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A New Mutation Operator for Evolution Strategies for Constrained Problems
Oliver Kramer
International Graduate School
University of Paderborn
D-33098 Paderborn, Germany
okramer@upb.de

Chuan-Kang Ting
International Graduate School
University of Paderborn
D-33098 Paderborn, Germany
ckting@upb.de

Abstract- We propose a new mutation operator - thebiased mutation operator (BMO) - for evolution strategies, which is capable of handling problems for constrained fitness landscapes. The idea of our approachis to bias the mutation ellipsoid in relation to the parent
and therefore lead the mutations into a beneficial direction self-adaptively. This helps to improve the successrate to reproduce better offspring. Experimental results
show this bias enhances the solution quality within constrained search domains. The number of the additional
strategy parameters used in our approach equals to thenumber of dimensions of the problem. Compared to thecorrelated mutation, the BMO needs much less memoryand supersedes the computation of the rotation matrixof the correlated mutation and the asymmetric probability density function of the directed mutation.

1 Introduction
Evolution strategies (ES) are the fourth main variant of evolutionary algorithms (EA) besides genetic algorithms (GA),evolutionary programming (EP) and genetic programming(GP). Evolution strategies are most appropriate to numerical
optimization [21], as their operators are especially designed
for numerical fitness landscapes. The main source for variation is the mutation operator, which is based on the Gaussian distribution to reproduce mutations in the vicinity of theparent, a consequence from the maximum entropy principle for unconstrained search spaces [5]. However, there arestill drawbacks, especially in multimodal and constrained
problems. There are three main principles for the design ofmutation operators which are proposed by Beyer [4]:* reachability,* unbiasedness,* scalability.

The first principle ensures that the whole search space andstrategy parameter space can be reached within a finite number of generations. The scalability condition ensures that the
mutation strength can adapt to values which guarantee improvements during the optimization process. The conditionof unbiasedness is appropriate to unconstrained real search
spaces. But for constrained problems evolutionary algorithms with a self-adaptive step size mechanism often sufferfrom a disadvantageous success probability at the boundaryof the feasible search space [14], also see section 3. This

0-7803-9363-5/05/$20.00 ©2005 IEEE.

Hans Kleine Buning
Department of Computer Science
University of Paderborn
D-33098 Paderbom, Germany
kbcsl@upb.de

results in premature step size reduction and fitness stagnation. In this paper we introduce a new mutation operator,
the biased mutation operator (BMO), which is capable ofbiasing the search into a certain direction and therefore increases the success rate to reproduce better offspring. The
number of additional strategy parameters only grows linearly with the number of problem dimensions. Experimental results show that our approach is capable of dealing withconstrained problems.
In section 2 we will review the most important mutation operators for evolution strategies: standard uncorrelated mutation with one and with N step sizes, correlated
mutation and directed mutation for the purpose of comparison. In the next section 3 constrained problems and premature fitness stagnation is described. In section 4 we will
introduce the new biased mutation operator (BMO). In section 5 we will show empirically that our approach is superior to standard mutation. In the last section 6 we summarize
the results and present the future work we plan to undertake.

2 Mutation operators for evolution strategies
For a comprehensive introduction to evolution strategies refer to [5]. Here it is important to keep in mind that an individual consists of a vector of objective and strategy variables. In real valued search spaces the objective variables
are unencoded real values XI, ..., XN representing the assignment of the variables of an N-dimensional optimization
problem. The strategy variables contain additional information which are usually important for the mutation operator.
There already exists a variety of mutation operators for evolution strategies. For the basic (, + A)-ES 1 standard uncorrelated mutation was introduced by Rechenberg and Schwefel as well as correlated mutation by Schwefel [20]. Many
extensions and variations were introduced within the lastyears, e.g. cumulative step size adaptation by Ostermeier,Gawelczyk and Hansen [18], covariance matrix adaption byHansen [8], cauchy mutation by Yauo and Liu [23] or directed mutation by Hildebrand [9]. In the following subsection the most important ones are reviewed.
2.1 Uncorrelated mutation with one step size

Mutation is the main source for variation in the evolutionary
process. In real-valued search spaces mutation means the
addition or subtraction of a small variation, so that a child
1The +-notation combines the notation for both the selection schemesplus- and comma-selection.

