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Content uploaded by Michael Affenzeller
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All content in this area was uploaded by Michael Affenzeller on Feb 01, 2019
Content may be subject to copyright.
SASEGASA: An Evolutionary Algorithm for
Retarding Premature Convergence by
Self-Adaptive Selection Pressure Steering
Michael Affenzeller, Stefan Wagner
Institute of Systems Science
Systems Theory and Information Technology
Johannes Kepler University
Altenbergerstrasse 69
A-4040 Linz - Austria
ma@cast.uni-linz.ac.at
Abstract. This paper presents a new generic Evolutionary Algorithm
(EA) for retarding the unwanted effects of premature convergence. This
is accomplished by a combination of interacting methods. To be intent on
this a new selection scheme is introduced, which is designed to maintain
the genetic diversity within the population by advantageous self-adaptive
steering of selection pressure. Additionally this new selection model en-
ables a quite intuitive condition to detect premature convergence. Based
upon this newly postulated basic principle the new selection mechanism
is combined with the already proposed Segregative Genetic Algorithm
(SEGA) [3], an advanced Genetic Algorithm (GA) that introduces par-
allelism mainly to improve global solution quality. As a whole, a new
generic evolutionary algorithm (SASEGASA) is introduced. The perfor-
mance of the algorithm is evaluated on a set of characteristic benchmark
problems. Computational results show that the new method is capable
of producing highest quality solutions without any problem-specific ad-
ditions.
1 Introduction
Evolutionary Algorithms (EAs) may be described as a class of bionic techniques
that imitate the evolution of a species. The most important representatives of
EAs are Genetic Algorithms (GAs) and Evolution Strategies (ES).
The fundamental principles of GAs were first presented by Holland [6]. Since
that time GAs have been successfully applied to a wide range of problems in-
cluding multimodal function optimization, machine learning, and the evolution
of complex structures such as neural networks. An overview of GAs and their
implementation in various fields is given by Goldberg [5] or Michalewicz [9].
Evolution Strategies, the second major representative of EAs, were intro-
duced by Rechenberg [10] and Schwefel [13]. Applied to problems of combinato-
rial optimization, ES tend to find local optima quite efficiently. Though, in the
case of multimodal test functions, global optima can be detected by ES only if
one of the starting values is located in the absorbing region of a global optimum.
The advantage of applying GAs to hard problems of combinatorial optimiza-
tion lies in the ability to search the solution space in a broader way than heuristic
methods based upon neighborhood search. Nevertheless, also GAs are frequently
faced with a problem which, at least in its impact, is quite similar to the problem
of stagnating in a local but not global optimum. This drawback, called prema-
ture convergence in the terminology of GAs, occurs when the population of a
GA reaches such a suboptimal state that the genetic operators can no longer
produce offspring that outperform their parents (e.g. [4]).
Inspired by Rechenberg’s 1/5 success rule for Evolution Strategies, we have
developed an advanced selection model for Genetic Algorithms that allows self-
adaptive control of selection pressure in a quite intuitive way. Based upon this
enhanced EA-model further generic extensions are being discussed.
The experimental part analyzes the new algorithms for the Traveling Sales-
man Problem (TSP) as a very well documented instance of a multimodal com-
binatorial optimization problem. In contrast to all other evolutionary heuristics
known to the authors that do not use any additional problem specific informa-
tion, we obtain the best known solution for all considered benchmarks.
2 The Self-Adaptive Selection Model
As there exists no manageable model for a controllable handling of selection
pressure within the theory of GAs[12], we have introduced some kind of inter-
mediate step (a ’virtual population’) into the selection process which provides a
handling of selection pressure very similar to that of ES [2]. As we have exemplar-
ily pointed out in contribution [2], the most common replacement mechanisms
can easily be implemented in this intermediate selection step. Furthermore, this
Evolution Strategy like variable selection pressure supported us to steer the de-
gree of population diversity. However, within this model it is necessary to adjust
a parameter for the actual selection pressure and in order to steer the search
process advantageously a lot of parameter tuning is essential.
Motivated by those considerations we have set up an advanced selection
model by introducing a new criterion abutted on Rechenberg’s 1/5 success rule.
The first selection step chooses the parents for crossover in the well-known way of
Genetic Algorithms by roulette wheel, linear-rank, or some kind of tournament
selection strategy. After having performed crossover with the selected parents
we introduce a further selection mechanism that considers the success of the
applied crossover in order to assure the proceeding of genetic search mainly with
successful offspring in that way that the used crossover operator was able to
create a child that surpasses its parents’ fitness.
