neural networks and genetic algorithms multiobjective acceleration

Dept. Polymer Engineering

University of Minho

Instituto Superior deEngenharia do Porto

Faculty of Science and TechnologyUniversity of Algarve

1

A Hybrid Multi-Objective Evolutionary Algorithm

Using an Inverse Neural Network

A. Gaspar-Cunha(1), A. Vieira(2), C.M. Fonseca(3)

(1)IPC- Institute for Polymers and Composites, Dept. of Polymer Engineering,

University of Minho, Guimarães, Portugal(2)ISEP and Computational Physics Centre,

Coimbra, Portugal(3)CSI- Centre for Intelligent Systems, Faculty of Science and Technology,

University of Algarve, Faro, Portugal

HYBRID METAHEURISTICS (HM 2004) ECAI 2004, Valencia, Spain

August, 2004


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2

Most real optimization problems are multiobjective

Example: Simultaneous minimization of the cost and maximization of the performance of a specific system

Performance

Cost

Single optimum(maximal performance)

Single optimum(minimal cost)

Multiple optima(both objectives optimized)

Dominated solution

PARETO FRONTIER

(set of non-dominated solutions)

INTRODUCTION


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Computation time required to evaluate the solutions

INTRODUCTION

Engineering problems:

Start

Initialise Population

Evaluation

Assign FitnessFi

Convergencecriterion satisfied?

Selection

Recombination

i = i + 1

Stop

no

yes

i = 0

Black Box

Numerical modelling

routines

• Finite elements• Finite differences• Finite volumes• etc

HIGH COMPUTATION TIMES


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4INTRODUCTION

OBJECTIVES:

• Develop an efficient multi-objective optimization algorithm

• Reduce the number of evaluations of objective functions necessary

• Compare performance with existing algorithms


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• Multi-Objective Evolutionary Algorithm (MOEA)

• Artificial Neural Networks (ANN)

• Hybrid Multi-Objective Algorithm (MOEA-IANN)

• Results and Discussion

• Conclusions

CONTENTS


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6

q

jjji FwFO

1

How to deal with multiple criteria (or objectives)?

10

10

1

10

i

j

j

j

FO

F

w

wSingle objective(for example, weighted sum)

Multiobjective optimization 1

4

63

52

160

170

180

190

200

500 1000 1500 2000Objective 1

Obj

ecti

ve 2

Pareto Frontier

Decision made before the search

Decision made after the search

MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA


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C1

C2

Density

Fitness

Archiving

Basic functions of a MOEA:


Guiding the population

towards the Pareto set

(Fitness assignment)

Maintaining a diverse nondominated set

(Density estimation)

Preventing nondominated solutions from being lost

(Elitist population - archiving)


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Start


Evaluation

Assign FitnessFi


Selection

Recombination

i = i + 1

Stop

no

yes

i = 0

Reduced Pareto Set G.A. with Elitism (RPSGAe)


a) Rank the individuals using a clustering

algorithm;

b) Calculate the fitness using a ranking

function;

c) Copy the best individuals to the external

population;

d) If the external population becomes full:

- Apply the clustering algorithm to the

external population;

- Copy the best individuals to the internal

population;

RPSGAe sorts the population individuals in a number of

pre-defined ranks using a clustering technique, in order

to reduce the number of solutions on the efficient

frontier.


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N=15; Nranks=3

C1

C2

1

1

1

1

1

r=1; NR=5

C1

C2

1

1

1

1

1

2

22

2

2

r=2; NR=10

Clustering algorithm example


ranksR N

NrN

Gaspar-Cunha, A., Covas, J.A. - RPSGAe - A Multiobjective Genetic Algorithm with Elitism: Application to Polymer Extrusion, in Metaheuristics for Multiobjective Optimisation, Lecture Notes in Economics and Mathematical Systems, Gandibleux, X.; Sevaux, M.; Sörensen, K.; T'kindt, V. (Eds.), Springer, 2004.


