
applied to the current population Px,g

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

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Insight into classic DE (DE/rand/1/bin)
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
uj,i,g=
xj,i,g otherwise
Replacement
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

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Illustration of classic DE
x2

x1

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x2
Target vector
xi,g

xr1,g
xr0,g

Base vector Two randomly
xr2,g
selected vectors

x1

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x2

xi,g

xr1,g
xr0,g

xr2,g

x1
Four operating vectors in 2D continuous space

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x2

xi,g

vi,g
xr1,g
xr0,g
F·(xr1,g-xr2,g)
xr2,g

x1
Mutation

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x2

xi,g

vi,g ui,g
xr1,g
xr0,g
F·(xr1,g-xr2,g)
xr2,g

x1

Crossover

-61-

x2

xi,g

xi,g+1
xr1,g
xr0,g

xr2,g

x1

Replacement

-62-

Differential vector distribution

A population of 5 vectors 20 generated difference vectors

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

-63-

DE variants
Modification of different components of DE can result in many DE
variants:
variants
Initialization
Uniform distribution and Gaussian distribution

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

-64-

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: Vi ,G = X r ,G + F ⋅ (X r ,G − X r ,G )
1 2 3

DE/best/1: Vi ,G = X best ,G + F ⋅ (X r ,G − X r ,G )
DE/current-to-best/1:Vi ,G = X i ,G + F ⋅ (X best ,G − X i ,G ) + F ⋅ (X r ,G − X r ,G )
1 2

Vi,G = Xr ,G + F ⋅ (Xr ,G − Xr ,G + Xr ,G − Xr ,G )
1 2

DE/rand/2:
Vi,G = Xbest,G + F ⋅ (Xr ,G − Xr ,G + Xr ,G − Xr ,G )
1 2 3 4 5

DE/best/2: 1 2 3 4

-65-

DE variants
Recombination
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)
x2 discrete
xb x1
x2
line
intermediate

xa discrete
x1
-66-

DE variants
Crossover rate CR∈[0,1]
Decomposable (small CR) and indecomposable functions (large CR)

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

Replacement
One-to-one replacement
Neighborhood replacement

-67-

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 effective
trial vectors during evolution

-68-

Related works
Practical 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 functions.
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.
-69-

DE with strategy and parameter self-adaptation (QS05, HQS06)
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
Steps:
Initialize selection probability pi=1/num_st, i=1,…,num_st for each strategy
According 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
-70-

ns1(1) … nsnum_st(1) ns1(2) … nsnum_st(2) ns1(3) … nsnum_st(3)
… … … …

ns1(L) … nsnum_st(L) ns1(L+1) … nsnum_st(L+1) ns1(L+2) … nsnum_st(L+2)

The selection probability pi is updated by: ∑ nsi (∑ nsi + ∑ nf i ). Go to 2nd step
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

-71-

Instantiations
In CEC’05, we use 2 strategies:
DE/rand/1/bin: (
Vi ,G = X r ,G + F ⋅ X r ,G − X r ,G )
DE/current-to-best/2/bin: Vi,G = Xi,G + F ⋅ (Xbest,G − Xi,G ) + F ⋅ (Xr ,G − Xr ,G + Xr ,G − Xr ,G )
1 2 3

1 2 3 4

LP = 50

In CEC’06, we employ 4 strategies:
DE/rand/1/bin: (
Vi,G = Xr ,G + F ⋅ Xr ,G − Xr G )
( )
1 2 3,

DE/rand/2/bin: Vi,G = Xr1 ,G + F ⋅ Xr2 ,G − Xr3,G + Xr4 ,G − Xr5,G
DE/current-to-best/2/bin: Vi,G = Xi,G + F ⋅ (Xbest,G − Xi,G ) + F ⋅ (Xr ,G − Xr ,G + Xr ,G − Xr ,G )
( )
1 2 3 4

DE/current-to-rand/1: ( 1
)
Vi,G = Xi,G + F ⋅ Xr ,G − Xi,G + F ⋅ Xr ,G − Xr ,G 2 3

LP = 50

-72-

Local search enhancement

To improve convergence speed, we apply a local search
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