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(a)

(b)

(c)

(d)

(e)

Figure 1: The mutation ellipsoids of the different mutation operators in a two-dimensional search space. For the purpose of
better understanding we assume all mutations fall into the ellipsoids instead of taking the probability density functions into
account. From left to the right: (a) uncorrelated mutation with one step size, (b) uncorrelated mutation with N = 2 step
sizes, (c) correlated mutation resulting in rotation, (d) directed mutation with different skewness, and (e) biased mutation
making use of the bias coefficient vector.
evolves which has got a high similarity to its parent, but is
not identical. The simple standard mutation for evolution
strategies makes use of only na = 1 endogenous strategy
variable a, which represents the standard deviation for thenormal distribution. For the purpose of better understanding
it can be seen as the radius for the sphere in which mutations
are created. For ES in real-valued search spaces, objective
variables are mutated in the following way:is achieved by the correlated mutation proposed by Schwefel [20]. For an N-dimensional problem k = N(N - 1)/2
additional strategy parameters, the angles for the rotationof the mutation ellipsoid, are introduced. Let a again be thevector of step sizes and M be the orthogonal rotation matrix.
The mutations are reproduced in the following way:

.x:=x + z

For a self-adaptation process the k angles al, ...oak have to
be mutated. Schwefel [20] proposed(1)with the mutation

a& = a±+ 3 A(0, l)=

IAN(O, 1)) (2)z := oj.(i (0, 1), ...,

where Ali (0, 1) provides a random number based on a Gaussian distribution with expected value 0 and standard deviation 1. The strategy variable itself is mutated with the lognormal rule:(3):= Oe((rA(0,1))This mechanism is the key to self-adaptation of the step
sizes.

2.2 Uncorrelated mutation with N step sizes
In the basic (, + A)-ES normally a vector of n, = N step
sizes is used, which results in mutation ellipsoids:z :=(o4Ai(0, 1),...JN.AN(0O 1))(4)

The corresponding strategy parameter vector is mutatedwith the extended log-normal rule:a:- e(TKo(Oi)) * (aie(r11(7i), .

Ne(

N())

(5)

The parameters To and ri have to be tuned. Comprising,an individual a consists of the object parameter set xi with
1 < i < N, the mutation strength vector and the assigned
fitness F(x). So it can be specified by
a= (xl, ..., XN,ia.

, N,F(x))(6)

2.3 Correlated mutation
For some fitness landscapes it is more beneficial to use arotated mutation ellipsoid for the purpose of an improvement of the success rate. Rotation of the mutation ellipsoid2601

z := M(91A"i(0, 1), ..., YNAVN(0, 1))

(7)

(8)

with ,B = 0.0873 corresponding to 5°.
2.4 Directed mutation
Similar to the idea of the correlated mutation, the assumption of the directed mutation is that in some parts of thelandscape a success rate improvement can be achieved byskewing the search into a certain direction. The idea ofthe directed mutation proposed by Hildebrand [9] is the usage of an asymmetrical mutation operator. This approachdemands an asymmetry parameter set of N additional parameters c' = (cli, . ..cN). These parameters determine themutation direction and therefore only cause linear growthof the strategy parameters instead of quadratic growth ofthe correlated mutation. The idea of skewing the mutations
into a certain direction is similar to our approach. But arather complex calculation of the asymmetric probability
density function is necessary, because it has to fulfill thecorresponding mathematical specifications of a probability
density function. Figure 1 visualizes the effect of the directed mutation operator when mutations are skewed intoa certain direction. The highest probability to reproduce
mutations is still in the neighborhood environment of thechild. In order to decouple asymmetry from the step sizes
a normalized directed mutation operator has recently been
proposed by Berlik [3].

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3 Constrained problems and premature fitness
stagnation

Direction
optimum- t

Infeasible
search space

3.1 The NLP Problem

Practical optimization problems, e.g. engineering problems,
are in most cases constrained. The constrained nonlinear
programming (NLP) problem is defined as follows: Find asolution
X = (XIiX2iXn),1(9)which minimizes f(x):Lines ofconstant F(x)...

f( ) -min.,inequalities
equalities

xEX

gi(x) < 0, i = 1, ..., khj(x') = 0, j = 1, ...,1

(10)

(11)
(12)

A feasible solution xF satisfies all k inequality and 1 equality
constraints.