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FO SP

SP N i

Ni

22 1 1

Fitness - Linear ranking :

FO(1) = 2.00

FO(2) = 1.87

FO(3) = 1.73

C1

C2

1

1

1

1

1

2

22

2

2

3

3

3

3

3

r=3; NR=15


Clustering algorithm example

• Number of Ranks - Nranks

• Limits of indifference of the clustering algorithm - limit

• N. of individuals copied to the external population - Next

RPSGAe

Parameters:


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Order of the RPSGAe: O(Nranks q N2)


Reduced Pareto Set G.A. with Elitism (RPSGAe)

Generation 1

Generation 2

Generation 3

Generation 4

Generation 5

Generation n

Internal population

Externalpopulation

Internalpopulation

(Generation n)

Externalpopulation

(Generation n)

Next

Next


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How the basic functions are accomplished in the RPSGAe :

1. Guiding the population towards the Pareto setFitness assignment: ranking function based on the reduction of the Pareto Set

2. Maintaining a diverse nondominated setDensity estimation: ranking function based on the reduction of the Pareto Set

3. Preventing nondominated solutions from being lostElitist population: periodic copy of the best solutions (to the main population), selected with the method of Pareto set reduction



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13

Artificial Neural Networks

ARTIFICIAL NEURAL NETWORKS – ANN

• A feed-forward neural network consists of an array of input nodes connected to an array of output nodes through successive intermediate layers;

• Each connection between nodes has a weight, initially random, which is adjusted during a training process;

• The output of each node of a specific layer is a function of the sum on the weighted signals coming from the previous layer;

P1

P2

Pi

C1

...

C2

Cj

...

InputLayer

OutputLayer

HiddenLayer

• ANN implemented by a Multilayer Preceptron is a flexible scheme capable of approximating an arbitrary complex function;

• The ANN builds a map between a set of inputs and the respective outputs;


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14HYBRID MULTI-OBJECTIVE ALGORITHM

Two possible approachs to reduce the computation time

1. During evaluation – Some solutions can be evaluated

using an approximate function, such as Fitness Inheritance,

Artificial Neural Networks, etc (this reduce the number of

exact evaluations necessary).

2. During recombination – Some individuals can be

generated using more efficient methods (this produce a fast

approximation to the optimal Pareto frontier, thus the

number of generations is reduced).


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15HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN

Start


Evaluation

Assign FitnessFi


Selection

Recombination

i = i + 1

Stop

no

yes

i = 0

Use of ANN to “Evaluate” some Solutions

P1

P2

Pi

C1

...

C2

Cj

...

Parametersto optimise Criteria

Artificial Neural Network


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16

Use of ANN to “Evaluate” some Solutions – Method A

HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN

p generations r generations

RPSGA with exact

function evaluation

Neural Network learning using some solutions

of the p generations

p generations r generations p generations... ...

RPSGA with exact

function evaluation

RPSGA with Neural

Network evaluation

RPSGA with exact

function evaluation

RPSGA with Neural

Network evaluation



Proposed by K. Deb et. al


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17HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN

p generations r generations

RPSGA with exact

function evaluation



RPSGA with:• All solutions

(N) evaluated by Neural Network

• M evaluated by exact function

p generations r generations p generations... ...



RPSGA with exact

function evaluation

RPSGA with exact

function evaluation

RPSGA with:• All solutions

(N) evaluated by Neural Network

• M evaluated by exact function

eNN > allowed error eNN > allowed error

Use of ANN to “Evaluate” some Solutions – Method B

M

S

CC

e

M

j

S

i

jiNN

ji

NN

1 1

2

,,


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Use of an Inverse ANN as “Recombination” operator

HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN

Start


Evaluation

Assign FitnessFi


Selection

Recombination

i = i + 1

Stop

no

yes

i = 0

Recombination operators:

• Crossover

• Mutation

• Inverse ANN (IANN)

C1

C2

Cq

V1

...

V2

VM

...

VariablesCriteria


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19

Set of Solutions Generated with the IANN


:criteria)ofnumbertheis(where,

...,,1For

q

qj

jjijjjijj CCCCC '

)('

)(

jjj CCC '

jjijjjjijj CCCCCC '

)('

)(

Point ej to a:

Point ej to b:

Point ej to c:

Criterion 1

Cri

teri

on 2

C1

C2

e2

e1

4

3

21

ac

b

a

bc

jjj CCC 'Points 1, 2, …, n:

Selection of n+q solutions from the

present population to generate:

• 3.q extreme solutions

• n interior solutions


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Set of Solutions Generated with the IANN