In doing so, a new parameter, called success ratio (SuccRatio ∈[0,1]), gives
the quotient of the next population members that have to be generated by
successful mating in relation to the total population size. Our adaptation of
Rechenberg’s success rule for GAs says that a child is successful if its fitness is
better than the fitness of its parents, whereby the meaning of ’better’ has to
be explained in more detail: Is a child better than its parents, if it surpasses
the fitness of the weaker, the better, or is it some kind of mean value of both?
For this problem we have decided to introduce a Simulated Annealing (SA) like
cooling strategy. Following the basic principle of SA we claim that a successful
descendent has to surpass the fitness value of the worse parent at the beginning
and while evolution proceeds the child has to be better than a fitness value con-
tinuously increasing in the range between the fitness of the weaker and the better
parent. Like in the case of SA this strategy effects a broader search at the begin-
ning whereas at the end of the search process this operator acts in a more and
more directed way. Having filled up the claimed ratio (SuccRatio) of the next
generation with successful individuals in the above meaning we simply fill up the
rest of the next generation ((1−SuccRatio)∗ |P O P |) with individuals arbitrarily
chosen from the pool of individuals that were also created by crossover but did
not reach the success critereon. The actual selection pressure ActSelP r ess at the
end of a single generation is defined by the quotient of individuals that had to
be considered until the success ratio was reached and the number of individuals
in the population ActS elP ress =|virtualP OP |+S uccRatio∗|P OP |
P OP . Fig. 1 shows the
operating sequence of the above described concepts.
child
'better' than
parents ?
yes
no
selection (roulette, linear rank, tournament, …)
crossover
mutation
fill up rest of next population after
(|POP| SuccRatio) is full
|POP|
population at
generation i
. . . . . . . .
|virtual POP|
. . . . . . . . virtual
population
|POP|
|POP| SuccRatio |POP| (1-SuccRatio)
population at
generation i+1
. . . . . . .. . . . .
Fig. 1. Flowchart for embedding the new selection principle into a GA
With an upper limit of selection pressure (M axSel P ress), defining the max-
imum number of children considered for the next generation that may be pro-
duced in order to fulfill the success ratio, this new model also acts as a pre-
cise detector of premature convergence: If it is not possible anymore to find
enough (SuccRatio ∗ |P O P |) children outperforming their own parents, even if
(M axSelP ress ∗|P OP |) candidates have been generated premature convergence
has occurred.
As a basic principle of this selection model a higher success ratio causes higher
selection pressure. Nevertheless, higher settings of success ratio and therefore of
selection pressure do not essentially cause premature convergence. This is be-
cause per definition the new selection step (after crossover) does not accept clones
that emanate from two identical parents. In traditional GAs such clones repre-
sent a major reason for premature convergence of the whole population around a
suboptimal value, whereas the new selection step specifically counteracts against
this phenomenon.
Experiments performed on the variable selection pressure model already in-
dicate the supremacy of this approach. Also the corresponding canonical Genetic
Algorithm is fully included in our new superstructure if the success ratio is sim-
ply set to 0. Furthermore, we are going to discuss new aspects and models built
up upon the described self adaptive selection model In the following section.
3 Generic GA-Concepts Based Upon the Self-Adaptive
Selection Model
When applying Genetic Algorithms to complex problems, one of the most fre-
quent difficulties is premature convergence. Concisely speaking, premature con-
vergence occurs when the population of a Genetic Algorithm reaches such a
suboptimal state that the genetic operators can no longer produce offspring that
outperform their parents (e.g. [4]).
A critical problem in studying premature convergence is the identification
of its occurrence and the measure of its extent. Srinivas and Patnaik [15], for
example, use the difference between the average and maximum fitness as a stan-
dard to measure premature convergence and adaptively vary the crossover and
mutation probabilities according to this measurement. On the other hand, as in
the present paper, the term ’population diversity’ has been used in many pa-
pers to study premature convergence (e.g. [14]) where the decrease of population
diversity is considered as the primary reason for premature convergence.
The following generic extensions, that are built up upon the self-adaptive
variable selection pressure model, aim to retard premature convergence in a
general way.
