Why Is Optimization Difficult?

Abstract

This chapter aims to address some of the fundamental issues that are often encountered in optimization problems, making them
difficult to solve. These issues include premature convergence, ruggedness, causality, deceptiveness, neutrality, epistasis,
robustness, overfitting, oversimplification, multi-objectivity, dynamic fitness, the No Free Lunch Theorem, etc. We explain
why these issues make optimization problems hard to solve and present some possible countermeasures for dealing with them.
By doing this, we hope to help both practitioners and fellow researchers to create more efficient optimization applications
and novel algorithms.

Full text

Why Is Optimization Difficult?
Thomas Weise, Michael Zapf, Raymond Chiong, and Antonio J. Nebro
Abstract This chapter aims to address some of the fundamental issues that
are often encountered in optimization problems, making them difficult to
solve. These issues include premature convergence, ruggedness, causality, de-
ceptiveness, neutrality, epistasis, robustness, overfitting, oversimplification,
multi-objectivity, dynamic fitness, the No Free Lunch Theorem, etc. We ex-
plain why these issues make optimization problems hard to solve and present
some possible countermeasures for dealing with them. By doing this, we hope
to help both practitioners and fellow researchers to create more efficient op-
timization applications and novel algorithms.
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Thomas Weise ·Michael Zapf
Distributed Systems Group, University of Kassel, Wilhelmsh¨oher Allee 73, 34121 Kassel,
Germany, e-mail: weise@vs.uni-kassel.de and zapf@vs.uni-kassel.de
Raymond Chiong
School of Computing & Design, Swinburne University of Technology (Sarawak Campus),
93350 Kuching, Sarawak, Malaysia, e-mail: rchiong@swinburne.edu.my
Antonio J. Nebro
Dept. Lenguajes y Ciencias de la Computaci´on, ETSI Inform´atica, University of M´alaga,
Campus de Teatinos, 29071 M´alaga, Spain, e-mail: antonio@lcc.uma.es
1

2 Weise, Zapf, Chiong, Nebro
1 Introduction
Optimization, in general, is concerned with finding the best solutions for a
given problem. Its applicability in many different disciplines makes it hard
to give an exact definition. Mathematicians, for instance, are interested in
finding the maxima and minima of a real function from within an allowable
set of variables. In computing and engineering, the goal is to maximize the
performance of a system or application with minimal runtime and resources.
In the business industry, people aim to optimize the efficiency of a production
process or the quality and desirability of their current products.
All these examples show that optimization is indeed part of our everyday
life. We often try to maximize our gain by minimizing the cost we need to
bear. However, are we really able to achieve an “optimal” condition? Frankly,
whatever problems we are dealing with, it is rare that the optimization pro-
cess will produce a solution that is truly optimal. It may be optimal for one
audience or for a particular application, but definitely not in all cases.
As such, various techniques have emerged for tackling different kinds of
optimization problems. In the broadest sense, these techniques can be classi-
fied into exact and stochastic algorithms. Exact algorithms, such as branch
and bound, A⋆search, or dynamic programming can be highly effective for
small-size problems. When the problems are large and complex, especially
if they are either NP-complete or NP-hard, i.e., have no known polynomial-
time solutions, the use of stochastic algorithms becomes mandatory. These
stochastic algorithms do not guarantee an optimal solution, but they are able
to find quasi-optimal solutions within a reasonable amount of time.
In recent years, metaheuristics, a family of stochastic techniques, has be-
come an active research area. They can be defined as higher level frameworks
aimed at efficiently and effectively exploring a search space [25]. The initial
work in this area was started about half a century ago (see [175, 78, 24], and
[37]). Subsequently, a lot of diverse methods have been proposed, and to-
day, this family comprises many well-known techniques such as Evolutionary
Algorithms, Tabu Search, Simulated Annealing, Ant Colony Optimization,
Particle Swarm Optimization, etc.
There are different ways of classifying and describing metaheuristic al-
gorithms. The widely accepted classification would be the view of nature-
inspired vs. non nature-inspired, i.e., whether or not the algorithm some-
how emulates a process found in nature. Evolutionary Algorithms, the most
widely used metaheuristics, belong to the nature-inspired class. Other tech-
niques with increasing popularity in this class include Ant Colony Optimiza-
tion, Particle Swarm Optimization, Artificial Immune Systems, and so on.
Scatter search, Tabu Search, and Iterated Local Search are examples of non
nature-inspired metaheuristics. Unified models of metaheuristic optimization
procedures have been proposed by Vaessens et al [220, 221], Rayward-Smith
[169], Osman [158], and Taillard et al [210].

Why Is Optimization Difficult? 3
In this chapter, our main objective is to address some fundamental issues
that make optimization problems difficult based on the nature-inspired class
of metaheuristics. Apart from the reasons of being large, complex, and dy-
namic, we present a list of problem features that are often encountered and
explain why some optimization problems are hard to solve. Some of the is-
sues that will be discussed, such as multi-modality and overfitting, concern
global optimization in general. We will also elaborate on other issues which
are often linked to Evolutionary Algorithms, e.g., epistasis and neutrality,
but can occur in virtually all metaheuristic optimization processes.
These concepts are important, as neglecting any one of them during the
design of the search space and operations or the configuration of the opti-
mization algorithms can render the entire invested effort worthless, even if
highly efficient optimization methods are applied. To the best of our knowl-
edge, to date there is not a single document in the literature comprising all
such problematic features. By giving clear definitions and comprehensive in-
troductions on them, we hope to create awareness among fellow scientists as
well as practitioners in the industry so that they could perform optimization
tasks more efficiently.
The rest of this chapter is organized as follows: In the next section, prema-
ture convergence to local minima is introduced as one of the major symptoms
of failed optimization processes. Ruggedness (Section 3), deceptiveness (Sec-
tion 4), too much neutrality (Section 5), and epistasis (Section 6), some of
which have been illustrated in Fig. 11, are the main causes which may lead
to this situation. Robustness, correctness, and generality instead are features
which we expect from valid solutions. They are challenged by different types
of noise discussed in Section 7 and the affinity of overfitting or overgeneral-
ization (see Section 8). Some optimization tasks become further complicated
because they involve multiple, conflicting objectives (Section 9) or dynami-
cally changing ones (Section 10). In Section 11, we give a short introduction
into the No Free Lunch Theorem, from which we can follow that no panacea,
no magic bullet can exist against all of these problematic features. We will
conclude our outline of the hardships of optimization with a summary in
Section 12.
1.1 Basic Terminology
In the following text, we will utilize a terminology commonly used in Evolu-
tionary Algorithms community and sketched in Fig. 2 based on the example
1We include in Fig. 1 different examples of fitness landscapes, which relate solution can-
didates (or genotypes) to their objective values. The small bubbles in Fig. 1 represent
solution candidates under investigation. An arrow from one bubble to another means that
the second individual is found by applying one search operation to the first one. The
objective values here are subject to minimization.

4 Weise, Zapf, Chiong, Nebro
objectivevaluesf(x)
x
Fig. 1.a: Best Case
objectivevaluesf(x)
x
Fig. 1.b: Low Total Variation
multiple(local)optima
objectivevaluesf(x)
x
Fig. 1.c: Multimodal
nousefulgradientinformation
? ??
objectivevaluesf(x)
Fig. 1.d: Rugged
regionwithmisleading
gradientinformation
objectivevaluesf(x)
x
Fig. 1.e: Deceptive
?
neutralarea
objectivevaluesf(x)
x
Fig. 1.f: Neutral
??
neutral area or
areawithoutmuch
information
needle
(isolated
optimum)
objectivevaluesf(x)
x
Fig. 1.g: Needle-In-A-Haystack
?
objectivevaluesf(x)
x
Fig. 1.h: Nightmare
Fig. 1: Different possible properties of fitness landscapes (minimization).

Why Is Optimization Difficult? 5
(1,3)
(3,3)
(0,2)
(0,2)
(0,2)
(0,2)
Genotype-PhenotypeMapping
ObjectiveFunction(s)
(0,0) (0,1) (0,2) (0,3)
(1,0) (1,1) (1,2) (1,3)
(2,0) (2,1) (2,2) (2,3)
(3,0) (3,1) (3,2) (3,3)
ProblemSpace X
SearchSpace G
0100
1010
1100 1101
0000 1000 1001
0101
0110 1110 1111
0111
0001
0010 0011 1011
ObjectiveSpace Rn
Genotype-PhenotypeMapping
ObjectiveFunction(s)
(0,0) (0,1) (0,3)
(1,0) (1,1) (1,2) (1,3)
(2,0) (2,1) (2,2) (2,3)
(3,0) (3,1) (3,2) (3,3)
Population(Phenotypes)
Population(Genotypes)
ObjectiveValues
1111
1111
1110
1000
0100
0111
0111
0010
0010
0010
0010
1111
f (x)
1
f (x)
1
fn
(xÎX)ÎR
TheInvolvedSpaces TheInvolvedSets/Elements
Fig. 2: The involved spaces and sets in optimization.
of a simple Genetic Algorithm. The possible solutions xof an optimization
problem are elements of the problem space X. Their utility as solutions is
evaluated by a set fof objective functions fwhich, without loss of general-
ity, are assumed to be subject to minimization. The set of search operations
utilized by the optimizers to explore this space does not directly work on
them. Instead, they are applied to the elements (the genotypes) of the search
space G(the genome). They are mapped to the solution candidates by a
genotype-phenotype mapping gpm : G7→ X. The term individual is used for
both, solution candidates and genotypes.