Parameter 1Pa

ram

eter

2

e1

ab

e2

a

bc

1 2

3

4

Criterion 1

Cri

teri

on 2

C1

C2

e2

e1

4

3

21

ac

b

a

bc

c

Use of IANN to generate

new solutions


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21

MOEA-IANN Algorithm Parameters


Number of Ranks - Nranks

N. of individuals copied to the external population - Next

Limits of indifference of the clustering algorithm – limit

Criteria variation at beginning - Cinit

Criteria variation at end - Cf

N. of generations which individuals are used to train the IANN – Ngen

Rate of individuals generated with the IANN – IR


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22RESULTS AND DISCUSSION – Test problems

.,,2,1for,Subject

,,,,,Minimize

,Minimize

,Minimize

,Minimize

112211

11

22

11

qixto

xgxfxfxfhxgxf

xf

xf

xf

ixi

qqqqq

qq

1

91,,where,

1,,

22

122

111

M

xxxg

gfgxxf

xxf

M

i i

M

M

f1

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1f2

K. Deb et. al - Test Problem Generator

2C-ZDT1 (Convex): M = 30; xi [0, 1]

2 Criteria


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1

91,,where,

1,,

22

2

122

111

M

xxxg

gfgxxf

xxf

M

i i

M

M

1

91,,where,

10sin1,,

22

111

22

111

M

xxxg

fgf

gfgxxf

xxf

M

i i

M

M

2 Criteria

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1

f1

f2

-1.00

-0.60

-0.20

0.20

0.60

1.00

0 0.2 0.4 0.6 0.8 1

f1f2

2C-ZDT3 (Discrete): M = 30; xi [0, 1]

2C-ZDT2 (Non-convex): M = 30; xi [0, 1]


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M

i iiM

M

xxMxxg

gfgxxf

xxf

2

22

122

111

4cos101101,,where,

1,,

25.0

22

2

122

16

111

191,,where,

1,,

)6(sin)4exp(1

M

xxxg

gfgxxf

xxxf

M

i i

M

M

2 Criteria

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 0.2 0.4 0.6 0.8 1

f1

f2

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1

f1

f2

2C-ZDT4 (Multimodal): M = 10; x1 [0, 1]; xi [-5, 5]

2C-ZDT6 (Non-uniform): M = 10; xi [0, 1]


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1

91,,where,

1,,

33

2133

222

111

M

xxxg

g

ffgxxf

xxf

xxf

M

i i

M

M

1

91,,where,

1,,

33

2

2133

222

111

M

xxxg

g

ffgxxf

xxf

xxf

M

i iM

M

3 Criteria

0.00.2

0.40.6

0.81.0

0.00.2

0.40.6

0.8

1.0

0.0

0.5

1.0

f3

f1f2

0.00.2

0.40.6

0.81.0

0.00.2

0.40.6

0.8

1.0

0.0

0.5

1.0

f3

f1f2

3C-ZDT1 (Convex): M = 30; xi [0, 1]

3C-ZDT2 (Non-convex): M = 30; xi [0, 1]


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1

91,,where,

10sin1,,

33

212121

33

222

111

M

xxxg

ffg

ff

g

ffgxxf

xxf

xxf

M

i i

M

M

3 Criteria

0.00.2

0.40.6

0.8

1.0

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.00.2

0.40.6

0.81.0

f3

f2f1

3C-ZDT3 (Discrete): M = 30; xi [0, 1]

0.00.2

0.40.6

0.81.0

0.00.2

0.4

0.6

0.8

1.0

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

f3

f1

f2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

-0.5000

-0.3125

-0.1250

0.06250

0.2500

0.4375

0.6250

0.8125

1.000

f1

f2


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25.0

33

2

2133

26

222

16

111

191,,where,

1,,

)6(sin)4exp(1

)6(sin)4exp(1

M

xxxg

g

ffgxxf

xxxf

xxxf

M

i i

M

M

3 Criteria

0.00.2

0.40.6

0.81.0

0.00.2

0.40.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

f1

f3

f2

3C-ZDT6 (Non-uniform): M = 10; xi [0, 1]

M

i iiM

M

xxMxxg

g

ffgxxf

xxf

xxf

3

23

2133

222

111

4cos101101,,where,

1,,

0.00.2

0.40.6

0.81.0

0.00.2

0.40.6

0.8

1.0

0

2

4

6

8

10

12

14

16

18

f3

f1f2

3C-ZDT4 (Multimodal): M = 10; x1,2 [0, 1]; xi [-5, 5]


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28RESULTS AND DISCUSSION – Metrics

Hypervolume Metric (Zitzler and Thiele - 1998)

This metric calculates the dominated space volume,

enclosed by the nondominated points and the origin.

