FOCUS
Memetic algorithms based on local search chains for large scale
continuous optimisation problems: MA-SSW-Chains
Daniel Molina • Manuel Lozano • Ana M. Sa´nchez •
Francisco Herrera
Published online: 18 September 2010
 Springer-Verlag 2010
Abstract Nowadays, large scale optimisation problems
arise as a very interesting field of research, because they
appear in many real-world problems (bio-computing,
data mining, etc.). Thus, scalability becomes an essential
requirement for modern optimisation algorithms. In a
previous work, we presented memetic algorithms based on
local search chains. Local search chain concerns the idea
that, at one stage, the local search operator may continue
the operation of a previous invocation, starting from the
final configuration reached by this one. Using this tech-
nique, it was presented a memetic algorithm, MA-CMA-
Chains, using the CMA-ES algorithm as its local search
component. This proposal obtained very good results for
continuous optimisation problems, in particular with
medium-size (with up to dimension 50). Unfortunately,
CMA-ES scalability is restricted by several costly opera-
tions, thus MA-CMA-Chains could not be successfully
applied to large scale problems. In this article we study the
scalability of memetic algorithms based on local search
chains, creating memetic algorithms with different local
search methods and comparing them, considering both the
error values and the processing cost. We also propose a
variation of Solis Wets method, that we call Subgrouping
Solis Wets algorithm. This local search method explores, at
each step of the algorithm, only a random subset of the
variables. This subset changes after a certain number of
evaluations. Finally, we propose a new memetic algorithm
based on local search chains for high dimensionality,
MA-SSW-Chains, using the Subgrouping Solis Wets’
algorithm as its local search method. This algorithm is
compared with MA-CMA-Chains and different reference
algorithms, and it is shown that the proposal is fairly
scalable and it is statistically very competitive for high-
dimensional problems.
Keywords Memetic algorithms  Continuous
optimisation  Large scale problems  Local search chains
1 Introduction
It is now well established that hybridisation of evolutionary
algorithms (EAs) with other techniques can greatly improve
the efficiency of search (Davis 1991; Goldberg and Voessner
1999). EAs that have been hybridised with local search
techniques (LS) are often called memetic algorithms (MAs)
(Krasnogor and Smith 2005; Merz 2000; Moscato 1989,
1999). One commonly used formulation of MAs applies LS
to members of the EA population after recombination and
mutation, with the aim of exploiting the best search regions
gathered during the global sampling done by the EA. Thus,
an important aspect of MAs is the trade-off between the
exploration abilities of the EA and the exploitation abilities
D. Molina (&)
Department of Computer Languages and Systems,
University of Ca´diz, Ca´diz, Spain
e-mail: daniel.molina@uca.es
M. Lozano  F. Herrera
Department of Computer Science and Artificial Inteligence,
University of Granada, Granada, Spain
e-mail: lozano@decsai.ugr.es
F. Herrera
e-mail: herrera@decsai.ugr.es
A. M. Sa´nchez
Department of Software Engineering,
University of Granada, Granada, Spain
e-mail: amlopez@ugr.es
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Soft Comput (2011) 15:2201–2220
DOI 10.1007/s00500-010-0647-2

of the LS technique used (Krasnogor and Smith 2001), i.e.,
MAs should combine their two ingredients following a
hybridisation scheme that allows them to work in a coop-
erative way, ensuring synergy between exploration and
exploitation.
Many real-world problems may be formulated as opti-
misation problems of parameters with variables in contin-
uous domains (continuous optimisation problems). Over
the past few years, an increasing interest has arisen in
solving this kind of problems using different EA models1
(Herrera and Lozano 2000; Herrera et al. 1998; Kennedy
and Eberhart 1995; Price et al. 2005). A common charac-
teristic of these EAs is that they evolve chromosomes that
are vectors of floating point numbers, directly representing
problem solutions (hence, they may be called real-coded
EAs). Nevertheless, for continuous optimisation, an
important difficulty must be addressed: solutions of high
precision must be obtained by the solvers (Kita 2001).