6 Weise, Zapf, Chiong, Nebro
1.2 The Term “Difficult”
Before we go more into detail about what makes these landscapes difficult,
we should establish the term in the context of optimization. The degree of
difficulty of solving a certain problem with a dedicated algorithm is closely
related to its computational complexity, i.e., the amount of resources such as
time and memory required to do so. The computational complexity depends
on the number of input elements needed for applying the algorithm. This
dependency is often expressed in the form of approximate boundaries with
the Big-O-family notations introduced by Bachmann [10] and made popular
by Landau [122]. Problems can be further divided into complexity classes. One
of the most difficult complexity classes owning to its resource requirements is
NP, the set of all decision problems which are solvable in polynomial time by
non-deterministic Turing machines [79]. Although many attempts have been
made, no algorithm has been found which is able to solve an NP-complete [79]
problem in polynomial time on a deterministic computer. One approach to
obtaining near-optimal solutions for problems in NP in reasonable time is to
apply metaheuristic, randomized optimization procedures.
As already stated, optimization algorithms are guided by objective func-
tions. A function is difficult from a mathematical perspective in this context
if it is not continuous, not differentiable, or if it has multiple maxima and
minima. This understanding of difficulty comes very close to the intuitive
sketches in Fig. 1.
In many real world applications of metaheuristic optimization, the charac-
teristics of the objective functions are not known in advance. The problems
are usually NP or have unknown complexity. It is therefore only rarely possi-
ble to derive boundaries for the performance or the runtime of optimizers in
advance, let alone exact estimates with mathematical precision.
Most often, experience, rules of thumb, and empirical results based on
the models obtained from related research areas such as biology are the only
guides available. In this chapter we discuss many such models and rules,
providing a better understanding of when the application of a metaheuristic
is feasible and when not, as well as with indicators on how to avoid defining
problems in a way that makes them difficult.
2 Premature Convergence
2.1 Introduction
An optimization algorithm has converged if it cannot reach new solution
candidates anymore or if it keeps on producing solution candidates from a

Why Is Optimization Difficult? 7
“small”2subset of the problem space. Global optimization algorithms will
usually converge at some point in time. One of the problems in global opti-
mization is that it is often not possible to determine whether the best solution
currently known is situated on a local or a global optimum and thus, if con-
vergence is acceptable. In other words, it is usually not clear whether the
optimization process can be stopped, whether it should concentrate on re-
fining the current optimum, or whether it should examine other parts of the
search space instead. This can, of course, only become cumbersome if there
are multiple (local) optima, i.e., the problem is multimodal as depicted in
Fig. 1.c.
A mathematical function is multimodal if it has multiple maxima or min-
ima [195, 246]. A set of objective functions (or a vector function) fis multi-
modal if it has multiple (local or global) optima – depending on the definition
of “optimum” in the context of the corresponding optimization problem.
2.2 The Problem
An optimization process has prematurely converged to a local optimum if it
is no longer able to explore other parts of the search space than the area
currently being examined and there exists another region that contains a
superior solution [192, 219]. Fig. 3 illustrates examples for prematurely con-
vergence.
globaloptimum
localoptimum
Fig. 3.a: Example 1: Maximization
objectivevaluesf(x)
x
Fig. 3.b: Example 2: Minimization
Fig. 3: Premature convergence in the objective space.
2according to a suitable metric like numbers of modifications or mutations which need to
be applied to a given solution in order to leave this subset

8 Weise, Zapf, Chiong, Nebro
The existence of multiple global optima itself is not problematic and the
discovery of only a subset of them can still be considered as successful in many
cases (see Section 9). The occurrence of numerous local optima, however, is
more complicated.
The phenomenon of domino convergence has been brought to attention by
Rudnick [184] who studied it in the context of his BinInt problem [184, 213].
In principle, domino convergence occurs when the solution candidates have
features which contribute significantly to different degrees of the total fitness.
If these features are encoded in separate genes (or building blocks) in the
genotypes, they are likely to be treated with different priorities, at least in
randomized or heuristic optimization methods.
Building blocks with a very strong positive influence on the objective val-
ues, for instance, will quickly be adopted by the optimization process (i.e.,
“converge”). During this time, the alleles of genes with a smaller contribution
are ignored. They do not come into play until the optimal alleles of the more
“important” blocks have been accumulated. Rudnick [184] called this sequen-
tial convergence phenomenon domino convergence due to its resemblance to
a row of falling domino stones [213].
In the worst case, the contributions of the less salient genes may almost
look like noise and they are not optimized at all. Such a situation is also an
instance of premature convergence, since the global optimum which would
involve optimal configurations of all blocks will not be discovered. In this
situation, restarting the optimization process will not help because it will
always turn out the same way. Example problems which are often likely to
exhibit domino convergence are the Royal Road [139] and the aforementioned
BinInt problem [184].
2.3 One Cause: Loss of Diversity
In biology, diversity is the variety and abundance of organisms at a given place
and time [159, 133]. Much of the beauty and efficiency of natural ecosystems
is based on a dazzling array of species interacting in manifold ways. Diversifi-
cation is also a good investment strategy utilized by investors in the economy
in order to increase their profit.
In population-based global optimization algorithms as well, maintaining a
set of diverse solution candidates is very important. Losing diversity means
approaching a state where all the solution candidates under investigation are
similar to each other. Another term for this state is convergence. Discus-
sions about how diversity can be measured have been provided by Routledge
[183], Cousins [49], Magurran [133], Morrison and De Jong [148], and Paenke
et al [159].

Why Is Optimization Difficult? 9
Preserving diversity is directly linked with maintaining a good balance be-
tween exploitation and exploration [159] and has been studied by researchers
from many domains, such as
•Genetic Algorithms [156, 176, 177],
•Evolutionary Algorithms [28, 29, 123, 149, 200, 206],
•Genetic Programming [30, 38, 39, 40, 53, 93, 94],
•Tabu Search [81, 82], and
•Particle Swarm Optimization [238].
The operations which create new solutions from existing ones have a very
large impact on the speed of convergence and the diversity of the populations
[69, 203]. The step size in Evolution Strategy is a good example of this issue:
setting it properly is very important and leads to the “exploration versus
exploitation” problem [102] which can be observed in other areas of global
optimization as well.3
In the context of optimization, exploration means finding new points in
areas of the search space which have not been investigated before. Since
computers have only limited memory, already evaluated solution candidates
usually have to be discarded. Exploration is a metaphor for the procedure
which allows search operations to find novel and maybe better solution struc-
tures. Such operators (like mutation in Evolutionary Algorithms) have a high
chance of creating inferior solutions by destroying good building blocks but
also a small chance of finding totally new, superior traits (which, however, is
not guaranteed at all).
Exploitation, on the other hand, is the process of improving and combin-
ing the traits of the currently known solution(s), as done by the crossover
operator in Evolutionary Algorithms, for instance. Exploitation operations
often incorporate small changes into already tested individuals leading to
new, very similar solution candidates or try to merge building blocks of dif-
ferent, promising individuals. They usually have the disadvantage that other,
possibly better, solutions located in distant areas of the problem space will
not be discovered.
Almost all components of optimization strategies can either be used for in-
creasing exploitation or in favor of exploration. Unary search operations that
improve an existing solution in small steps can be built, hence being exploita-
tion operators (as is done in Memetic Algorithms, for instance). They can
also be implemented in a way that introduces much randomness into the indi-
viduals, effectively making them exploration operators. Selection operations
in Evolutionary Computation choose a set of the most promising solution
candidates which will be investigated in the next iteration of the optimizers.
They can either return a small group of best individuals (exploitation) or a
wide range of existing solution candidates (exploration).
3More or less synonymously to exploitation and exploration, the terms intensifications
and diversification have been introduced by Glover [81, 82] in the context of Tabu Search.