3.2 Constraint handling techniques
Various constraint handling techniques exist. For a survey of constraint handling techniques for evolutionary algorithms have a look at [7] or [16]. Most methods fall

into the category of penalty functions. The idea of penalty
functions is to allow the evolutionary process to discover
the whole search space, but to penalize the infeasible part.
Static penalty functions use defined penalties [10], [15],while dynamic penalties depend on the number of generations, e.g. see [12]. Annealing penalty functions take anexternal cooling scheme into account [12], while adaptive
methods tune the penalties according to features determining the progress of the evolutionary search [1]. Constraint
handling techniques exist that do not use the penalty approach. The decoder method [16], [13] builds up a relation between the constraint search space and an unconstrained one using a decoder function. Repair algorithms
repair infeasible solutions using heuristics like minimizing
the constraint violation [2], [7]. Multiobjective methods canbe used as constraint handling methods when treating theconstraints as separate objectives [11], [6], [22], [19]. Recently, Coello introduced a technique based on a multimembered evolution strategy combining a feasibility comparison
mechanism with several modifications of the standard ES[17].

3.3 Premature fitness stagnation
Premature fitness stagnation is caused by a disadvantageous
success probability near the infeasible part of the search
space [14]. In the vicinity of the constraint boundary the
success area can be shortened by the infeasible search space.
The success area is denoted as the part of the mutation ellipsoid, in which infeasible solutions have got a better fitness than their parent. If the step sizes are bigger than the
distance from the center of the mutation ellipsoid to theconstraint boundary the success area is shortened. Figure
2 illustrates the situation. However, for smaller step sizes
the success area is not shortened. So the process of selfadaptivity favors individuals with smaller step sizes. The

Feasiblesearch space

Figure 2: Premature fitness stagnation in the vicinity of theconstraint boundary. We assume that all mutations fall into
the circles in the case of uncorrelated mutation. The marked
success area increases for smaller step sizes (right) in comparison to bigger ones (left).
consequence of this premature step size reduction is premature fitness stagnation before reaching the optimum. Experimentally premature fitness stagnation is shown in theexperiments of section 5.1.

Figure 3: Principle of the BMO in two dimensions: The
center of the mutation ellipsoid is shifted by the bias coefficient vector b within the bounds of the step sizes v!.

4 The Biased mutation operator (BMO)
Unlike directed mutation, the BMO does not change theskewness, but biases the mean of Gaussian distribution tolead the search into a more beneficial direction. This is reflected in the success rate of reproducing superior offspring.
For the BMO we introduce a bias coefficient vector (, whichindicates the level of bias relative to the standard deviationa~.

. (.li ---i&) With=For every idefined by:

E 1,

...,

I <.j < I(13)

N the bias vector b = (bl, ..., bN) is

bi

(14)
Since the absolute value of bias coefficient (i is less thanor equal to 1, the bias will be bound to the step sizes oi.
This restriction prevents the search from being biased too
far away from the parent. Figure 3 illustrates the BMO. The
BMO follows the standard way of mutation:
=

j-%

(15)

x' :=X±Z.

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The mutation in the BMO works as follows:z:= (ulKl(0, 1) + bl, *-,UNKN(O, 1) + bN)(16)- (uAl (O, 1) + 61 1, ..N,ONKN(O, 1) + &NTUN) (17)- (alJ1fi/ 1, 1), .., NKN(N,
)N)
1
(18)
In terms of modifying the mutation strength, the aforementioned log-normal rule is applied. Furthermore, in the BMO
the bias coefficients are mutated in the following way:.j'=.j+-y-X(Ojl) i=l)...,N.

(19)

The parameter -y is a new parameter introduced for the BMO
to determine the mutation strength on the bias. In section
5 recommendations will be proposed to tune this parameter. The BMO biases the mean of mutation and enables thelike evolution strategies, it is not easy to find an optimum
which lies on the boundary of the feasible search space due
to premature step size reduction. This results in premature
fitness stagnation before approximating the optimum, alsosee [14]. For big step sizes the constrained search space cuts
off a big area of success to reproduce better mutations thanthe parent. So the step sizes reduce self-adaptively before
reaching the area of the optimum.
5.1 Experimental results