-73-

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

ui,g will replace xi,g 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

-74-

SaDE for constrained optimization

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

∑im 1 wi Gi (x )
v(x ) = = m
∑i =1 wi

wi = 1
G maxi is a weighting factor

G max i is the maximum violation of the ith constraint obtained so far

-75-

Overview of DE research trends

Number of publications per year in ISI web of knowledge database

-76-

Overview of DE research trends
DE Applications
Digital Filter Design Optimal Design of Shell-and-Tube Heat
Exchangers
Multiprocessor synthesis
Optimization of an Alkylation's Reaction
Neural network learning
Optimization of Thermal Cracker Operation
Diffraction grating design
Optimization of Non-Linear Chemical
Crystallographic characterization Processes
Beam weight optimization in Optimum planning of cropping patterns
radiotherapy
Optimization of Water Pumping System
Heat transfer parameter estimation
in a trickle bed reactor Optimal Design of Gas Transmission Network
Electricity market simulation Differential Evolution for Multi-Objective
Optimization
Scenario-Integrated Optimization of
Dynamic Systems Bioinformatics

-77-

DE Resources
Books

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

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

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

Blog http://dealgorithm.blogspot.com/
-78-

Estimation of distribution algorithm (EDA)
New approach to evolutionary computation, which was proposed in
1996 [MP96] and described in details in [LL01, LLIB06]
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

-79-

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
(MIMICGc) [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]

-80-

rEDA resources
Books

2001 [LL01] 2006 [LLIB06]
Websites
http://medal.cs.umsl.edu/
http://www.iba.k.u-tokyo.ac.jp/english/EDA.htm
http://www.evolution.re.kr/

-81-

Benchmarking Evolutionary Algorithms

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

CEC06 comparison results (Single obj + general const.)

-82-

CEC’05 Comparison Results
Algorithms involved in the comparison:
BLX-GL50 (Garcia-Martinez & Lozano, 2005 ): Hybrid Real-Coded Genetic
Algorithms with Female and Male Differentiation
BLX-MA (Molina et al., 2005): Adaptive Local Search Parameters for Real-Coded
Memetic Algorithms
CoEVO (Posik, 2005): Mutation Step Co-evolution
DE (Ronkkonen et al.,2005):Differential Evolution
DMS-L-PSO: Dynamic Multi-Swarm Particle Swarm Optimizer with Local Search
EDA (Yuan & Gallagher, 2005): Estimation of Distribution Algorithm
G-CMA-ES (Auger & Hansen, 2005): A restart Covariance Matrix Adaptation
Evolution Strategy with increasing population size
K-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 Strategy
L-SaDE (Qin & Suganthan, 2005): Self-adaptive Differential Evolution algorithm
with Local Search
SPC-PNX (Ballester et al.,2005): A steady-state real-parameter GA with PNX
crossover operator

-83-

Problems: 25 minimization problems (Suganthan et al. 2005)
Dimensions: D=10, 30
Runs / 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.

-84-

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)

-85-

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.
-86-


First three algorithms:
1. G-CMA-ES
2. DE
3. DMS-L-PSO

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.)
-87-

Success Rates of the 11 algorithms for 30-D

*In the comparison, only the problems which at least one algorithm
succeeded once are considered. -88-


First three algorithms:
1. G-CMA-ES
2. DMS-L-PSO
3. L-CMA-ES

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.)
-89-

Involved Algorithms
DE (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

-90-

Problems: 24 minimization problems with constraints
(Liang, 2006b)
Runs / problem: 25 (total runs)
Max_FES: 500,000
Feasible Rate = (# of feasible runs) / total runs
Success Rate = (# of successful runs) / total runs
Success 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 f (x) − f (x*) ≤ 0.0001

-91-


Algorithms’ Parameters
DE NP, F, CR
DMS-PSO ω , c1, c2 , Vmax, n, ns, R, L, L_FES
ε_DE N, F, CR, Tc, Tmax, cp, Pg, Rg, Ne
GDE NP, F, CR
jDE-2 NP, F, CR, k, l
MDE µ, CR, Max_Gen, λ, Fα, Fβ,
MPDE F, CR, np1, np2
PCX N , λ, r (a different N is used for g02),
PESO+ ω , c1, c2 , n, not sensitive toω , c1, c2
SaDE NP, LP, LS_gap

-92-

Success Rate for Problems 1-11

-93-

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

-94-

1st ε_DE
2nd DMS-PSO
3rd MDE, PCX
5th SaDE
6th MPDE
7th DE
8th jDE-2
9th GDE, PESO+

Empirical distribution over all functions( 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.)
-95-

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

-96-

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