10 Weise, Zapf, Chiong, Nebro
Optimization algorithms that favor exploitation over exploration have
higher convergence speed but run the risk of not finding the optimal solution
and may get stuck at a local optimum. Then again, algorithms which per-
form excessive exploration may never improve their solution candidates well
enough to find the global optimum or it may take them very long to discover
it “by accident”. A good example for this dilemma is the Simulated Anneal-
ing algorithm [117]. It is often modified to a form called simulated quenching
which focuses on exploitation but loses the guaranteed convergence to the
optimum [110]. Generally, optimization algorithms should employ at least
one search operation of explorative character and at least one which is able
to exploit good solutions further. There exists a vast body of research on the
trade-off between exploration and exploitation that optimization algorithms
have to face [7, 57, 66, 70, 103, 152].
2.4 Countermeasures
As we have seen, global optimization algorithms are optimization methods
for finding the best possible solution(s) of an optimization problem instead
of prematurely converging to a local optimum. Still, there is no general ap-
proach to ensure their success. The probability that an optimization process
prematurely converges depends on the characteristics of the problem to be
solved and the parameter settings and features of the optimization algorithms
applied [215].
A very crude and yet, sometimes effective measure is restarting the opti-
mization process at randomly chosen points in time. One example for this
method is GRASP s, Greedy Randomized Adaptive Search Procedures [71, 72],
which continuously restart the process of creating an initial solution and re-
fining it with local search. Still, such approaches are likely to fail in domino
convergence situations.
In order to extend the duration of the evolution in Evolutionary Algo-
rithms, many methods have been devised for steering the search away from
areas which have already been frequently sampled. This can be achieved by
integrating density metrics into the fitness assignment process. The most
popular of such approaches are sharing and niching based on the Euclidean
distance of the solution candidates in objective space [55, 85, 104, 138]. Using
low selection pressure furthermore decreases the chance of premature conver-
gence but also decreases the speed with which good solutions are exploited.
Another approach against premature convergence is to introduce the ca-
pability of self-adaptation, allowing the optimization algorithm to change its
strategies or to modify its parameters depending on its current state. Such
behaviors, however, are often implemented not in order to prevent prema-
ture convergence but to speed up the optimization process (which may lead
to premature convergence to local optima) [185, 186, 187].

Why Is Optimization Difficult? 11
3 Ruggedness and Weak Causality
3.1 The Problem: Ruggedness
Optimization algorithms generally depend on some form of gradient in the
objective or fitness space. The objective functions should be continuous and
exhibit low total variation4, so the optimizer can descend the gradient easily.
If the objective functions are unsteady or fluctuating, i.e., going up and down,
it becomes more complicated for the optimization process to find the right
directions to proceed to. The more rugged a function gets, the harder it
becomes to optimize it. From a simplified point of view, ruggedness is multi-
modality plus steep ascends and descends in the fitness landscape. Examples
of rugged landscapes are Kauffman’s fitness landscape [113, 115], the p-Spin
model [6], Bergman and Feldman’s jagged fitness landscape [19], and the
sketch in Fig. 1.d.
3.2 One Cause: Weak Causality
During an optimization process, new points in the search space are created
by the search operations. Generally we can assume that the genotypes which
are the input of the search operations correspond to phenotypes which have
previously been selected. Usually, the better or the more promising an indi-
vidual is, the higher are its chances of being selected for further investigation.
Reversing this statement suggests that individuals which are passed to the
search operations are likely to have a good fitness. Since the fitness of a solu-
tion candidate depends on its properties, it can be assumed that the features
of these individuals are not so bad either. It should thus be possible for the
optimizer to introduce slight changes to their properties in order to find out
whether they can be improved any further5. Normally, such modifications
should also lead to small changes in the objective values and, hence, in the
fitness of the solution candidate.
Definition 1 (Strong Causality). Strong causality (locality) means that
small changes in the properties of an object also lead to small changes in its
behavior [170, 171, 180].
This principle (proposed by Rechenberg [170, 171]) should not only hold
for the search spaces and operations designed for optimization, but applies
to natural genomes as well. The offspring resulting from sexual reproduction
of two fish, for instance, has a different genotype than its parents. Yet, it is
4http://en.wikipedia.org/wiki/Total_variation [accessed 2008-04-23]
5We have already mentioned this under the subject of exploitation.

12 Weise, Zapf, Chiong, Nebro
far more probable that these variations manifest in a unique color pattern of
the scales, for example, instead of leading to a totally different creature.
Apart from this straightforward, informal explanation here, causality has
been investigated thoroughly in different fields of optimization, such as Evolu-
tion Strategy [170, 65], structure evolution [129, 130], Genetic Programming
[65, 107, 179, 180], genotype-phenotype mappings [193], search operators [65],
and Evolutionary Algorithms in general [65, 182, 207].
In fitness landscapes with weak (low) causality, small changes in the so-
lution candidates often lead to large changes in the objective values, i.e.,
ruggedness. It then becomes harder to decide which region of the problem
space to explore and the optimizer cannot find reliable gradient information
to follow. A small modification of a very bad solution candidate may then
lead to a new local optimum and the best solution candidate currently known
may be surrounded by points that are inferior to all other tested individuals.
The lower the causality of an optimization problem, the more rugged its
fitness landscape is, which leads to a degradation of the performance of the
optimizer [120]. This does not necessarily mean that it is impossible to find
good solutions, but it may take very long to do so.
3.3 Countermeasures
To our knowledge, no viable method which can directly mitigate the effects of
rugged fitness landscapes exists. In population-based approaches, using large
population sizes and applying methods to increase the diversity can decrease
the influence of ruggedness, but only up to a certain degree. Utilizing the
Baldwin effect [13, 100, 101, 233] or Lamarckian evolution [54, 233], i.e.,
incorporating a local search into the optimization process, may further help
to smoothen out the fitness landscape [89].
Weak causality is often a home-made problem: it results from the choice
of the solution representation and search operations. Thus, in order to apply
Evolutionary Algorithms in an efficient manner, it is necessary to find repre-
sentations which allow for iterative modifications with bounded influence on
the objective values.

Why Is Optimization Difficult? 13
4 Deceptiveness
4.1 Introduction
Especially annoying fitness landscapes show deceptiveness (or deceptivity).
The gradient of deceptive objective functions leads the optimizer away from
the optima, as illustrated in Fig. 1.e.
The term deceptiveness is mainly used in the Genetic Algorithm commu-
nity in the context of the Schema Theorem. Schemas describe certain areas
(hyperplanes) in the search space. If an optimization algorithm has discov-
ered an area with a better average fitness compared to other regions, it will
focus on exploring this region based on the assumption that highly fit areas
are likely to contain the true optimum. Objective functions where this is not
the case are called deceptive [20, 84, 127]. Examples for deceptiveness are the
ND fitness landscapes [17], trap functions [1, 59, 112] like the one illustrated
in Fig. 4, and the fully deceptive problems given by Goldberg et al [86, 60].
u(x)
f(x) globaloptimium
withsmallbasin
ofattraction
localoptimium
withlargebasin
ofattraction
Fig. 4: Ackley’s “Trap” function [1, 112].
4.2 The Problem
If the information accumulated by an optimizer actually guides it away from
the optimum, search algorithms will perform worse than a random walk or
an exhaustive enumeration method. This issue has been known for a long
time [228, 140, 141, 212] and has been subsumed under the No Free Lunch
Theorem which we will discuss in Section 11.

14 Weise, Zapf, Chiong, Nebro
4.3 Countermeasures
Solving deceptive optimization tasks perfectly involves sampling many indi-
viduals with very bad features and low fitness. This contradicts the basic ideas
of metaheuristics and thus, there are no efficient countermeasures against de-
ceptivity. Using large population sizes, maintaining a very high diversity, and
utilizing linkage learning (see Section 6.3.2) are, maybe, the only approaches
which can provide at least a small chance of finding good solutions.
5 Neutrality and Redundancy
5.1 The Problem: Neutrality
We consider the outcome of the application of a search operation to an el-
ement of the search space as neutral if it yields no change in the objective
values [15, 172]. It is challenging for optimization algorithms if the best solu-
tion candidate currently known is situated on a plane of the fitness landscape,
i.e., all adjacent solution candidates have the same objective values. As illus-
trated in Fig. 1.f, an optimizer then cannot find any gradient information and
thus, no direction in which to proceed in a systematic manner. From its point
of view, each search operation will yield identical individuals. Furthermore,
optimization algorithms usually maintain a list of the best individuals found,
which will then overflow eventually or require pruning.
The degree of neutrality νis defined as the fraction of neutral results
among all possible products of the search operations Op applied to a specific
genotype [15]. We can generalize this measure to areas Gin the search space
Gby averaging over all their elements. Regions where νis close to one are
considered as neutral.
∀g1∈G⇒ν(g1) = |{g2|P(g2=Op(g1))>0∧f(gpm(g2))=f(gpm(g1))}|
|{g2|P(g2=Op(g1)) >0}| (1)
∀G⊆G⇒ν(G) = 1
|G|X
g∈G
ν(g) (2)
5.2 Evolvability
Another metaphor in global optimization borrowed from biological systems
is evolvability [52]. Wagner [225, 226] points out that this word has two uses
in biology: According to Kirschner and Gerhart [118], a biological system is
evolvable if it is able to generate heritable, selectable phenotypic variations.

