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


Constraint violation and e-level comparisons
The e constrained differential evolution (eDEag)

differential evolution (DE) with an archive

gradient-based mutation

control of the e-level

Experimental results

Conclusions
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Constrained Optimization Problems

objective function f , decision variables xi

inequality constraints gj, equality constraints hj

lower bound li, upper bound ui
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e constrained method

Algorithm transformation method
 algorithm
for unconstrained optimization
→ algorithm for constrained optimization
 e-level comparison
 comparison
between pairs of objective value
and constraint violation
replacing ordinary comparisons to e-level
comparisons in unconstrained optimization
algorithm
 by
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Constraint Violation

Constraint Violation f (x)
f ( x )  0, if x is infeasible
f ( x )  0, if x is feasible

max
f ( x)  max{ max{0, g j ( x)}, max | h j ( x) |}

j
sum
j
f ( x )   || max{ 0, g j ( x )} ||   || h j ( x ) ||
p
j
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p
j
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e-level comparison

Function value and constraint violation（f ,f）
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 precedes
constraint violation usually
 precedes
function value if violation is small
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Definition of e constrained method
∥

Constrained problems can be solved by replacing
ordinary comparisons with e level comparisons in
unconstrained optimization algorithm

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＜→＜e，
→  e
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Differential Evolution (DE)

simple operation avoiding step size control
crossover
parent
population
base vector
-
F
difference vector
(CR)
+
trial vector

trial vector (child) will survive if the child is better

robust to non-convex, multi-modal problems
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eDEa: eDE with an archive (1)

A small population and a large archive are adopted


N
Generate M initial individuals


Small population is good for search efficiency
but is bad for diversity
A={ xk | k=1,2,...,M } (M=100n)
Select top N individuals from A
as an initial population
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P

P={ xi | i=1,2,...,N } (N=4n)

A=A-P
A
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M-N
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eDE with an archive (2)

DE/rand/1/exp operation
correction of Fig.2
 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
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eDE 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),f(xtrial)) ＜e (f(xi), f(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
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Gradient-based mutation (1)


adopts the gradient of constraints to reach
feasible region
Constraint vector and constraint violation vector
C( x )  (g1 ( x ) , ,g q ( x ) ,h q 1( x ) , h m ( x ))T
 C( x )  ( g1 ( x ) , , g q ( x ) ,h q 1( x ) , h m ( x ))T
 g j ( x )  max{ 0, g j ( x )}

Gradient of constraint vector
 C( x )x   C( x )
if satisfied, x  x will be feasible
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Gradient-based mutation (2)


inverse  C( x ) 1 cannot be defined generally
Moore-Penrose inverse (pseudoinverse)  C(x ) 
 approximate

or best (LSE) solution
x'  x   C( x)   C( x)
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
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Control of e-level

Small feasible region and e-level
small feasible region
f=0
f≦e(Tf)
f≦e(0)
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Control scheme of e-level
 e-level
(t  0)
f ( x )
cp



t
e (t )  e (0)1  
(0  t  Tc )
T
c 


(t  Tc )
0
x is the top - th individual in f
e(t)
e0
0
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should converge to 0 gradually
Tc
Tmax
t
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control of cp

instead of specifying cp, specify e-level at Tl
,


To search better objective value
 generation
 enlarge
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from Tl to Tc
e-level and scaling factor F
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Effectiveness of e constrained method


The e level comparison does not need
objective values if one of the constraint
violations is larger than e-level
Lazy evaluation
 objective
function is evaluated only when
needed
 evaluation
of objective function can be often
omitted when feasible region is small
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Conditions of experiments

18 constrained problems, 25 trials per a problem

eDEag/rand/1/exp


Max. FEs: 20,000n

M=100n, N=4n, F=0.5, CR=0.9

e level control: =0.9, Tc=1,000, Tl=0.95Tc
Gradient-based mutation
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
mutation rate: Pg=0.1, max. iterations: Rg=3

applied only in every n generations
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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:
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C12 (1)
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10D (C01-C06)
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10D (C07-C012)
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10D (C13-C018)
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30D (C01-C06)
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30D (C07-C12)
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30D (C13-C18)
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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
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Conclusions

eDE 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
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Future works

To find better objective values
 dynamic
control of e level
e level according to the number of
feasible points
 changing
 mechanism
for maintaining diversity
 subpopulations
or species to search various
regions
 adaptive
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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
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30D problems
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Moore-Penrose inverse

A : pseudo inverse of A
singlar va lue decomposit ion
A  UV


T
A  V U
T

 : inverting non -zero elements
on the diagonal of 
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