First of all we show the behavior of a (p + A)-ES with standard uncorrelated mutation operator on our four selected
constrained test problems, see the appendix A. The experimental conditions are shown in table 1. We follow the recommendation for the parameters To and Tr
1

To =
Ti

-4

-2

b

2

Figure 4: Comparison of standard mutation (solid), directed
mutation (dotted), and biased mutation (dashed) with bias b.
evolution strategies to reproduce offspring outside the standard mutation ellipsoid. To direct the search, the BMO enables the mutation ellipsoid to move within the bounds ofthe regular step sizes. Without the BMO the success rate
to reproduce better offspring is relatively low because many
mutation lie beyond the feasible search space or have got
a worse fitness, as described in section 3.3. The bias coefficient vector ( improves the success rate situation as the
success area increases. The BMO approach is as flexible
as correlated and directed mutation, but is less computational expensive than both methods. In comparison to correlated mutation N (N-1) additional strategy parameters canbe saved. Furthermore, the rotation of the mutation ellipsoid which demands O(N2) steps can be saved. Concerning the directed mutation the computation of the random
numbers of an asymmetric probability density function isusually more computationally expensive than the computation of Gaussian random numbers. Especially, in practise
the implementation is less complex. Figure 4 shows a comparison of the different probability density functions of thethree mentioned approaches.
5

The BMO on constrained numerical problems

For many optimization problems the search space is constrained due to a variety of practical conditions. For evolutionary algorithms with a self-adaptive step size mechanism
2603

\12N

(20)

and

1
=

(21)

V2VrN

Intermediate recombination was used for objective andstrategy variables. As constraint handling method the simple death penalty, i.e. the rejection of infeasible solutions,
was used. Table 2 shows the results from 100 runs. The
experiments show that the evolution strategies with uncorrelated standard mutation are not able to approximate anyof the optima of the constrained problems within the given
number of generations. Instead, the evolution strategies suffer from fitness stagnation before reaching the optimum.
problem
2.40
2.41
gO4
g09

ES-type(15,300)(15,300)(15,100)(15,100)

0.1

0.1
0.1
0.1

generations

runs

1000
500
200
500

100
100
100
100

Table 1: Parameter settings for the experiments on the test
problems, the parameter -y is used only for the BMO.

The experimental results of our biased mutation opera-

tor (BMO) on the constrained test problems are presented
in table 3. The results show that due to the BMO approach

the ES are capable of approximating the optimum in every run on the three problems 2.40, 2.41 and HB. Experiments with different settings for the mutation strength pa-

rameter a for the mutation of the bias coefficient vector (showed that the setting 7y = 0.1 is a reasonable recommendation. The experiments on problem gO9 show no significant improvement. Here the BMO is not capable of guiding the search into a more beneficial direction to prevent
fitness stagnation. Moreover, we conduct one-tailed t-test
to examine whether or not there exists statistically significant improvement of the BMO over uncorrelated standard
mutation in solution quality. Table 4 shows the t-test results of the three of the four experiments: The extremely
small p-values (<< 0.05) in this table show that the BMO
can improve the solution quality over standard uncorrelated

2604

problem
optimum
best
meanworst std.dev
2.40-5000.00-4999.74 -4911.51-4691.61
3.53
2.41-17857.14 -17857.12 -17340.42 -15912.11
2.66
g04-30665.539 -30665.539 -30660.15 -30631.48
0.54
680.63g09
680.630680.66680.79 0.00279
Table 2: Experimental results of the evolution strategies (see table 1) with uncorrelated standard mutation and N step sizeson the four constrained test problems.

optimum
best
mean
worst
std.dev-5000.00 -5000.00 -5000.00 -5000.00 1.4 10-10-17857.14 -17857.14 -17857.14 -17857.14 3.3 .10-12-30665.539 -30665.54 -30665.54 -30665.49 3.2 i10-4
680.630 680.6310
680.67680.780.001599

problem
2.40
2.41
g04
g09

Table 3: Experimental results of the evolution strategies (see table 1) with the BMO on the constrained test problems.

mutation significantly. This fact validates the effectivenessof the BMO on these problems. However, no improvement
is achieved on problem g09. Here the BMO is obviously notcapable of improving the success rate by biasing the mutation self-adaptively. Further experiments showed that boththe correlated and the directed mutation are capable of approximating the optimum of the constrained problems similar to the BMO.
1 0000.