Why Is Optimization Difficult? 15
Such properties can then be evolved and changed by natural selection. In its
second sense, a system is evolvable if it can acquire new characteristics via
genetic change that help the organism(s) to survive and to reproduce. The-
ories about how the ability of generating adaptive variants has evolved have
been proposed by Riedl [174], Altenberg [3], Wagner and Altenberg [227],
and Bonner [26], amongst others. The idea of evolvability can be adopted for
global optimization as follows:
Definition 2 (Evolvability). The evolvability of an optimization process in
its current state defines how likely the search operations will lead to solution
candidates with new (and eventually, better) objectives values.
The direct probability of success [170, 22], i.e., the chance that search op-
erators produce offspring fitter than their parents, is also sometimes referred
to as evolvability in the context of Evolutionary Algorithms [2, 5].
5.3 Neutrality: Problematic and Beneficial
The link between evolvability and neutrality has been discussed by many
researchers. The evolvability of neutral parts of a fitness landscape depends
on the optimization algorithm used. It is especially low for Hill Climbing
and similar approaches, since the search operations cannot directly provide
improvements or even changes. The optimization process then degenerates
to a random walk, as illustrated in Fig. 1.f. The work of Beaudoin et al [17]
on the ND fitness landscapes shows that neutrality may “destroy” useful
information such as correlation.
Researchers in molecular evolution, on the other hand, found indications
that the majority of mutations have no selective influence [77, 106] and that
the transformation from genotypes to phenotypes is a many-to-one mapping.
Wagner [226] states that neutrality in natural genomes is beneficial if it con-
cerns only a subset of the properties peculiar to the offspring of a solution
candidate while allowing meaningful modifications of the others. Toussaint
and Igel [214] even go as far as declaring it a necessity for self-adaptation.
The theory of punctuated equilibria in biology introduced by Eldredge and
Gould [67, 68] states that species experience long periods of evolutionary
inactivity which are interrupted by sudden, localized, and rapid phenotypic
evolutions [47, 134, 12]. It is assumed that the populations explore neutral
layers during the time of stasis until, suddenly, a relevant change in a genotype
leads to a better adapted phenotype [224] which then reproduces quickly.
The key to differentiating between “good” and “bad” neutrality is its de-
gree νin relation to the number of possible solutions maintained by the
optimization algorithms. Smith et al [204] have used illustrative examples
similar to Fig. 5 showing that a certain amount of neutral reproductions can
foster the progress of optimization. In Fig. 5.a, basically the same scenario

16 Weise, Zapf, Chiong, Nebro
of premature convergence as in Fig. 3.a is depicted. The optimizer is drawn
to a local optimum from which it cannot escape anymore. Fig. 5.b shows
that a little shot of neutrality could form a bridge to the global optimum.
The optimizer now has a chance to escape the smaller peak if it is able to
find and follow that bridge, i.e., the evolvability of the system has increased.
If this bridge gets wider, as sketched in Fig. 5.c, the chance of finding the
global optimum increases as well. Of course, if the bridge gets too wide, the
optimization process may end up in a scenario like in Fig. 1.f where it cannot
find any direction. Furthermore, in this scenario we expect the neutral bridge
to lead to somewhere useful, which is not necessarily the case in reality.
globaloptimum
localoptimum
Fig. 5.a: Premature Con-
vergence
Fig. 5.b: Small Neutral
Bridge
Fig. 5.c: Wide Neutral
Bridge
Fig. 5: Possible positive influence of neutrality.
Examples for neutrality in fitness landscapes are the ND family [17], the
NKp [15] and NKq [155] models, and the Royal Road [139]. Another common
instance of neutrality is bloat in Genetic Programming [131].
5.4 Redundancy: Problematic and Beneficial
Redundancy in the context of global optimization is a feature of the genotype-
phenotype mapping and means that multiple genotypes map to the same
phenotype, i.e., the genotype-phenotype mapping is not injective. The role
of redundancy in the genome is as controversial as that of neutrality [230].
There exist many accounts of its positive influence on the optimization pro-
cess. Shackleton et al [194, 197], for instance, tried to mimic desirable evolu-
tionary properties of RNA folding [106]. They developed redundant genotype-
phenotype mappings using voting (both, via uniform redundancy and via a
non-trivial approach), Turing machine-like binary instructions, Cellular au-

Why Is Optimization Difficult? 17
tomata, and random Boolean networks [114]. Except for the trivial voting
mechanism based on uniform redundancy, the mappings induced neutral net-
works which proved beneficial for exploring the problem space. Especially the
last approach provided particularly good results [194, 197]. Possibly converse
effects like epistasis (see Section 6) arising from the new genotype-phenotype
mappings have not been considered in this study.
Redundancy can have a strong impact on the explorability of the prob-
lem space. When utilizing a one-to-one mapping, the translation of a slightly
modified genotype will always result in a different phenotype. If there ex-
ists a many-to-one mapping between genotypes and phenotypes, the search
operations can create offspring genotypes different from the parent which
still translate to the same phenotype. The optimizer may now walk along a
path through this neutral network. If many genotypes along this path can be
modified to different offspring, many new solution candidates can be reached
[197]. The experiments of Shipman et al [198, 196] additionally indicate that
neutrality in the genotype-phenotype mapping can have positive effects.
Yet, Rothlauf [182] and Shackleton et al [194] show that simple uniform
redundancy is not necessarily beneficial for the optimization process and
may even slow it down. There is no use in introducing encodings which, for
instance, represent each phenotypic bit with two bits in the genotype where
00 and 01 map to 0and 10 and 11 map to 1.
5.5 Summary
Different from ruggedness which is always bad for optimization algorithms,
neutrality has aspects that may further as well as hinder the process of find-
ing good solutions. Generally we can state that degrees of neutrality νvery
close to 1 degenerate optimization processes to random walks. Some forms
of neutral networks [14, 15, 27, 105, 208, 222, 223, 237] accompanied by low
(nonzero) values of νcan improve the evolvability and hence, increase the
chance of finding good solutions.
Adverse forms of neutrality are often caused by bad design of the search
space or genotype-phenotype mapping. Uniform redundancy in the genome
should be avoided where possible and the amount of neutrality in the search
space should generally be limited.

18 Weise, Zapf, Chiong, Nebro
6 Epistasis
6.1 Introduction
In biology, epistasis is defined as a form of interaction between different genes
[163]. The term was coined by Bateson [16] and originally meant that one
gene suppresses the phenotypical expression of another gene. In the context
of statistical genetics, epistasis was initially called “epistacy” by Fisher [74].
According to Lush [132], the interaction between genes is epistatic if the ef-
fect on the fitness of altering one gene depends on the allelic state of other
genes. This understanding of epistasis comes very close to another biological
expression: Pleiotropy, which means that a single gene influences multiple
phenotypic traits [239]. In global optimization, such fine-grained distinctions
are usually not made and the two terms are often used more or less synony-
mously.
Definition 3 (Epistasis). In optimization, Epistasis is the dependency of
the contribution of one gene to the value of the objective functions on the
allelic state of other genes. [4, 51, 153]
We speak of minimal epistasis when every gene is independent of every
other gene. Then, the optimization process equals finding the best value for
each gene and can most efficiently be carried out by a simple greedy search
[51]. A problem is maximally epistasic when no proper subset of genes is
independent of any other gene [205, 153]. Examples of problems with a high
degree of epistasis are Kauffman’s fitness landscape [113, 115], the p-Spin
model [6], and the tunable model of Weise et al [232].
6.2 The Problem
As sketched in Fig. 6, epistasis has a strong influence on many of the pre-
viously discussed problematic features. If one gene can “turn off” or affect
the expression of many other genes, a modification of this gene will lead to
a large change in the features of the phenotype. Hence, the causality will be
weakened and ruggedness ensues in the fitness landscape. On the other hand,
subsequent changes to the “deactivated” genes may have no influence on the
phenotype at all, which would then increase the degree of neutrality in the
search space. Epistasis is mainly an aspect of the way in which we define the
genome Gand the genotype-phenotype mapping gpm. It should be avoided
where possible.
Generally, epistasis and conflicting objectives in multi-objective optimiza-
tion should be distinguished from each other. Epistasis as well as pleiotropy
is a property of the influence of the elements (the genes) of the genotypes

Why Is Optimization Difficult? 19
ruggedness multi-
modality
weakcausality
high
epistasis
ºcauses
neutrality
Fig. 6: The influence of epistasis on the fitness landscape.
on the phenotypes. Objective functions can conflict without the involvement
of any of these phenomena. We can, for example, define two objective func-
tions f1(x) = xand f2(x) = −xwhich are clearly contradicting regardless of
whether they are subject to maximization or minimization. Nevertheless, if
the solution candidates xas well as the genotypes are simple real numbers
and the genotype-phenotype mapping is simply an identity mapping, neither
epistatic nor pleiotropic effects can occur.
Naudts and Verschoren [154] have shown for the special case of length-
two binary string genomes that deceptiveness does not occur in situations
with low epistasis and also that objective functions with high epistasis are
not necessarily deceptive. Another discussion about different shapes of fitness
landscapes under the influence of epistasis is given by Beerenwinkel et al [18].
6.3 Countermeasures
6.3.1 General
We have shown that epistasis is a root cause for multiple problematic fea-
tures of optimization tasks. General countermeasures against epistasis can be
divided into two groups. The symptoms of epistasis can be mitigated with
the same methods which increase the chance of finding good solutions in the
presence of ruggedness or neutrality – using larger populations and favor-
ing explorative search operations. Epistasis itself is a feature which results
from the choice of the search space structure, the search operations, and the
genotype-phenotype mapping. Avoiding epistatic effects should be a major
concern during their design. This can lead to a great improvement in the
quality of the solutions produced by the optimization process [231]. General
advice for good search space design is given in [84, 166, 178] and [229].

