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Constrained Optimization by the e Constrained Differential Evolution with an Archive and Gradient-Based Mutation. Tetsuyuki TAKAHAMA ( Hiroshima City University ) Setsuko SAKAI ( Hiroshima Shudo University). Outline. Constrained optimization problems The e constrained method - PowerPoint PPT Presentation
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Constrained Optimization by the Constrained Differential Evolution with an Archive and Gradient-Based Mutation
Tetsuyuki TAKAHAMA
( Hiroshima City University )Setsuko SAKAI
( Hiroshima Shudo University)
2010/07/21 T.Takahama and S.Sakai in CEC2010 2
Outline
Constrained optimization problems The constrained method
Constraint violation and -level comparisons The constrained differential evolution (DEag)
differential evolution (DE) with an archive gradient-based mutation control of the -level
Experimental results Conclusions
2010/07/21 T.Takahama and S.Sakai in CEC2010 3
Constrained Optimization Problems
objective function f , decision variables xi
inequality constraints gj, equality constraints hj
lower bound li, upper bound ui
2010/07/21 T.Takahama and S.Sakai in CEC2010 4
constrained method
Algorithm transformation method algorithm for unconstrained optimization
→ algorithm for constrained optimization -level comparison
comparison between pairs of objective value and constraint violation
by replacing ordinary comparisons to -level comparisons in unconstrained optimization algorithm
2010/07/21 T.Takahama and S.Sakai in CEC2010 5
Constraint Violation
Constraint Violation (x)
max
sum
feasible is if ,0)(infeasible is if ,0)(
xxxx
|})(|max)},(,0{maxmax{)( xxx jj
jj
hg
j
pj
p
jj hg ||)(||||)}(,0max{||)( xxx
2010/07/21 T.Takahama and S.Sakai in CEC2010 6
-level comparison
Function value and constraint violation ( f , ) precedes constraint violation usually precedes function value if violation is small
2010/07/21 T.Takahama and S.Sakai in CEC2010 7
Definition of constrained method
Constrained problems can be solved by replacing ordinary comparisons with level comparisons in unconstrained optimization algorithm <→< , →
∥∥
2010/07/21 T.Takahama and S.Sakai in CEC2010 8
Differential Evolution (DE)
simple operation avoiding step size control
trial vector (child) will survive if the child is better
robust to non-convex, multi-modal problems
population
difference vector - F
base vector
parent crossover(CR)
+
trial vector
2010/07/21 T.Takahama and S.Sakai in CEC2010 9
DEa:DE with an archive (1)
A small population and a large archive are adopted Small population is good for search efficiency
but is bad for diversity Generate M initial individuals
A={ xk | k=1,2,...,M } (M=100n)
Select top N individuals from Aas an initial population P={ xi | i=1,2,...,N } (N=4n)
A=A-P
A
PN
M-N
2010/07/21 T.Takahama and S.Sakai in CEC2010 10
DE with an archive (2)
DE/rand/1/exp operation mutant vector: and are selected from P is selected from P A w.p. 0.95 or P w.p. 0.05 exponential crossover
Uniform convergence of individuals When a parent generates a child and
the child is not better than the parent,the parent can generate another child
correction of Fig.2
2010/07/21 T.Takahama and S.Sakai in CEC2010 11
DE with an archive (3)
Direct replacement for efficiency Continuous generation model If the child is better than the parent,
the parent is directly replaced by the child (f(xtrial),(xtrial)) < (f(xi), (xi))
Perturb scaling factor F in small probability to escape from local minima F is a fixed value (0.5) w.p 0.95 F=1+|C(0,0.05)| truncated to 1.1 w.p. 0.05
2010/07/21 T.Takahama and S.Sakai in CEC2010 12
Gradient-based mutation (1)
adopts the gradient of constraints to reach feasible region
Constraint vector and constraint violation vector
Gradient of constraint vector
