Neural Networks and Genetic Algorithms Multiobjective acceleration
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Transcript of 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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
3
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
4INTRODUCTION
OBJECTIVES:
• Develop an efficient multi-objective optimization algorithm
• Reduce the number of evaluations of objective functions necessary
• Compare performance with existing algorithms
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
5
• Multi-Objective Evolutionary Algorithm (MOEA)
• Artificial Neural Networks (ANN)
• Hybrid Multi-Objective Algorithm (MOEA-IANN)
• Results and Discussion
• Conclusions
CONTENTS
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
7
C1
C2
Density
Fitness
Archiving
Basic functions of a MOEA:
MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – 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)
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
8
Start
Initialise Population
Evaluation
Assign FitnessFi
Convergencecriterion satisfied?
Selection
Recombination
i = i + 1
Stop
no
yes
i = 0
Reduced Pareto Set G.A. with Elitism (RPSGAe)
MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA
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.
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
9
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
MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA
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.
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
10
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
MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA
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:
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
11
Order of the RPSGAe: O(Nranks q N2)
MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
12
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
MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM – MOEA
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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;
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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).
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
15HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN
Start
Initialise Population
Evaluation
Assign FitnessFi
Convergencecriterion satisfied?
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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
Neural Network learning using some solutions
of the p generations
Proposed by K. Deb et. al
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
17HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-ANN
p generations r generations
RPSGA with exact
function evaluation
Neural Network learning using some solutions
of the p generations
RPSGA with:• All solutions
(N) evaluated by Neural Network
• M evaluated by exact function
p generations r generations p generations... ...
Neural Network learning using some solutions
of the 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
,,
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
18
Use of an Inverse ANN as “Recombination” operator
HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN
Start
Initialise Population
Evaluation
Assign FitnessFi
Convergencecriterion satisfied?
Selection
Recombination
i = i + 1
Stop
no
yes
i = 0
Recombination operators:
• Crossover
• Mutation
• Inverse ANN (IANN)
C1
C2
Cq
V1
...
V2
VM
...
VariablesCriteria
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
19
Set of Solutions Generated with the IANN
HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
20
Set of Solutions Generated with the IANN
HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
21
MOEA-IANN Algorithm Parameters
HYBRID MULTI-OBJECTIVE ALGORITHM – MOEA-IANN
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
22RESULTS AND DISCUSSION – Test problems
.,,2,1for,Subject
,,,,,Minimize
,Minimize
,Minimize
,Minimize
112211
11
22
11
qixto
xgxfxfxfhxgxf
xf
xf
xf
ixi
qqqqq
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
23RESULTS AND DISCUSSION – Test problems
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]
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
24RESULTS AND DISCUSSION – Test problems
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]
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
25RESULTS AND DISCUSSION – Test problems
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]
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
26RESULTS AND DISCUSSION – Test problems
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
27RESULTS AND DISCUSSION – Test problems
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]
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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;
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
32
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
0 100 200 300Generations
S m
etr
ic
IANN
RPSGAe
2C-ZDT4
0
0.05
0.1
0.15
0 100 200 300Generations
S m
etr
ic
IANN
RPSGAe
2C-ZDT6
0
0.2
0.4
0.6
0.8
0 100 200 300Generations
S m
etr
ic
IANN
RPSGAe
RESULTS AND DISCUSSION – 2 Criteria Test Problems
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
33
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
RESULTS AND DISCUSSION – 3 Criteria Test Problems
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
34
MOEA - Inverse ANN3C-ZDT2
0
0.2
0.4
0.6
0.8
0 100 200 300Generations
S m
etr
ic
IANN
RPSGAe
3C-ZDT3
0
0.3
0.6
0.9
1.2
1.5
1.8
0 100 200 300Generations
S m
etr
ic
IANN
RPSGAe
3C-ZDT4
0
0.02
0.04
0.06
0 100 200 300Generations
S m
etr
ic
IANN
RPSGAe
3C-ZDT6
0
0.1
0.2
0.3
0.4
0 100 200 300Generations
S m
etr
ic
IANN
RPSGAe
RESULTS AND DISCUSSION – 3 Criteria Test Problems
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
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
Dept. Polymer Engineering
University of Minho
Instituto Superior deEngenharia do Porto
Faculty of Science and TechnologyUniversity of Algarve
36
ANY QUESTION!?