20 Weise, Zapf, Chiong, Nebro
6.3.2 Linkage Learning
According to Winter et al [240], linkage is “the tendency for alleles of different
genes to be passed together from one generation to the next” in genetics. This
usually indicates that these genes are closely located in the same chromosome.
In the context of Evolutionary Algorithms, this notation is not useful since
identifying spatially close elements inside the genotypes is trivial. Instead,
we are interested in alleles of different genes which have a joint effect on the
fitness [150, 151].
Identifying these linked genes, i.e., learning their epistatic interaction, is
very helpful for the optimization process. Such knowledge can be used to pro-
tect building blocks from being destroyed by the search operations. Finding
approaches for linkage learning has become an especially popular discipline
in the area of Evolutionary Algorithms with binary [99, 150, 46] and real
[63] genomes. Two important methods from this area are the messy Genetic
Algorithm (mGA) by Goldberg et al [86] and the Bayesian Optimization
Algorithm (BOA) [162, 41]. Module acquisition [8] may be considered as a
similar effort in the area of Genetic Programming.
Let us take the mGA as an illustrative example for this family of ap-
proaches. By explicitly allowing the search operations to rearrange the genes
in the genotypes, epistatically linked genes may get located closer to each
other by time. As sketched in Fig. 7, the tighter the building blocks are
packed, the less likely are they to be destroyed by crossover operations which
usually split parent genotypes at randomly chosen points. Hence, the opti-
mization process can strengthen the causality in the search space.
destroyedin6outof9casesbycrossover
destroyedin1outof9casesbycrossover
rearrange
Fig. 7: Two linked genes and their destruction probability under single-point
crossover.
7 Noise and Robustness
7.1 Introduction – Noise
In the context of optimization, three types of noise can be distinguished. The
first form is noise in the training data used as basis for learning (i). In many

Why Is Optimization Difficult? 21
applications of machine learning or optimization where a model mfor a given
system is to be learned, data samples including the input of the system and its
measured response are used for training. Some typical examples of situations
where training data is the basis for the objective function evaluation are
•the usage of global optimization for building classifiers (for example for
predicting buying behavior using data gathered in a customer survey for
training),
•the usage of simulations for determining the objective values in Genetic
Programming (here, the simulated scenarios correspond to training cases),
and
•the fitting of mathematical functions to (x, y)-data samples (with artificial
neural networks or symbolic regression, for instance).
Since no measurement device is 100% accurate and there are always random
errors, noise is present in such optimization problems.
Besides inexactnesses and fluctuations in the input data of the optimiza-
tion process, perturbations are also likely to occur during the application of
its results. This category subsumes the other two types of noise: perturbations
that may arise from inaccuracies in (ii) the process of realizing the solutions
and (iii) environmentally induced perturbations during the applications of
the products.
This issue can be illustrated using the process of developing the perfect
tire for a car as an example. As input for the optimizer, all sorts of material
coefficients and geometric constants measured from all known types of wheels
and rubber could be available. Since these constants have been measured or
calculated from measurements, they include a certain degree of noise and
imprecision (i).
The result of the optimization process will be the best tire construction
plan discovered during its course and it will likely incorporate different ma-
terials and structures. We would hope that the tires created according to
the plan will not fall apart if, accidently, an extra 0.0001% of a specific rub-
ber component is used (ii). During the optimization process, the behavior of
many construction plans will be simulated in order to find out about their
utility. When actually manufactured, the tires should not behave unexpect-
edly when used in scenarios different from those simulated (iii) and should
instead be applicable in all driving scenarios likely to occur.
The effects of noise in optimization have been studied by various re-
searchers; Miller and Goldberg [136, 137], Lee and Wong [125], and Gurin
and Rastrigin [92] are some of them. Many global optimization algorithms
and theoretical results have been proposed which can deal with noise. Some
of them are, for instance, specialized
•Genetic Algorithms [75, 119, 188, 189, 217, 218],
•Evolution Strategies [11, 21, 96], and
•Particle Swarm Optimization [97, 161] approaches.

22 Weise, Zapf, Chiong, Nebro
7.2 The Problem: Need for Robustness
The goal of global optimization is to find the global optima of the objective
functions. While this is fully true from a theoretical point of view, it may
not suffice in practice. Optimization problems are normally used to find good
parameters or designs for components or plans to be put into action by human
beings or machines. As we have already pointed out, there will always be noise
and perturbations in practical realizations of the results of optimization.
Definition 4 (Robustness). A system in engineering or biology is robust if
it is able to function properly in the face of genetic or environmental pertur-
bations [225].
Therefore, a local optimum (or even a non-optimal element) for which
slight deviations only lead to gentle performance degenerations is usually
favored over a global optimum located in a highly rugged area of the fitness
landscape [31]. In other words, local optima in regions of the fitness landscape
with strong causality are sometimes better than global optima with weak
causality. Of course, the level of this acceptability is application-dependent.
Fig. 8 illustrates the issue of local optima which are robust vs. global optima
which are not. More examples from the real world are:
•When optimizing the control parameters of an airplane or a nuclear power
plant, the global optimum is certainly not used if a slight perturbation can
have hazardous effects on the system [218].
•Wiesmann et al [234, 235] bring up the topic of manufacturing tolerances
in multilayer optical coatings. It is no use to find optimal configurations
if they only perform optimal when manufactured to a precision which is
either impossible or too hard to achieve on a constant basis.
•The optimization of the decision process on which roads should be pre-
cautionary salted for areas with marginal winter climate is an example
of the need for dynamic robustness. The global optimum of this problem
is likely to depend on the daily (or even current) weather forecast and
may therefore be constantly changing. Handa et al [98] point out that it is
practically infeasible to let road workers follow a constantly changing plan
and circumvent this problem by incorporating multiple road temperature
settings in the objective function evaluation.
•Tsutsui et al [218, 217] found a nice analogy in nature: The phenotypic
characteristics of an individual are described by its genetic code. Dur-
ing the interpretation of this code, perturbations like abnormal tempera-
ture, nutritional imbalances, injuries, illnesses and so on may occur. If the
phenotypic features emerging under these influences have low fitness, the
organism cannot survive and procreate. Thus, even a species with good
genetic material will die out if its phenotypic features become too sensi-
tive to perturbations. Species robust against them, on the other hand, will
survive and evolve.

Why Is Optimization Difficult? 23
globaloptimum
robustlocaloptimum
f(x)
X
Fig. 8: A robust local optimum vs. a “unstable” global optimum.
7.3 Countermeasures
For the special case where the problem space corresponds to the real vectors
(X⊆Rn), several approaches for dealing with the problem of robustness
have been developed. Inspired by Taguchi methods6[209], possible distur-
bances are represented by a vector δ= (δ1, δ2, .., δn)T, δi∈Rin the method
of Greiner [87, 88]. If the distribution and influence of the δiare known,
the objective function f(x) : x∈Xcan be rewritten as ˜
f(x,δ) [235]. In
the special case where δis normally distributed, this can be simplified to
˜
f(x1+δ1, x2+δ2, .., xn+δn)T. It would then make sense to sample the
probability distribution of δa number of ttimes and to use the mean val-
ues of ˜
f(x,δ) for each objective function evaluation during the optimization
process. In cases where the optimal value yof the objective function fis
known, Equation 3 can be minimized. This approach is also used in the work
of Wiesmann et al [234, 235] and basically turns the optimization algorithm
into something like a maximum likelihood estimator.
f′(x) = 1
t
t
X
i=1
y−˜
f(x,δi)2
(3)
This method corresponds to using multiple, different training scenarios
during the objective function evaluation in situations where X6⊆ Rn. By
adding random noise and artificial perturbations to the training cases, the
chance of obtaining robust solutions which are stable when applied or realized
under noisy conditions can be increased.
6http://en.wikipedia.org/wiki/Taguchi_methods [accessed 2008-07-19]