)}(,0max{)(g
))(h)(h)(g)(g()C(
))(h)(h)(g)((g)C(
m1qq1
m1qq1
xx
xxxxx
xxxxx
jj
T
T
g
,,,,
,,,,
feasible be will satisfied, if
)C()C(
xx
xxx
2010/07/21 T.Takahama and S.Sakai in CEC2010 13
Gradient-based mutation (2)
inverse cannot be defined generally Moore-Penrose inverse (pseudoinverse)
approximate or best (LSE) solution
Modifications Numerical gradient (costs n+1 FEs) Mutation is applied only in every n generations Skipped w.p. 0.5, if num. of violated constraints
is one
1)C( x
)C()C(' xxxx
)C(x
2010/07/21 T.Takahama and S.Sakai in CEC2010 14
Control of -level
Small feasible region and -level
small feasible region=0≦ (Tf)
≦ (0)
2010/07/21 T.Takahama and S.Sakai in CEC2010 15
Control scheme of -level
-level should converge to 0 gradually
in individualth - top theis
)(0
)0(1)0(
)0()(
)(
x
x
c
c
cp
c
Tt
TtTt
t
t
t0
(t)
Tc Tmax
0
2010/07/21 T.Takahama and S.Sakai in CEC2010 16
control of cp
instead of specifying cp, specify -level at T
, To search better objective value
generation from Tto Tc
enlarge -level and scaling factor F
2010/07/21 T.Takahama and S.Sakai in CEC2010 17
Effectiveness of constrained method
The level comparison does not need objective values if one of the constraint violations is larger than -level
Lazy evaluation objective function is evaluated only when
needed evaluation of objective function can be often
omitted when feasible region is small
2010/07/21 T.Takahama and S.Sakai in CEC2010 18
Conditions of experiments
18 constrained problems, 25 trials per a problem DEag/rand/1/exp
Max. FEs: 20,000n M=100n, N=4n, F=0.5, CR=0.9
level control: =0.9, Tc=1,000, T=0.95Tc
Gradient-based mutation mutation rate: Pg=0.1, max. iterations: Rg=3
applied only in every n generations
2010/07/21 T.Takahama and S.Sakai in CEC2010 19
Summary of Results
Feasible and stable solutions in all runs 10D: C01-C07, C09, C10, C12-C14, C18 (13) 30D: C01, C02, C05-C08, C10, C13-C16 (11)
Feasible solutions in all runs 10D: C08, C11, C15, C16, C17 (5) 30D: C03, C04, C09, C11, C17, C18 (6)
Often infeasible solutions 30D: C12 (1)
2010/07/21 T.Takahama and S.Sakai in CEC2010 20
10D (C01-C06)
2010/07/21 T.Takahama and S.Sakai in CEC2010 21
10D (C07-C012)
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10D (C13-C018)
2010/07/21 T.Takahama and S.Sakai in CEC2010 23
30D (C01-C06)
2010/07/21 T.Takahama and S.Sakai in CEC2010 24
30D (C07-C12)
2010/07/21 T.Takahama and S.Sakai in CEC2010 25
30D (C13-C18)
2010/07/21 T.Takahama and S.Sakai in CEC2010 26
Computational Complexity
T1: Time (seconds) of 10,000 function
evaluations for a problem on average T2: Time (seconds) of 10,000 function
evaluation with algorithm for a problem
2010/07/21 T.Takahama and S.Sakai in CEC2010 27
Conclusions
DE with a large archive and gradient-based mutation can find feasible solutions in all run and all
problems except for C12 of 30D can often omit evaluation of objective values
and find solutions efficiently and very fast
2010/07/21 T.Takahama and S.Sakai in CEC2010 28
Future works
To find better objective values dynamic control of level
changing level according to the number of feasible points
mechanism for maintaining diversity subpopulations or species to search various
regions adaptive control of F and CR
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Thank you for your kind attention
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10D problems
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10D problems
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30D problems
2010/07/21 T.Takahama and S.Sakai in CEC2010 33
30D problems
2010/07/21 T.Takahama and S.Sakai in CEC2010 34
Moore-Penrose inverse
of diagonal on the
elements zero-non inverting :
iondecomposit luesinglar va
of inverse pseudo :
T
T
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