3.1 SASEGASA: The Core Algorithm
In principle, the new SASEGASA (Self Adaptive SEgregative Genetic Algorithm
with Simulated Annealing aspects) introduces two enhancements to the basic
concept of Genetic Algorithms. Firstly, we bring in a variable selection pressure,
as described in section 2, in order to control the diversity of the evolving popu-
lation. The second concept introduces a separation of the population to increase
the broadness of the search process and joins the subpopulation after their evo-
lution in order to end up with a population including all genetic information
sufficient for locating a global optimum.
The aim of dividing the whole population into a certain number of subpopu-
lations (segregation) that grow together in case of stagnating fitness within those
subpopulations (reunification) is to combat premature convergence which is the
source of GA-difficulties. This segregation and reunification approach is an effi-
cient method to overcome premature convergence [1] called the SEGA algorithm
(SEgregative GA).
The principle idea of SEGA is to divide the whole population into a certain
number of subpopulations at the beginning of the evolutionary process. These
subpopulations evolve independently from each other until the fitness increase
stagnates in all subpopulations because of too similar individuals within the
subpopulations, i.e. local premature convergence. Then a reunification from n
to (n−1) subpopulations is performed by joining an appropriate number of
adjacent subpopulation members.
Metaphorically speaking this means, that the a certain number of villages
(subpopulations) at the beginning of the evolutionary process are slowly grow-
ing together to bigger cities, ending up with one big town containing the whole
population at the end of evolution. By this approach of width-search essential
building blocks can evolve independently in different regions of the search space
at the beginning and during the evolutionary process. In the case of a standard
GA those building blocks are likely to disappear early and, therefore, their ge-
netic information can not be provided at a later phase of evolution, when the
search for a global optimum is of paramount importance.
Within the classical SEGA algorithm there is no criterion to detect prema-
ture convergence and there is also no self-adaptive selection pressure steering
mechanism. Even if the results of SEGA are quite good with regard to global
convergence it requires an experienced user to adjust the selection pressure steer-
ing parameters and as there is no criterion to detect premature convergence the
dates of reunification have to be implemented statically.
Equipped with our new self adaptive selection technique we have both: A
self-adaptive selection pressure (depending on the given success ratio) as well as
an automated detection of local premature convergence, if the current selection
pressure becomes higher then the given maximal selection pressure parameter
(M axSelP ress). Therefore, a date of reunification has to be set, if local pre-
mature convergence has occurred within all subpopulations in order to increase
genetic diversity again.
Again, like in the context of the new selection model which is included in
SASEGASA as well, it should be pointed out that a corresponding Genetic
Algorithm is unrestrictedly included in SASEGASA, when the number of sub-
populations (villages) is set to 1 and the success ratio is set to 0 at the beginning
of the evolutionary process. Moreover, the introduced techniques also do not use
any problem specific information.
4 Experimental Results
Empirical studies with different problem classes and instances are the most ef-
fective way to analyze the potential of heuristic optimization searches like Evo-
lutionary Algorithms.
In our experiments, all computations are performed on a Pentium 4 PC with
1 GB RAM. The programs are written in the C# programming language. For
the tests we have selected the Traveling Salesman Problem (TSP) as a well doc-
umented instance of a typical multimodal combinatorial optimization problem.
We have tested the new concepts on a selection of symmetric as well as asym-
metric TSP benchmark problem instances taken from the TSPLIB [11] using
updated results for the best or at least the best known solutions. In all exper-
iments, the results are represented as the relative difference to the best known
solution defined as relativeDifference = ( Result
Optimal −1) ∗100%.
Especially we aim to point out the main effect of the present contribution
- namely that an increasing number of subpopulations at the beginning of the
evolutionary process allows to achieve scalable improvements in terms of global
convergence. As Tab. 1 shows, the global solution can be scaled up to highest
quality by just increasing the number of evolving subpopulations. This definitely
represents a new achievement in the area of Evolutionary Algorithms with dis-
tributed subpopulations. For all the experiments in Tab. 1 the starting size of
one subpopulation is fixed at a value of 100, mutation rate is 5%, and the upper
limit of selection pressure is set to a value of 10 for the TSPs respectively 15 for
the ATSPs.
The results in Tab. 1 present the best solution quality of five runs of each
test-instance as well as the average best result-value of the five runs in terms of
average difference to the best known solution.
Indeed, as Tab. 1 shows, the optimal solution could be found for all bench-
mark test cases if the initial number of subpopulations is set high enough.
