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A Multiobjective Memetic Algorithm

Based on Particle Swarm Optimization

Dr. Liu Dasheng

James Cook University, Singapore

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Outline of Talk

1. Particle Swam Optimization

2. Multiobjective Particle Swarm Optimization

3. A Multiobjective Memetic Algorithm Based on

Particle Swarm Optimization

4. Research Ideas

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Particle Swarm Optimization

▪ Particle swarm optimization (PSO) was

first introduced by James Kennedy (a

social psychologist) and Russell Eberhart

(an electrical engineer) in 1995, which

originates from the simulation of behavior

of bird flocks.

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Particle Swarm Optimization

▪ There are a number of algorithms to

simulate the movement of a bird flock or

fish school.

▪ Kennedy and Eberhart became particularly

interested in the models developed by

Heppner (a zoologist) [62].

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Heppner’s Model

▪ In Heppner’s model, birds would begin by

flying around with no particular destination

and in spontaneously formed flocks until

one of the birds flew over the roosting

area.

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Particle Swarm Optimization

▪ To Eberhart and Kennedy, finding a roost

is analogous to finding a good solution in

the field of possible solutions.

▪ They revised Heppner’s methodology so

that particles will fly over a solution space

and try to find the best solution depending

on their own discoveries and past

experiences of their neighbors.

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Working Principle of PSO

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Original Version

▪ In the original version of PSO, each

individual is treated as a volume-less

particle in the D dimensional solution

space.

▪ The equations for calculating velocity and

position of particles are shown below:

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Adjustable Step Size

▪ Further research shows that to adjust

velocity not by a fixed step size but

according to the distance between current

position and best position can improve

performance.

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Vmax

▪ One parameter Vmax is introduced, and the

particle’s velocity on each dimension

cannot exceed Vmax.

▪ If Vmax is too large, particle may fly past

good solutions.

▪ If Vmax is too small, particle may not

explore sufficiently beyond locally good

regions.

▪ Vmax is usually set at 10-20% of the

dynamic range of each dimension.

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Inertial Weight

▪ To better control the exploration and

exploitation in particle swarm optimization,

the concept of inertial weight (w) was

developed.

1

, , 1 1 , ,

2 2 , ,

( )

( )

k k k k k

i d i d i d i d

k k k

g d i d

v w v c r p x

c r p x

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1

, , 1 1 , ,

2 2 , ,

( )

( )

k k k k k

i d i d i d i d

k k k

g d i d

v w v c r p x

c r p x

1 1

, , ,

k k k

i d i d i dx x v

w is the inertia weight; c1 is the cognition weight and c2 is the

social weight; r1 and r2 are two random values uniformly

distributed in the range of [0, 1].

Each individual in PSO is assigned a random velocity and

flies across the solution space with a memory of its own

best position called pbest and a knowledge of the whole

swarm’s global best position called gbest.

Particle Swarm Optimization Formula

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Terminology

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Terminology

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▪ Real world problems usually involve simultaneous

optimization of several competing objectives.

▪ Solution exists in the form of alternative tradeoffs.

A minimization problem

✓ Non-inferior solutions are known as

nondominated solutions

✓ The set of nondominated solutions

form the Pareto solution set

f1

f2Unfeasible

Region

Trade-off Curve

Multiobjective Optimization

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▪ Multi-objective particle swarm optimization (MOPSO) is a

powerful tool for solving MO optimization problems.

✓ Capable of searching for the global

trade-off.

✓ Maintain a diverse set of solutions

✓ Robust and applicable to a wide

variety of problems.

f1

f2

▪ Conventional optimization search techniques

✓ Hardly handle multiple objectives

✓ The gradients need to be well-defined and differentiable

✓ May trap in “local optima”

MOPSO

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MOPSO

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Performance Assessments

▪ For MOO, performance metrics

must be able to measure

quality in terms of:

✓ Diversity.

✓ Proximity between the

generated and true Pareto

front. f1

f2

Minimization

Min

imiz

atio

n

Non-dominated solution

Pareto Frontier

Non-dominated set

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▪ Generational Distance (GD) (Veldhuizen, 1999)

✓ Represents how far the evolved solution set is from the

true Pareto front.

▪ Spacing (S) (Schott, 1995)

✓ Measures how “evenly” evolved solutions distribute itself.

▪ Maximum Spread (MS) (Zitzler, 1999)

✓ Measures how well the true Pareto front is covered by the

evolved solution set.

Performance Measures

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Problem Test Suite

Test

Problem

Features

1 ZDT1 Pareto front is convex.

2 ZDT2 Pareto front is non-convex.

3 ZDT3 Pareto front consists of several noncontiguous convex

parts.

4 ZDT4 Pareto front is highly multi-modal where there are 219

local Pareto fronts.

5 ZDT6 The Pareto optimal solutions are non-uniformly

distributed along the global Pareto front. The density of

the solutions is low near the Pareto front and high

away from the front.

ZDT1

ZDT2

ZDT3 ZDT4 ZDT6

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

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1

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 10

0.2

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1

0.2 0.4 0.6 0.8 10

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1

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Test

Problem

Features

6 FON Pareto front is non-convex.

7 KUR Pareto front consists of several noncontiguous convex

parts.

