How parameters have been chosen in metaheuristics: A survey
Eddy Janneth Mesa Delgado
Applied computation. Systems School. Faculty of Mines.
National University of Colombia. Medellín. Cra. 80 x Cl. 65. Barrio Robledo.
ejmesad@unal.edu.co
Abstract:
Keywords: Global Optimization, Heuristic Programming, nonlinear programming,
parameters.
1. INTRODUCTION
The difficult of real-world optimization problems has
increased over time (shorter times to find solutions,
more variables, greater accuracy is requires, etc.).
Classic methods of optimization (based on
derivatives, indirect methods) are not enough to solve
all optimization problems. So, direct strategies were
developed, despite they can’t guarantee an optimal
solution. The first direct methods implemented were
systematic searches over solutions spaces. Step by
step, they evolved to be more and more effective
methods until to arrive to artificial intelligence.
Those new methods were called heuristics and
metaheuristics (Himmelblau, 1972).
Heuristics mean “to find” or “to discover by trial and
error”. Metaheuristics mean a better way “to find”
(Yang, 2010). Besides to intelligence, metaheuristics
differs from other kind of direct methods because
they use parameters to adjust the search and the
strategy to enhance them. Parameters let algorithms
to get good answers in a bigger spam of problems
than initial direct methods with less effort (Zanakis
and Evans, 1981; Zilinskas and Törn, 1989) e.g.
evolutionary strategies are used to get an initial
solution for an unknown problem. By the way,
parameters are a weakness too, because they need to
be tuning and controlling for every problem. I.e. a
wrong set of parameters can make metaheuristics fail
(Angeline, 1995; E. Eiben, et al., 1999; Kramer,
2008; Yang, 2010).
The aim of this paper is to summarize the
information available for parameterization of
metaheuristics, how them are controlled and tuned
and the effect over performance in some cases. To do
that, initially, three key words (parameters,
metaheuristics and heuristics) were used to search in
SCOPUS database. After that, a filter was made
manually including papers that contain:
 Applied metaheuristics with adaptive and
self-adaptive parameters
 Presentation and comparison of new
methods to adapt or to automate parameters.
 Classifications and formalization of methods
to control and tune parameters.
With these criteria 49 papers were chosen.
This paper is organized as follows: parameters are
introduced in section 2. The parameters classification
found in the literature are described in section 3 and
Section 4, the first part present the parameters
relevance in the heuristics to categorize the
parameters choice and the second part uses the
technique to choose the parameters to indicate the
type of setting. An example of adaptation and selfadaptation are shown in Section 6. Finally, general
conclusions and future works are presented in section
7.
2.
PARAMETERS
Stochastic methods have not the same pathway each
run of algorithms. Also, the intelligence immerse in
metaheuristics change dynamically from the
characteristics found in the search. Parameters give
to metaheuristics robustness and flexibility (Talbi,
2009).
To choose parameters is not a simple task because a
wrong set of parameters would make that
metaheuristics have premature convergence or even
no convergence at all (Bäck, 1996; Talbi, 2009).
Metaheuristics need to have enough parameters to be
flexible, but each parameter increase the complexity
to the tune and control of the method and the
necessary changes to optimize a new problem. Each
metaheuristic is a complete different world, has
different parameters, and they influence the
metaheuristic in different ways. There is not a unique
right to choice parameters in metaheuristics. In
literature two points of view are related, but they are
only for Evolutionary Computing (EC).
Traditionally, EC includes genetic algorithms (GA),
evolution strategies (ES), and evolutionary
programming (EP). By the way, other metaheuristics
met the conditions propose by Angeline (1995) to be
an EC metaheuristic, method that iterated until
specific stopping criteria. So, we would extend this
classification to another metaheuristics and heuristics
in general, but they do not use the same operators
that EC ( Fogel, D. B., et al, 1991). Specifically, most
part of another heuristics has not operators
formalized as operators, they use a complete rules
with parameters inside, so maybe it is not a direct
analogy (Zapfel, et al., 2010). In the next two
sections the formal parameter classification proposed
in literature was discussed and follow an example of
other heuristic adaptation are presented.