Even if the achieved results are clearly superior to most of the results re-
ported for applications of Evolutionary Algorithms to the TSP [8], it has to be
pointed out again, that all introduced and applied additions to a standard evo-
lutionary algorithm are generic and absolutely no problem specific local pre- or
post-optimization techniques have been applied in our experiments. Additional
experiments performed on non-standardized scheduling problems (job-shop and
multiprocessor) show comparable potential and underscore the generic potential
of the new techniques in various fields of applications.
5 Conclusion
In this paper an enhanced Genetic Algorithm and two upgrades have been pre-
sented and exemplarily tested on some TSP benchmarks. The proposed EA-
based techniques couple aspects from Evolution Strategies (selection pressure
and success rule in our new selection procedure), Simulated Annealing (growing
selective pressure) as well as a special segregation and reunification strategy with
crossover and mutation in the general model of a Genetic Algorithm. Therefore,
established crossover and mutation operators for certain problems may be used
analogously to the corresponding Genetic Algorithm. The investigations in this
paper have mainly focused on the avoidance of premature convergence and on
Table 1. Experimental results of the SASEGASA-algorithm for TSPLIB benchmark
problems with variable number of subpopulations for tuning global solution quality.
Problem Parameters and Operators Results (in %)
noOf SubP opul ations SuccessRatio Crossover Iterations Best Average
berlin52 1 0,8 OX,ERX,COSA 139 6.7 8.6
berlin52 5 0,8 OX,ERX,COSA 218 0.0 0.0
berlin52 10 0,8 OX,ERX,COSA 301 0.0 0.0
ch130 1 0,8 OX,ERX,COSA 295 52.9 59.4
ch130 5 0,8 OX,ERX,COSA 584 12.2 13.24
ch130 10 0,8 OX,ERX,COSA 783 4.8 6.11
ch130 20 0,8 OX,ERX,COSA 1024 1.6 2.5
ch130 40 0,8 OX,ERX,COSA 1426 0.63 0.89
ch130 80 0,8 OX,ERX,COSA 2067 0.45 0.74
ch130 160 0,8 OX,ERX,COSA 3518 0.0 0.15
kroA200 1 0,8 OX,ERX,COSA 584 77.6 87.3
kroA200 5 0,8 OX,ERX,COSA 1035 22.5 24.9
kroA200 10 0,8 OX,ERX,COSA 1310 12.4 13.1
kroA200 20 0,8 OX,ERX,COSA 1604 4.7 7.4
kroA200 40 0,8 OX,ERX,COSA 2243 0.9 2.6
kroA200 80 0,8 OX,ERX,COSA 2842 0.6 1.3
kroA200 160 0,8 OX,ERX,COSA 4736 0.0 0.3
rbg323 1 0,8 OX,ERX,COSA 1463 20.59 26.24
rbg323 5 0,8 OX,ERX,COSA 2690 4.37 8.02
rbg323 10 0,8 OX,ERX,COSA 5991 2.11 2.84
rbg323 20 0,8 OX,ERX,COSA 13456 0.30 0.53
rbg323 40 0,8 OX,ERX,COSA 40762 0.15 0.20
rbg323 80 0,8 OX,ERX,COSA 100477 0.00 0.06
rbg323 160 0,8 OX,ERX,COSA 212418 0.00 0.00
the introduction of methods which make the algorithm more open for scalability
in terms of solution quality versus computation time.
Concerning the speed of SASEGASA, it has to be pointed out that the supe-
rior performance concerning global convergence requires a higher computation
time, mainly because of the greater total population size |P O P |and because of
the increase of generated individuals in our new self adaptive selection mecha-
nism under higher selection pressure. Nevertheless, in contrast to other imple-
mentations in the field of evolutionary computation, it is absolutely remarkable,
that it has become possible to almost linearly improve the global solution qual-
ity by introducing greater population sizes and an accordingly greater number
of subpopulations, whereas the computational costs are ’only’ increasing lin-
early due to the greater number of individuals having to be taken into account.
This allows to transfer already developed GA-concepts to increasingly power-
ful computer systems in order to achieve better results. Using parallel computer
architectures seems especially suited to increase the performance of SASEGASA.
Anyway, with special parameter settings the corresponding Genetic Algo-
rithm is fully included within the introduced concepts achieving a performance
only marginally worse than the performance of the equivalent Genetic Algorithm.
In other words, the introduced models can be interpreted as a superstructure
of the GA model or as a technique upwards compatible to Genetic Algorithms.
Therefore, an implementation of the new algorithm(s) for a certain problem
should be quite easy to do, presumed that the corresponding Genetic Algorithm
(coding, operators) is known.
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