24 Weise, Zapf, Chiong, Nebro
8 Overfitting and Oversimplification
In all scenarios where optimizers evaluate some of the objective values of the
solution candidates by using training data, two additional phenomena with
negative influence can be observed: overfitting and oversimplification.
8.1 Overfitting
8.1.1 The Problem
Definition 5 (Overfitting). Overfitting is the emergence of an overly com-
plicated model (solution candidate) in an optimization process resulting from
the effort to provide the best results for as much of the available training data
as possible [64, 80, 190, 202].
A model (solution candidate) m∈Xcreated with a finite set of training
data is considered to be overfitted if a less complicated, alternative model
m′∈Xexists which has a smaller error for the set of all possible (maybe
even infinitely many), available, or (theoretically) producible data samples.
This model m′may, however, have a larger error in the training data.
The phenomenon of overfitting is best known and can often be encountered
in the field of artificial neural networks or in curve fitting [124, 128, 181, 191,
211]. The latter means we that have a set Aof ntraining data samples
(xi, yi) and want to find a function fthat represents these samples as well
as possible, i.e., f(xi) = yi∀(xi, yi)∈A.
There exists exactly one polynomial of the degree n−1 that fits to each
such training data and goes through all its points. Hence, when only polyno-
mial regression is performed, there is exactly one perfectly fitting function of
minimal degree. Nevertheless, there will also be an infinite number of poly-
nomials with a higher degree than n−1 that also match the sample data
perfectly. Such results would be considered as overfitted.
In Fig. 9, we have sketched this problem. The function f1(x) = xshown in
Fig. 9.b has been sampled three times, as sketched in Fig. 9.a. There exists
no other polynomial of a degree of two or less that fits to these samples than
f1. Optimizers, however, could also find overfitted polynomials of a higher
degree such as f2which also match the data, as shown in Fig. 9.c. Here, f2
plays the role of the overly complicated model mwhich will perform as good
as the simpler model m′when tested with the training sets only, but will fail
to deliver good results for all other input data.
A very common cause for overfitting is noise in the sample data. As we
have already pointed out, there exists no measurement device for physical
processes which delivers perfect results without error. Surveys that represent
the opinions of people on a certain topic or randomized simulations will ex-

Why Is Optimization Difficult? 25
x
y
Fig. 9.a: Three sample
points of f1.
x
y
m`
Fig. 9.b: m′≡f1(x) = x.
x
y
m
Fig. 9.c: m≡f2(x).
Fig. 9: Overfitting due to complexity.
hibit variations from the true interdependencies of the observed entities, too.
Hence, data samples based on measurements will always contain some noise.
In Fig. 10 we have sketched how such noise may lead to overfitted re-
sults. Fig. 10.a illustrates a simple physical process obeying some quadratic
equation. This process has been measured using some technical equipment
and the 100 noisy samples depicted in Fig. 10.b has been obtained. Fig. 10.c
shows a function resulting from an optimization that fits the data perfectly.
It could, for instance, be a polynomial of degree 99 that goes right through
all the points and thus, has an error of zero. Although being a perfect match
to the measurements, this complicated model does not accurately represent
the physical law that produced the sample data and will not deliver precise
results for new, different inputs.
m`
x
y
Fig. 10.a: The original
physical process.
x
y
Fig. 10.b: The measure-
ment/training data.
x
y
m
Fig. 10.c: The overfitted re-
sult.
Fig. 10: Fitting noise.
From the examples we can see that the major problem that results from
overfitted solutions is the loss of generality.
Definition 6 (Generality). A solution of an optimization process is general
if it is not only valid for the sample inputs a1, a2,...,anwhich were used

26 Weise, Zapf, Chiong, Nebro
for training during the optimization process, but also for different inputs
a6=ai∀i: 0 < i ≤nif such inputs aexist.
8.1.2 Countermeasures
There exist multiple techniques that can be utilized in order to prevent over-
fitting to a certain degree. It is most efficient to apply multiple such techniques
together in order to achieve best results.
A very simple approach is to restrict the problem space Xin a way that
only solutions up to a given maximum complexity can be found. In terms
of function fitting, this could mean limiting the maximum degree of the
polynomials to be tested. Furthermore, the functional objective functions
which solely concentrate on the error of the solution candidates should be
augmented by penalty terms and non-functional objective functions putting
pressure in the direction of small and simple models [64, 116].
Large sets of sample data, although slowing down the optimization pro-
cess, may improve the generalization capabilities of the derived solutions. If
arbitrarily many training datasets or training scenarios can be generated,
there are two approaches which work against overfitting:
1. The first method is to use a new set of (randomized) scenarios for each eval-
uation of a solution candidate. The resulting objective values may differ
largely even if the same individual is evaluated twice in a row, introducing
incoherence and ruggedness into the fitness landscape.
2. At the beginning of each iteration of the optimizer, a new set of (random-
ized) scenarios is generated which is used for all individual evaluations
during that iteration. This method leads to objective values which can be
compared without bias.
In both cases it is helpful to use more than one training sample or scenario per
evaluation and to set the resulting objective value to the average (or better
median) of the outcomes. Otherwise, the fluctuations of the objective values
between the iterations will be very large, making it hard for the optimizers
to follow a stable gradient for multiple steps.
Another simple method to prevent overfitting is to limit the runtime of the
optimizers [190]. It is commonly assumed that learning processes normally
first find relatively general solutions which subsequently begin to overfit be-
cause the noise “is learned”, too.
For the same reason, some algorithms allow to decrease the rate at which
the solution candidates are modified by time. Such a decay of the learning
rate makes overfitting less likely.
If only one finite set of data samples is available for training/optimization,
it is common practice to separate it into a set of training data Atand a set
of test cases Ac. During the optimization process, only the training data is
used. The resulting solutions are tested with the test cases afterwards. If their

Why Is Optimization Difficult? 27
behavior is significantly worse when applied to Acthan when applied to At,
they are probably overfitted.
The same approach can be used to detect when the optimization process
should be stopped. The best known solution candidates can be checked with
the test cases in each iteration without influencing their objective values
which solely depend on the training data. If their performance on the test
cases begins to decrease, there are no benefits in letting the optimization
process continue any further.
8.2 Oversimplification
8.2.1 The Problem
Oversimplification (also called overgeneralization) is the opposite of overfit-
ting. Whereas overfitting denotes the emergence of overly-complicated so-
lution candidates, oversimplified solutions are not complicated enough. Al-
though they represent the training samples used during the optimization pro-
cess seemingly well, they are rough overgeneralizations which fail to provide
good results for cases not part of the training.
A common cause for oversimplification is sketched in Fig. 11: The training
sets only represent a fraction of the set of possible inputs. As this is normally
the case, one should always be aware that such an incomplete coverage may
fail to represent some of the dependencies and characteristics of the data,
which then may lead to oversimplified solutions. Another possible reason
is that ruggedness, deceptiveness, too much neutrality, or high epistasis in
the fitness landscape may lead to premature convergence and prevent the
optimizer from surpassing a certain quality of the solution candidates. It then
cannot completely adapt them even if the training data perfectly represents
the sampled process. A third cause is that a problem space which does not
include the correct solution was chosen.
Fig. 11.a shows a cubic function. Since it is a polynomial of degree three,
four sample points are needed for its unique identification. Maybe not know-
ing this, only three samples have been provided in Fig. 11.b. By doing so,
some vital characteristics of the function are lost. Fig. 11.c depicts a square
function – the polynomial of the lowest degree that fits exactly to these sam-
ples. Although it is a perfect match, this function does not touch any other
point on the original cubic curve and behaves totally differently at the lower
parameter area.
However, even if we had included point Pin our training data, it would
still be possible that the optimization process would yield Fig. 11.c as a
result. Having training data that correctly represents the sampled system
does not mean that the optimizer is able to find a correct solution with
perfect fitness – the other, previously discussed problematic phenomena can

28 Weise, Zapf, Chiong, Nebro
prevent it from doing so. Furthermore, if it was not known that the system
which was to be modeled by the optimization process can best be represented
by a polynomial of the third degree, one could have limited the problem space
Xto polynomials of degree two and less. Then, the result would likely again
be something like Fig. 11.c, regardless of how many training samples are used.
x
y
P
Fig. 11.a: The “real sys-
tem” and the points de-
scribing it.
x
y
Fig. 11.b: The sampled
training data.
x
y
Fig. 11.c: The oversimpli-
fied result.
Fig. 11: Oversimplification.
8.2.2 Countermeasures
In order to counter oversimplification, its causes have to be mitigated. Gen-
erally, it is not possible to have training scenarios which cover the complete
input space of the evolved programs. By using multiple scenarios for each
individual evaluation, the chance of missing important aspects is decreased.
These scenarios can be replaced with new, randomly created ones in each
generation, which will decrease this chance even more. The problem space,
i.e., the representation of the solution candidates, should further be chosen
in a way which allows constructing a correct solution to the problem defined.
Then again, releasing too many constraints on the solution structure increases
the risk of overfitting and thus, careful proceeding is recommended.
9 Multi-Objective Optimization
9.1 Introduction
Many optimization problems in the real world have k, possibly contradictory
objectives fiwhich must be optimized simultaneously. Furthermore, the so-
lutions must satisfy minequality constraints gand pequality constraints h.