-problem
2.40
2.41
g04

degree offreedom
250.67989
99
1942.5564
999999.814815
t-valuep-value
6.912. 10-141
6.861 . 10- 229
1.7996. 10-101

Table 4: The t-test verifies that the BMO is superior to standard uncorrelated mutation on three of the constrained problems.
ness does not develop logarithmically linear, but comprises
phases of slower convergence.

.Standard"
.bmo"

100

6 Conclusions and future work
o1

1,I

0,01

0.0001--,.
1 e-06

iX

F

1e-08

I41

.-u,

50

100

150

200

250

300

generations

Figure 5: Fitness development of an (15,100)-ES with uncorrelated mutation with N steps sizes (standard) and with
the BMO (bmo) on problem g04. On the y-axis the development of the difference between fitness and the optimum
is shown on a logarithmic scale.

We have introduced a new biased mutation operator (BMO)
which is based on shifting the mutation ellipsoid selfadaptively to improve the success rate for reproducing better
offspring. The BMO approach can be as flexible as correlated mutation and directed mutation, but is less computationally expensive. The adaptation process of the bias ofmutation is more efficient than the adaptation of the correlated mutation angles because only N additional endogenous strategy variables are necessary. In comparison to directed mutation, the computational effort of calculating an
asymmetric probability density function is not necessary.
Experimental results showed that the BMO approach is
helpful to improve the ability of convergence. Further investigation is necessary to evaluate the efficiency of convergence on constrained and multimodal problems as well.

5.2 Convergence Speed

Bibliography

Figure 5 shows the fitness development of 20 runs of a(15,100)-ES on problem g04 with uncorrelated mutation (Nstep sizes) and with the BMO. On the y-axis the difference
between the fitness in each generation and the optimum isvisualized on a logarithmic scale. As the figure shows, theBMO converges faster than the standard approach. Overthe whole run the BMO-fitness develops faster than the fitness of the standard mutation. For both approaches the fit-

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A Test problems
Problem 2.40 - Schwefel's problem 2.40 [21]Minimize:

5f(x) = -Exi
Cr1

Constraints:

gJ()t- E (9 + i)xi + 50000 > 0,
{

, +

j

i=l

Minimum:x2605= (5000, 0, 0, 0, 0)T

forj-1,...,5
for j

6

2606

f(x*) = -5000
92 bis 96 active.

Problem 2.41 - Schwefel's problem 2.41 [211
Minimize:

5

F(x) =-(ixz)
Constraints like problem 2.40.
Minimum:X=(0,0,0,0,)4250000f (X* ) = _ 500
14g, active forj = 1, 2, 3, 4, 6.

Problem g04 - Himmelblau's nonlinear optimization problem. Minimize:f () = 5.3578547X34+0.8356891xlx5+37.293239x1-40792.141
Constraints:=1()=85.334407+0.0056858x2x5+0.00026xlX4-0.0022053x3x5
92(x) = 80.51249+0.0071317x2x5+0.0029955xlx2+0.0021813x32
93(X) = 9.300961+0.0047026x3x3+0.0012547xlX3+0.0019085x3x4
0 < g (7) < 92
90 < 92(x) < 110

20 < 93(x) < 25
78 < xi < 102
33 < X2 < 45
27 < xi < 45 (i = 3,4,5)

Optimum:x = (78.000, 33.000, 29.995, 45.000, 36.776)

f (x*) = -30665.539
Problem g09 Minimize:f(x) = (Xl - 10)2 + 5(X2 - 12)2 + X4 + 3(x4 -11)2+1ox5 + 7x6 + x4

4X6X7 - 1OX6 -8x7

Constraints:2

42

g () = -127 + 2X1 + 3x2 + X3 + 4x4 + 5X5 < 092(X) = -282 + 7x1 + 3X2 + 1OX23 + X4-X5 < 0
196 + 23x, + x2+ 6x2-8x7 < 0
93(X)94(x) = 4x, + X2 - 3x1x2 + 2X3 + 5x6 - 11X7 < 0x* = (78.000,33.000, 29.995,-45000, 36.776)-10 < xi < 10(i= 1.7)
x* = (2.330499, 1.951372, -0,4775414, 4.365726,-0.6244870, 1.038131, 1.594227)

f (x*) = 680.630

2606