S metric:Volume of the space dominated by the set of objective vectors

C1

C2

Criteria C1 and C2 to maximize

Hypervolume

However, is not possible to say

that one set is better than other


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29RESULTS AND DISCUSSION – Algorithm Parameters

Influence of algorithm parameters on performance

Parameter Tested values(*) Best results Influence Selected

limit 0.01; 0.05; 0.1; 0.2 [0.01; 0.2] Small 0.01

Cinit 0.3; 0.4; 0.5; 0.6 [0.3; 0.5] Small 0.5

Cf 0.0; 0.1; 0.2; 0.3 [0.0; 0.3] Small 0.2

Ngen 5; 10; 15; 20 [5; 10] Small 5

IR 0.35; 0.50; 0.65; 0.80 [0.35; 0.8] Small 0.8(*) 5 runs for each tested parameter value

• The influence of the algorithm parameters on its

performance is very small.

• Each optimisation run was carried out 21 times

using the algorithm parameters selected and

different seed values.

Algorithm Parameters:- N = 100

- Ne = 100

- Nranks = 30

- Next = 3N/Nranks = 10

- cR = 0.8

- mR = 0.05


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30RESULTS AND DISCUSSION – Method B

Use of ANN to “Evaluate” some Solutions – Method B

Test problem

S metric, 22000 evaluations Number of evaluations

Method B RPSGAe Decrease (%) Method B RPSGAe Decrease (%)

ZDT1 0.851 0.849 0.24 10000 19000 47.4

ZDT2 0.786 0.773 1.68 15300 22000 30.5

ZDT3 2.736 2.554 7.13 18000 22000 18.2

ZDT4 0.1116 0.0807 38.29 5000 22000 77.3

ZDT6 0.599 0.571 4.90 12500 22000 43.2

• The S metric after 22000 evaluations decrease when Method B is

used

• The number of evaluations necessary to attain identical level of the

S metric decreases considerably when Method B is used


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31RESULTS AND DISCUSSION – 2 Criteria Test Problems

MOEA - Inverse ANN

2C-ZDT1

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300Generations

S m

etri

c

IANNRPSGAe

• The Inverse ANN approach has the largest improvement during the

first generations, i.e., when the solution is far from the optimum;


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MOEA - Inverse ANN2C-ZDT2

0

0.2

0.4

0.6

0.8

1

0 100 200 300Generations

S m

etr

ic

IANN

RPSGAe

2C-ZDT3

0

0.5

1

1.5

2

2.5

3


S m

etr

ic

IANN

RPSGAe

2C-ZDT4

0

0.05

0.1

0.15


S m

etr

ic

IANN

RPSGAe

2C-ZDT6

0

0.2

0.4

0.6

0.8


S m

etr

ic

IANN

RPSGAe

RESULTS AND DISCUSSION – 2 Criteria Test Problems


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MOEA - Inverse ANN

3C-ZDT1

0

0.2

0.4

0.6

0.8

0 50 100 150 200 250 300

Generations

S m

etri

c

IANN

RPSGAe



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MOEA - Inverse ANN3C-ZDT2

0

0.2

0.4

0.6

0.8


S m

etr

ic

IANN

RPSGAe

3C-ZDT3

0

0.3

0.6

0.9

1.2

1.5

1.8


S m

etr

ic

IANN

RPSGAe

3C-ZDT4

0

0.02

0.04

0.06


S m

etr

ic

IANN

RPSGAe

3C-ZDT6

0

0.1

0.2

0.3

0.4


S m

etr

ic

IANN

RPSGAe



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35CONCLUSIONS

• Algorithm parameters have a limited influence on its performance

• Good performance of the proposed algorithm

• The number of generations needed to reach identical level of performance is reduced thus, the computation time is reduced by more than 50%.

• Most improvements of the IANN approach are accomplished during the first generations


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ANY QUESTION!?

neural networks and genetic algorithms multiobjective acceleration

Education

portofaculty of science

single objective

clustering algorithm

population individuals

polymer extrusion

best individuals

number of solutions

clustering technique