3.
PARAMETERS CLASSIFICATION
Each metaheuristic has a parameter’s set to control
the search. Although there isn’t consensus about a
unique classification to metaheuristic parameters, one
approximation had been developed for EC (Angeline,
1995; Hinterding,et al., 1997; Kramer, 2008).
According with the definition provided by Angeline
(1995) this classification could be extended to others
paradigms.
The most complete classification is proposed by
Kramer (2008) who links Angeline (1995) point of
view and Hinterding et al. (1997)
2.1. Exogenous.
These parameters are those whose are affected by
metaheuristic performance, but they are external to it,
for example: constrain changes, problems with parts
functions.
2.2. Endogenous
These are internal to the method and could be change
by the user or method itself.
Even though,
Endogenous parameters are our focus, we cannot
forget the exogenous ones because they will affect
the choice.
Population level: In this level the parameters are
global for the optimization. Operator of EC that use
this type or parameters control the next generation.
For example: population size, stopping criteria, etc.
Individual level .This kind of parameters only affects
each individual, for example: It could be the pass for
each individual.
Component level. In this level, the parameters affect
part of the individual like a gene of a chromosome in
Generic Algorithms (GA).
Also, it is important to notice that, authors propose
this classification, but they do not talk about a right
or unique manner to adapt and automatize each level
of parameters. The techniques are addressed by next
classification.
4.
PARAMETERS SETTINGS
In last section, parameters were changed according to
their level of sort in the metaheuristic. In this case,
the parameters are chosen in two stages the first is
before to use metaheuristic, and it is called “to tune”
and the second one is called “to control” and both
have different ways to be select.
2.1. Tuning.
Parameters are tuned before to use metaheuristic.
Those initial parameters could be chosen by three
different levels accord Eiben, et al (1999).
Manually: The initial parameters could be chosen by
and expert ad hoc it is a right manner but lately are
not the most recommended (Kramer, 2008; Talbi,
2009)
Design of experiments (DOE): It implies design test
to show the behavior, and use or define a metric
analyze the result and take a decision about a value
or a range of values for each parameter. (Kramer,
2008; Talbi, 2009).
Metaevolutionary: It means that parameters are
chosen by other metaheuristics, in literature a good
example is given by bacterial chemotaxis method
proposed in (Müller, et al., 2000). Additionally,
Kramer (2008) extend this procedure to control like
in Hyper-metaheuristics (Hamadi, et al., 2011)
2.2. Control
Parameters can change while metaheuristic is
running. Control process is this change in “real
time”. Accord to Hinterding et al. (1997) and Eiden
et al (1999), Kramer (2008) describes and discusses
about three methods to control parameters.
Deterministic: It could be static or dynamics. Statics
means there is no change at all and dynamic it
change with a specific rule like a dynamic penalty
than change with the distance to feasible zone
(Kramer, 2008; Mesa, 2010)
Adaptive: In this case, parameter changes agree a rule
like if it happens, then do this. A classic example of
adaptive parameters is the 1/5 th rule for mutation use
in ES (Bäck, 1996; Kramer, 2008).
Self-adaptive: In this type of control, parameters
evolves agree to the problem for example the selfadaptation of the mutation step-size with the
direction to mutation (Kramer, 2008; Schwefel,
1995)
5. AN EXAMPLE OF THE ADAPTATIVE AND
SELF-ADAPTATIVE PARAMETERS FOR
DIFFERENTIAL EVOLUTION (DE).
Besides the metaheuristics covered by EC, there are
at least twenty different metaheuristics. Additionally,
they have different approaches and hybrids. In the
sake of brevity, only Differential Evolution (DE)
parameters adaptation is review and just one
approach of self-adaptation is present.
DE is a metaheuristic proposed by Price (1996). It
was classified as evolutionary metaheuristic, but DE
does not use formal operators (crossover, mutation)
like other EC algorithms. This metaheuristic have
certain characteristics that make a good example to
study: conceptually are near to EC, have one
complete rule but they identify same operators that
have EC, mutation and crossover; have few
parameters, it is widely known and have selfadaptive parameter approach.