Memetic algorithms comprising efficient local imp-
rovement processes in continuous domains (continuous LS
methods) have been presented to deal with this problem
(Hart 1994; Renders and Flasse 1996). In this paper, they
will be named MACOs (MAs for continuous optimisation
problems). MACO instances employing a real-coded EA as
the EA component and invoking a continuous LS method
have been presented to address the difficulty of obtaining
reliable solutions of high precision for complex continuous
optimisation problems (Hart 1994; Lozano et al. 2004;
Molina et al. 2010; Nguyen et al. 2009; Noman and Iba
2008).
There is a kind of continuous optimisation problems that
is receiving much attention, large scale optimisation prob-
lems, appearing in many real-world problems (bio-com-
puting, data mining, etc.). Unfortunately, the performance of
most available optimisation algorithms deteriorates very
quickly when the dimensionality increases (van den Bergh
and Engelbrencht 2004). Thus, scalability for high-dimen-
sional problems becomes an essential requirement for
modern optimisation algorithms.
In recent years, it has been increasingly recognised that
the influence of the employed continuous LS algorithm has
a major impact on the search performance of MACOs (Ong
and Keane 2004). In particular, the LS method is a com-
ponent directly affected by a high dimensionality. Because
the improvement method explores a region close to the
current solution, with a higher dimension the domain
search increases, and so does the region to explore. This
larger area suggests it is advisable to increase the number
of fitness function evaluations required by the LS algorithm
during its search, called LS intensity. However, a high LS
intensity increases the cost of the LS process, thus, a
MACO should adjust carefully the LS intensity to be
successful for large scale problems.
In a previous work, we proposed a MACO model, MA
with LS Chains (Molina et al. 2010) that employs the
concept of LS chain to adjust the LS intensity, assigning to
each individual an LS intensity that depends on its features,
by chaining different LS invocations. In that model, an
individual resulting from an LS invocation may later
become the initial point of a subsequent LS invocation,
adopting the final strategy parameter values achieved by
the former as its initial ones. In this way, the continuous LS
method may adaptively fit its strategy parameters to the
particular features of the search zones, increasing the LS
effort over the most promising solutions and regions.
An instance of this MACO was experimentally studied,
MA-CMA-Chains, which employed the Covariance Matrix
Adaptation Evolution Strategy, CMA-ES (Hansen and
Ostermeier 1996), as its local optimiser. The results
showed that it was very competitive with the state of the art
in both MACOs and EAs for continuous optimisation
problems. Particularly, significant improvements were
obtained for the problems with the highest dimensionality
among those considered for the empirical study (in par-
ticular, 30 and 50 dimensions), which suggests that the
application of this MACO approach to optimisation prob-
lems with higher dimensionality (large scale optimisation
problems) is indeed worth of further investigation (Molina
et al. 2009).
Unfortunately, CMA-ES is an algorithm that uses sev-
eral operations with a complexity of O(n3), where n is the
dimension value, and although there are versions that try to
reduce this problem, it has not been actually resolved
(Hansen 2009). This behaviour makes CMA-ES an LS
method that does not scale well for large scale optimisation
problems.
In this work, we create different instances of MAs based
on LS chains that differ from MA-CMA-Chains in the LS
method used. By applying a different and more scalable LS
method we can avoid the scalability problem from the
previous LS method and test whether scalable MAs based
on LS chains can be created. We carry out several exper-
iments and compare the different instances among them to
identify which one achieves the best results, both in error
values and in processing cost. The final objective is to
present a new MA based on LS chains that can effectively
tackle large scale problems.
For the scalability study of the LS methods, we have
selected several LS methods from the literature and we
have also proposed a variation of the Solis Wets’ method,
Subgrouping Solis Wets’ algorithm (SSW). This LS
method, instead of exploring all variables at the same time,
it explores, for a certain number of evaluations, a random
1 See the website http://sci2s.ugr.es/EAMHCO/ for a large repository
of approaches to this kind of problems.
2202 D. Molina et al.
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