Why Is Optimization Difficult? 29
A solution candidate xis feasible, if and only if gi(x)≥0∀i= 1,2, .., m and
hi(x) = 0 ∀i= 1,2, .., p holds. A multi-objective optimization problem (MOP)
can then be formally defined as follows:
Definition 7 (MOP). Find a solution candidate x⋆in Xwhich minimizes
(or maximizes) the vector function f(x⋆) = (fi(x⋆), f2(x⋆), .., fk(x⋆))Tand is
feasible, (i.e., satisfies the minequality constraints gi(x⋆)≥0∀i= 1,2, .., m,
the pequality constraints hi(x⋆) = 0 ∀i= 1,2, .., p).
As in single-objective optimization, nature-inspired algorithms are popu-
lar techniques to solve such problems. The fact that there are two or more
objective functions implies additional difficulties. Due to the contradictory
feature of the functions in a MOP and the fact that there exists no total
order in Rnfor n > 1, the notions of “better than” and “optimum” have to
be redefined. When comparing any two solutions x1and x2, solution x1can
have a better value in objective fi, i.e., fi(x1)< fi(x2), while solution x2can
have a better value in objective fj. The concepts commonly used here are
Pareto dominance and Pareto optimality.
Definition 8 (Pareto Dominance). In the context of multi-objective
global optimization, a solution candidate x1is said to dominate another so-
lution candidate x2(denoted by x14x2) if and only if f(x1) is partially less
than f(x2), i.e., ∀i∈ {1, .., k}fi(x1)≤fi(x2)∧ ∃j∈ {1, .., k}:fj(x1)<
fj(x2).
f1
f2
8
0
1
2
3
4
5
6
7
1 2 345 6 7 8 9 10 11
9
10
16
7
2
5
3
4
ParetoFront
1 2
⋠
12 ⋠
15
´
6
1´
1⋠
´
⋠
10
9
9
7
10
8
Fig. 12: Some examples for the dominance relation.
The dominance notion allows us to assume that if solution x1dominates
solution x2, then x1is preferable to x2. If both solution are non-dominated
(such as candidate ①and ②in Fig. 12), some additional criteria have to be
used to choose one of them.

30 Weise, Zapf, Chiong, Nebro
Definition 9 (Pareto Optimality). A feasible point x⋆∈Xis Pareto-
optimal if and only if there is no feasible xb∈Xwith xb4x⋆.
This definition states that x⋆is Pareto-optimal if there is no other feasible
solution xbwhich would improve some criterion without causing a simul-
taneous worsening in at least one other criterion. The solution to a MOP,
considering Pareto optimality, is the set of feasible, non-dominated solutions
which is known as Pareto-optimal set:
Definition 10 (Pareto-Optimal Set). For a given MOP f(x), the Pareto
optimal set is defined as P⋆={x⋆∈X|¬∃x∈X:x4x⋆}.
When the solutions in the Pareto-optimal set are plotted in the objective
space (as sketched in Fig. 12), they are collectively known as the Pareto front:
Definition 11 (Pareto Front). For a given MOP f(x) and its Pareto-
optimal set P⋆, the Pareto front is defined as PF ⋆={f(x)|x∈ P⋆}.
Obtaining the Pareto front of a MOP is the main goal of multi-objective
optimization. In a real scenario, the solutions in the Pareto front are sent
to an expert in the MOP, the decision maker, who will be responsible for
choosing the best tradeoff solution among all of them. Fig. 13 depicts the
Pareto front of a bi-objective MOP. In a real problem example, f1could
represent the time required by a car to cover a given distance, while f2could
be the fuel consumption.
f1
f2
Fig. 13: Example of Pareto front of a bi-objective MOP.
The Pareto front of a MOP can contain a large (possibly infinite) number
of points. Usually, the goal of optimization is to obtain a fixed-size set of
solutions called the Pareto front approximation set. Population-based algo-
rithms, such as Genetic Algorithms, are very popular to solve MOPs because
they can provide an approximation set in a single run.

Why Is Optimization Difficult? 31
Given that the goal is to find a Pareto front approximation set, two is-
sues arise. First, the optimization process should converge to the true Pareto
front and return solutions as close to it as possible. Second, they should be
uniformly spread along this front.
f1
f2
trueParetofront
frontreturned
bytheoptimizer
Fig. 14.a: Bad Convergence and Good
Spread
f1
f2
Fig. 14.b: Good Convergence and Bad
Spread
f1
f2
Fig. 14.c: Good Convergence and Spread
Fig. 14: Pareto front approximation sets.
Let us examine the three fronts included in Fig. 14. The first picture
(Fig. 14.a) shows an approximation set having a very good spread7of solu-
tions, but the points are far away from the true Pareto front. Such results
are not attractive because they do not provide Pareto-optimal solutions. The
second example (Fig. 14.b) contains a set of solutions which are very close to
the true Pareto front but cover it only partially, so the decision maker could
lose important trade-off solutions. Finally, the front depicted in Fig. 14.c has
the two desirable properties of good convergence and spread.
7In MO optimization, this property is usually called diversity. In order to avoid confusion
with the (related) diversity property from Section 2.3, we here use the term spread instead.

32 Weise, Zapf, Chiong, Nebro
9.2 The Problem
Features such as multi-modality, deceptiveness, or epistasis found in single-
objective optimization also affect MOPs, making them more difficult to solve.
However, there are some characteristics that are particular to MOPs. Here
we comment on two of them: geometry and dimensionality.
The Pareto front in Fig. 13 has a convex geometry, but there are other
different shapes as well. In Fig. 15 we show some examples, including non-
convex (concave), disconnected, linear, and non-uniformly distributed Pareto
fronts. Besides Pareto optimization, there is a wide variety of other concepts
for defining what optima are in the presence of multiple objective functions
[45]. The simplest approach is maybe to use a weighted sum of all objective
values and set v(x) = Pk
i=1 fi(x). Then the optima would be the element(s)
x⋆with ¬∃x∈X:v(x)< v(x⋆). However, an optimization process driven
by such a linear aggregating function will not find portions of Pareto fronts
with non-convex geometry as shown by Das and Dennis [50].
f1
f2
Fig. 15.a: Non-Convex (Concave)
f1
f2
Fig. 15.b: Disconnected
f1
f2
Fig. 15.c: linear
f1
f2
Fig. 15.d: Non-Uniformly Distributed
Fig. 15: Examples of Pareto fronts.

Why Is Optimization Difficult? 33
Many studies in the literature consider mainly bi-objective MOPs. As a
consequence, many algorithms are designed to deal with that kind of prob-
lems. However, MOPs having a higher number of objective functions are
common in practice, leading to the so-called many-objective optimization
[165], which is currently a hot research topic. Most of the optimization al-
gorithms applied today utilize the Pareto dominance relation. When the di-
mension of the MOPs increases, the majority of solution candidates are non-
dominated. As a consequence, traditional nature-inspired algorithms have to
be redesigned.
9.3 Countermeasures
In order to obtain an accurate approximation to the true Pareto front, many
nature-inspired multi-objective algorithms apply a fitness assignment scheme
based on the concept of Pareto dominance, as commented before. For exam-
ple, NSGA-II [61, 62], the most well-known multi-objective technique, assigns
to each solution a rank depending on the number of solutions dominating it.
Thus, solutions with rank 1 are non-dominated, solutions with rank 2 are
dominated by one solution, and so on. Other algorithms, such as SPEA2
[247, 248] introduce the concept of strength, which is similar to the ranking
but also considers the number of dominated solutions.
While the use of Pareto-based ranking methods allows the techniques to
search in the direction of finding approximations with good convergence, addi-
tional strategies are needed to promote spread. The most commonly adopted
approach is to include a kind of density estimator in order to select those
solutions which are in the less crowded regions of the objective space. Thus,
NSGA-II employs the crowding distance [61] and SPEA2 the distance to the
k-nearest neighbor [62].
9.4 Constraint Handling
How the constraints mentioned in Definition 7 are handled is a whole research
area in itself with roots in single-objective optimization. Maybe one of the
most popular approach for dealing with constraints goes back to Courant [48]
who introduced the idea of penalty functions [73, 44, 201] in 1943: Consider,
for instance, the term f′(x) = f(x) + v[h(x)]2where fis the original objec-
tive function, his an equality constraint, and v > 0. If f′is minimized, an
infeasible individual will always have a worse fitness than a feasible one with
the same objective values.
Besides such static penalty functions, dynamic terms incorporating the
generation counter [111, 157] or adaptive approaches utilizing additional

