Tetsuyuki TAKAHAMA （ Hiroshima City University ） Setsuko SAKAI （ Hiroshima Shudo University)

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


Constrained Optimization Problems

objective function f , decision variables xi

inequality constraints gj, equality constraints hj

lower bound li, upper bound ui


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


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


-level comparison

Function value and constraint violation （ f , ） precedes constraint violation usually precedes function value if violation is small


Definition of constrained method

Constrained problems can be solved by replacing ordinary comparisons with level comparisons in unconstrained optimization algorithm ＜→＜， →

∥∥


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


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


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


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


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


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


Control of -level

Small feasible region and -level

small feasible region=0≦ (Tf)

≦ (0)


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


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


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


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


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)


10D (C01-C06)


10D (C07-C012)


10D (C13-C018)


30D (C01-C06)


30D (C07-C12)


30D (C13-C18)


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


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


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


Thank you for your kind attention


10D problems


30D problems


Moore-Penrose inverse

of diagonal on the

elements zero-non inverting :

iondecomposit luesinglar va

of inverse pseudo :

T

T

UVA

VUA

AA

Tetsuyuki TAKAHAMA （ Hiroshima City University ） Setsuko SAKAI （ Hiroshima Shudo University)

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