This metaheuristic (DE) has different versions; five
classic versions for this metaheuristics are present in
(Mezura-Montes, et al., 2006). Brest, et al.( 2006)
use the version called DE/rand/1/bin to propose their
idea of self-adaptive parameters.
DE parameters for DE/rand/1/bin version are:
 Size of population 𝑁𝑃(population-level)
 Scale factor 𝐹(Individual-level)
 Crossover parameters 𝐶𝑅(component-level)
 Minimal border 𝐱𝐦𝐢𝐧
 Maximal border 𝐱𝐦𝐚𝐱
 Number of generation 𝐺 (population-level)
 Dimensions 𝐷
 Trial vector 𝐯 (individual-level)
 Randomly chosen index(an random integer
∈ [1, 𝐷]) 𝑟𝑛(𝑖), 𝑟1, 𝑟2, 𝑟3
 Random number 𝑟(𝑗)
Figure 1 shows the pseudocode for DE, 𝐱 is a
solution vector and 𝑓(⋅) is the value of goal function.
Line 04 is mutation operator (individual-level). Line
06 is the complete heuristic rule. Lines 07-09 are a
borders handle when there a component outside the
feasible zone is forced to go in feasible solution
(exogenous-level).
01 Start
02 for 𝑡 = 1 until 𝐺
03 for 𝑖 = 1 until 𝑁𝑃
04
𝐯𝑖 = 𝐱 𝑟1 + 𝐹(𝐱 𝑟2 − 𝐱 𝑟3 ) ∴ 𝑟1 ≠ 𝑟2 ≠ 𝑟3 ≠ 𝑖
05
for 𝑗 = 1 until 𝐷
𝑣𝑖𝑗 if 𝑟(𝑗) ≤ 𝐶𝑅 ∨ 𝑗 = 𝑟𝑛(𝑖)
06
𝑢𝑖𝑗 = {
𝑥𝑖𝑗 if 𝑟(𝑗) > 𝐶𝑅 ∧ 𝑗 ≠ 𝑟𝑛(𝑖)
07
if 𝑢𝑖𝑗 < 𝑥𝑚𝑖𝑛𝑗 ∨ 𝑢𝑖𝑗 > 𝑥𝑚𝑎𝑥𝑗 then
08
𝑢𝑖𝑗 = 𝑟𝑎𝑛𝑑[𝑥𝑚𝑖𝑛𝑗 , 𝑥𝑚𝑎𝑥𝑗 ]
09
end if
08
𝑗 =𝑗+1
09
end for
𝐯 if 𝑓(𝐯𝑖 ) < 𝑓(𝐱 𝑖 )
10
𝐱 𝑖 (𝑡 + 1) = { 𝑖
𝐱𝑖
otherwise
11 end for
12 end for
13 end algorithm
Fig. 1. Pseudocode of DE.
Original algorithm tuned parameter manually based
on a few test, they recommended 𝐶𝑅 = [0,1] ∈ ℝ ,
𝑁𝑃 = [10,50] ∈ ℤ, 𝐹 = [0,1.2] ∈ ℝ, the other
parameters are random or given by the problem. The
parameters control deterministically (statics).
Later, one of the approaches to control parameters is
proposed by Ali and A. Törn (2004) they fixed
𝑁𝑃 = 10 ∗ 𝐷, and 𝐶𝑅 = 0.5 and adapt 𝐹 using:
𝐹={
𝑓
𝑓
𝑓𝑚𝑖𝑛
𝑓𝑚𝑖𝑛
𝑓𝑚𝑖𝑛
max (𝑙𝑚𝑖𝑛 , 1 − | 𝑚𝑎𝑥 |) if | 𝑚𝑎𝑥 | < 1
max (𝑙𝑚𝑖𝑛 , 1 − |
𝑓𝑚𝑎𝑥
|)
(1)
framework to set the parameters is important because
it could help to determine which parameters need
more control than others before long tests. Anyway,
this review just gives an initial idea of the
possibilities. There is a lot of work to do in this way.
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otherwise
Where 𝑓𝑚𝑖𝑛 and 𝑓𝑚𝑎𝑥 are the minimum and
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