34 Weise, Zapf, Chiong, Nebro
population statistics [95, 199] have been proposed. Rigorous discussions on
penalty functions have been contributed by Fiacco and McCormick [73] and
Smith and Coit [201].
During the last fifteen years, many approaches have been developed
which incorporate constraint handling and multi-objectivity. Instead of using
penalty terms, Pareto ranking can also be extended by additionally com-
paring individuals according to their feasibility, for instance. Examples for
this approach are the Method of Inequalities (MOI) Zakian [245] as used
by Pohlheim [164] and the Goal Attainment method defined in [76]. Deb
[56, 58] even suggested to simply turn constraints into objective functions in
his MOEA version of Goal Programming.
10 Dynamically Changing Fitness Landscape
It should also be mentioned that there exist problems with dynamically
changing fitness landscapes [33, 32, 36, 147, 173]. The task of an optimization
algorithm is, then, to provide solution candidates with momentarily optimal
objective values for each point in time. Here we have the problem that an
optimum in iteration twill possibly not be an optimum in iteration t+ 1
anymore.
The moving peaks benchmarks by Branke [33, 32] and Morrison and De
Jong [147] are good examples for dynamically changing fitness landscapes.
Such problems with dynamic characteristics can, for example, be tackled with
special forms [244] of
•Evolutionary Algorithms [9, 34, 35, 145, 146, 216, 236],
•Genetic Algorithms [83, 119, 142, 143, 144],
•Particle Swarm Optimization [23, 42, 43, 126, 160],
•Differential Evolution [135, 243], and
•Ant Colony Optimization [90, 91]
11 The No Free Lunch Theorem
By now, we know the most important problems that can be encountered when
applying an optimization algorithm to a given problem. Furthermore, we
have seen that it is arguable what actually an optimum is if multiple criteria
are optimized at once. The fact that there is most likely no optimization
method that can outperform all others on all problems can, thus, easily be
accepted. Instead, there exist a variety of optimization methods specialized
in solving different types of problems. There are also algorithms which deliver
good results for many different problem classes, but may be outperformed by
highly specialized methods in each of them.

Why Is Optimization Difficult? 35
These facts have been formalized by Wolpert and Macready [241, 242]
in their No Free Lunch Theorems (NFL) for search and optimization algo-
rithms. Wolpert and Macready [242] focus on single-objective optimization
and prove that the sum of the values of any performance measure (such as the
objective value of the best solution candidate discovered until a time step m)
over all possible objective functions fis always identical for all optimization
algorithms.
From this theorem, we can immediately follow that, in order to outper-
form the optimization method a1in one optimization problem, the algorithm
a2will necessarily perform worse in another. Fig. 16 visualizes this issue.
The higher the value of the performance measure illustrated there, the faster
will the corresponding problem be solved. The figure shows that general op-
timization approaches (like Evolutionary Algorithms) can solve a variety of
problem classes with reasonable performance. Hill Climbing approaches, for
instance, will be much faster than Evolutionary Algorithms if the objective
functions are steady and monotonous, that is, in a smaller set of optimization
tasks. Greedy search methods will perform fast on all problems with matroid
structure. Evolutionary Algorithms will most often still be able to solve these
problems, it just takes them longer to do so. The performance of Hill Climb-
ing and greedy approaches degenerates in other classes of optimization tasks
as a trade-off for their high utility in their “area of expertise”.
allpossibleoptimizationproblems
performance
randomwalkorexhaustiveenumerationor...
generaloptimizationalgorithm-anEA,forinstance
specializedoptimizationalgorithm1;ahillclimber,forinstance
specializedoptimizationalgorithm2;adepth-firstsearch,forinstance
verycrudesketch
Fig. 16: A visualization of the No Free Lunch Theorem.

36 Weise, Zapf, Chiong, Nebro
One interpretation of the No Free Lunch Theorem is that it is impossi-
ble for any optimization algorithm to outperform random walks or exhaus-
tive enumerations on all possible problems. For every problem where a given
method leads to good results, we can construct a problem where the same
method has exactly the opposite effect (see Section 4). As a matter of fact,
doing so is even a common practice to find weaknesses of optimization algo-
rithms and to compare them with each other.
Another interpretation is that every useful optimization algorithm utilizes
some form of problem-specific knowledge. Radcliffe [167] states that without
such knowledge, search algorithms cannot exceed the performance of simple
enumerations. Incorporating knowledge starts with relying on simple assump-
tions like “if xis a good solution candidate, than we can expect other good
solution candidates in its vicinity”, i.e., strong causality. The more (correct)
problem specific knowledge is integrated (correctly) into the algorithm struc-
ture, the better will the algorithm perform. On the other hand, knowledge
correct for one class of problems is, quite possibly, misleading for another
class. In reality, we use optimizers to solve a given set of problems and are
not interested in their performance when (wrongly) applied to other classes.
Today, there exists a wide range of work on No Free Lunch Theorems
for many different aspects of machine learning. The website http://www.
no-free-lunch.org/8gives a good overview about them. Further summaries
and extensions have been provided by K¨oppen et al [121] and Igel and Tou-
ssaint [108, 109]. Radcliffe and Surry [168] discuss the NFL in the context of
Evolutionary Algorithms and the representations used as search spaces. The
No Free Lunch Theorem is furthermore closely related to the Ugly Duckling
Theorem proposed by Watanabe [228] for classification and pattern recogni-
tion.
12 Concluding Remarks
The subject of this introductory chapter was the question about what makes
optimization problems hard, especially for metaheuristic approaches. We have
discussed numerous different phenomena which can affect the optimization
process and lead to disappointing results. If an optimization process has con-
verged prematurely, it has been trapped in a non-optimal region of the search
space from which it cannot “escape” anymore (Section 2). Ruggedness (Sec-
tion 3) and deceptiveness (Section 4) in the fitness landscape, often caused
by epistatic effects (Section 6), can misguide the search into such a region.
Neutrality and redundancy (Section 5) can either slow down optimization
because the application of the search information does not lead to a gain in
information or may also contribute positively by creating neutral networks
8accessed: 2008-03-28

Why Is Optimization Difficult? 37
from which the search space can be explored and local optima can be escaped
from. The solutions that are derived, even in the presence of noise, should
be robust (Section 7). Also, they should neither be too general (oversimpli-
fication, Section 8.2) nor too specifically aligned only to the training data
(overfitting, Section 8.1). Furthermore, many practical problems are multi-
objective, i.e., involve the optimization of more than one criterion at once
(Section 9), or concern objectives which may change over time (Section 10).
In the previous section, we discussed the No Free Lunch Theorem and
argued that it is not possible to develop the one optimization algorithm, the
problem-solving machine which can provide us with near-optimal solutions
in short time for every possible optimization task. This must sound very
depressing for everybody new to this subject.
Evolutionary
Algorithms
ACO
PSO
Extremal
Optimiz.
Simulated
Annealing
EDA
Tabu
Search
Branch&
Bound Dynamic
Program.
A«
Search
IDDFS
Hill
Climbing
Memetic
Algorithms
Downhill
Simplex
GA,GP,ES,
DE,EP,...
LCS
RFD
Random
Optimiz.
Fig. 17: The puzzle of optimization algorithms.
Actually, quite the opposite is the case, at least from the point of view of a
researcher. The No Free Lunch Theorem means that there will always be new
ideas, new approaches which will lead to better optimization algorithms to
solve a given problem. Instead of being doomed to obsolescence, it is far more
likely that most of the currently known optimization methods have at least
one niche, one area where they are excellent. It also means that it is very likely
that the “puzzle of optimization algorithms” will never be completed. There
will always be a chance that an inspiring moment, an observation in nature,
for instance, may lead to the invention of a new optimization algorithm which
performs better in some problem areas than all currently known ones.
Acknowledgements We gratefully acknowledge comments on early drafts of this chap-
ter by Peter J Bentley and Patrick Siarry. The last author acknowledges support from

38 Weise, Zapf, Chiong, Nebro
the “Consejer´ıa de Innovaci´on, Ciencia y Empresa”, Junta de Andaluc´ıa under contract
P07-TIC-03044 DIRICOM project, http://diricom.lcc.uma.es.
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Why Is Optimization Difficult? 51
@incollection{WZCN2009WIOD,
author = {Thomas Weise and Michael Zapf and Raymond Chiong
and Antonio Jes{\’u}s {Nebro Urbaneja}},
title = {Why Is Optimization Difficult?},
chapter = {1},
booktitle = {Nature-Inspired Algorithms for Optimisation},
publisher = {Springer},
year = {2009},
pages = {1--50},
ISBN = {978-3-642-00266-3},
month = apr # {~30},
issn = {1860-949X (Print) 1860-9503 (Online)},
series = {Studies in Computational Intelligence},
volume = {193},
editor = {Raymond Chiong},
url = {http://www.it-weise.de/documents/files/WZCN2009WIOD.pdf},
url = {http://www.springer.com/engineering/book/978-3-642-00266-3},
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