8 POL Pareto front and Pareto optimal solutions consist of

several noncontiguous convex parts.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

FON

-20 -19 -18 -17 -16 -15 -14-15

-10

-5

0

5

KUR

0 5 10 15 200

5

10

15

20

25

POL

Problem Test Suite

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Two modification introduced to

improve performance

▪ Fuzzy global best

▪ Synchronous particle local search

Memetic algorithm : evolution algorithm with

local improvement technique

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Two Modifications

▪ Fuzzy Global Best (f-gbest)

✓ A new particle updating strategy is proposed based upon the

concept of fuzzy global-best to deal with the problem of premature

convergence and diversity maintenance within the swarm.

▪ Synchronous Particle Local Search (SPLS)

✓ Hybridized with a directed local search operator for local fine tuning,

which helps to discover a well-distributed Pareto front.

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Fuzzy gbest

▪ Fuzzy Global Best (f-gbest) accounts for the uncertainty of

global best knowledge to prevent premature convergence

✓ Incorporates fuzzy number

to represent global best.

✓ F-gbest is characterized by

normal distribution.

✓ Degree of uncertainty

reduces with generations

synonymous with

information gain.

x2

x1

x3

Possible location

of gbest

Particle

Search Region

incorporating f-gbest Search trajectory using

conventional gbest

gbest

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The calculation of particle velocity can be rewritten as

, ,( , )

k k

c d g dp N p

( )f k

1

, , 1 1 , , 2 2 , ,( ) ( )

k k k k k k k k

i d i d i d i d c d i dv w v c r p x c r p x

,( , )

k

g dN p f-gbest is characterized by a normal distribution, , where

representing the degree of uncertainty about the optimality of the global-best.

Formula for Fuzzy gbest

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▪ SPLS of assimilated particles along x1 and x3

x2

x1

Possible location of

assigned gbest

A

x3

Assimilated Particle A'

Assimilated Particle B'

Trajectory along assigned

search direction, x1

B

Trajectory along assigned

search direction, x3

Synchronous Particle Local Search

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▪ SPLS is performed in the vicinity of the particles.

SPLS:

Select LSS particles randomly from particle swarm

Select LSN nondominated particles from the archive with

Assign an arbitrary dimension to each of the LSS particles

update the position of particles in the desion space

Assimilation: With the exception of the assigned dimension,

with the selected gbest position

the best niche count into a selection pool

Assign an arbitrary nondominated solution from the selection

pool to each of theLS

S

LSS

particles as gbest

Update the position of allLSS assimilated particles using fuzzy

gbest along the pre-assigned dimension

Synchronous Particle Local Search

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Update Particle Position

using f-gbest

Select Personal

BestNo

Initialize Particle

Swarm

cycle = max_cycles?Yes

Return Archive

Select Global Best

Evaluate Particles

SPLS

Select particles

for SPLSLS

S

Archiving

Implementation

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(d) (e) (f) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Evolved tradeoffs by a) FMOPSO, b) CMOPSO, c) SMOPSO, d) IMOEA, e)

NSGA II, and f) SPEA2 for ZDT1

Simulation Results

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1 2 3 4 5 6

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1 2 3 4 5 6012345678

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(g) (i) (h)

Statistical performance of the different algorithms: a) GD, b) MS, c) S for ZDT4; d)

GD, e) MS, f) S for ZDT6; and g) GD, i) MS, h) S for FON

Simulation Results

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Research Ideas

▪ Handling of high dimensional problems.

✓ Use of dimensional reduction techniques.

✓ Use of learning techniques to gain information on

the shape and position of Pareto front/Pareto set.

▪ Apply surrogates to reduce evaluation time.

✓ Surrogate models are cheap and approximate

evaluation models.

▪ Solving real world problems of your interest

Researchers are facing the challenge of increasing

dimensionality and computational cost of today’s

applications.

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▪ Zhinzhong Ding, Fuqiang Lu, and Hualing Bi, “A TWO-STAGE PARTICLE SWARM

OPTIMIZATION FOR VIRTUAL ENTERPRISE RISK MANAGEMENT”, International Journal of

Innovative Computing, Information and Control, vol. 10, no. 4, pp. 1495-1508, 2014.

▪ Marco Corazza, Giovanni Fasano, S. Y., and Riccardo Gusso, “Particle Swarm Optimization with

non-smooth penalty reformulation, for a complex portfolio selection problem”, Applied

Mathematics and Computation 244, pp. 611-624, 2013.

▪ Kuo, R. J. and Hong, C. K. “Integration of Genetic Algorithm and Particle Swarm Optimization for

Investment Portfolio Optimization”, Applied Mathematics & Information Sciences, vol. 7, no. 6, pp.

2397-2408, 2013.

▪ Jui-Fang Chang, Peng Shi, “Using investment satisfaction capability index based particle swarm

optimization to construct a stock portfolio”, Information Sciences 181, pp. 2989-2999, 2011

▪ Liu, D. S., Tan, K. C., Goh, C. K. and Ho, W. K., “A Multiobjective Memetic Algorithm Based on

Particle Swarm Optimization,” IEEE Transactions on Systems, Man and Cybernetics: Part B

(Cybernetics), vol. 37, no. 1, pp. 42-50, 2007.

The Papers