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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.

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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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theoriginal.

http://www.it-weise.de/

Original available at http://www.springer.com/engineering/book/

978-3-642-00266-3

See citation suggestion at the end of this document

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

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)

Fig. 1.a: Best Case

objectivevaluesf(x)

Fig. 1.b: Low Total Variation

multiple(local)optima

objectivevaluesf(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)

Fig. 1.e: Deceptive

neutralarea

objectivevaluesf(x)

Fig. 1.f: Neutral

neutral area or

areawithoutmuch

information

needle

(isolated

optimum)

objectivevaluesf(x)

Fig. 1.g: Needle-In-A-Haystack

objectivevaluesf(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)

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 R n

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

1110

1000

0100

0111

0010

1111

f (x)

(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 x of an optimization

problem are elements of the problem space X . Their utility as solutions is

evaluated by a set f of objective functions f which, 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 : G 7→ 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"2 subset 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) f is 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)

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 G in the search space

Gby averaging over all their elements. Regions where ν is 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 ν in 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 0 and 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 ν very

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 ν can 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 G and 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 ) = x and f2 (x ) = −x which are clearly contradicting regardless of

whether they are subject to maximization or minimization. Nevertheless, if

the solution candidates x as 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 m for 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)

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 ∈R in the method

of Greiner [87, 88]. If the distribution and inﬂuence of the δi are known,

the objective function f (x ) : x∈X can be rewritten as ˜

f(x ,δ) [235]. In

the special case where δ is 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 δ a number of t times 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 y of the objective function f is

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

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 X 6⊆ 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∈X created 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 A of n training data samples

(xi , yi ) and want to ﬁnd a function f that 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 ) = x shown 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 f2 which also match the data, as shown in Fig. 9.c. Here, f2

plays the role of the overly complicated model m which 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

Fig. 9.a: Three sample

points of f1 .

Fig. 9.b: m′ ≡ f1 (x ) = x.

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.

Fig. 10.a: The original

physical process.

Fig. 10.b: The measure-

ment/training data.

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 ,...,a nwhich were used

26 Weise, Zapf, Chiong, Nebro

for training during the optimization process, but also for diﬀerent inputs

a6= ai ∀ i: 0 < i ≤ n if such inputs a exist.

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 X in 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 At and 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 Ac than 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 P in 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.

Fig. 11.a: The "real sys-

tem" and the points de-

scribing it.

Fig. 11.b: The sampled

training data.

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 fi which must be optimized simultaneously. Furthermore, the so-

lutions must satisfy m inequality constraints g and p equality constraints h.

Why Is Optimization Diﬃcult? 29

A solution candidate x is 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 X which minimizes

(or maximizes) the vector function f ( x⋆ ) = (fi (x⋆ ) , f2 ( x⋆ ) , .., fk (x⋆ ))T and is

feasible, (i.e., satisﬁes the m inequality constraints gi ( x⋆ )≥ 0 ∀i = 1, 2, .., m,

the p equality 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 Rn for n > 1, the notions of "better than" and "optimum" have to

be redeﬁned. When comparing any two solutions x1 and x2 , solution x1 can

have a better value in objective fi , i.e., fi (x1 ) < fi ( x2 ), while solution x2 can

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 x1 is said to dominate another so-

lution candidate x2 (denoted by x1 4x2 ) 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 ).

1 2 3 4 5 6 7 8 9 10 11

ParetoFront

1 2

⋠

12 ⋠

1´

1⋠

⋠

Fig. 12: Some examples for the dominance relation.

The dominance notion allows us to assume that if solution x1 dominates

solution x2 , then x1 is preferable to x2 . If both solution are non-dominated

(such as candidate ① and ② in 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⋆ ∈X is Pareto-

optimal if and only if there is no feasible xb ∈X with xb 4x⋆ .

This deﬁnition states that x⋆ is Pareto-optimal if there is no other feasible

solution xb which 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 :x4 x⋆ } .

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, f1 could

represent the time required by a car to cover a given distance, while f2 could

be the fuel consumption.

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.

trueParetofront

frontreturned

bytheoptimizer

Fig. 14.a: Bad Convergence and Good

Spread

Fig. 14.b: Good Convergence and Bad

Spread

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 spread7 of 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 ) = P k

i=1 f i (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].

Fig. 15.a: Non-Convex (Concave)

Fig. 15.b: Disconnected

Fig. 15.c: linear

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)]2 where f is the original objec-

tive function, h is 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 t will 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 f is always identical for all optimization

algorithms.

From this theorem, we can immediately follow that, in order to outper-

form the optimization method a1 in one optimization problem, the algorithm

a2 will 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 x is 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/8 gives 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

Branch&

Bound Dynamic

Program.

A«

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},

}

... The search space created by a real-world single-or multi-objective OP is not free of variable linkage, epistasis, rotation and other relations between their decision variables (Weise, Zapf et al. 2009;Weise, Chiong et al. 2012). Moreover, these properties could change throughout the search space and would make the optimization process uncertain, which must be managed. ...

In this paper, we propose a novel hybrid fuzzy–metaheuristic approach with the aim of overcoming premature convergence when solving multimodal single and multi-objective optimization problems. The metaheuristic algorithm used in our proposed approach is based on the imperialist competitive algorithm (ICA), a population-based method for optimization. The ICA divides its population into sub-populations, known as empires. Each empire is composed of a high fitness solution—the imperialist—and some lower fitness solutions—the colonies. Colonies move towards their associated imperialist in each empire to achieve better status (higher fitness). The most powerful empire tends to attract weaker colonies. These competitions and movements can be enhanced for better algorithm performance. In our hybrid approach, a global learning strategy is devised for each colony to learn from its best-known position, its associated imperialist and the global best imperialist. A fast-evolutionary elitism local search is used to enhance the collaborative search mechanism (competition) in each empire, and thus the overall optimization performance may be improved. Other main evolutionary operators include velocity adaptation and velocity divergence. To address parameterization and computational cost evaluation issues, two fuzzy inferencing mechanisms are designed and used in parallel: one is a learning strategy adaptor in each run, and the other is a smart evolution selector in each running window. For Pareto front approximation, fast-elitism non-dominated sorting is applied to the solutions, and a novel penalized sigma diversity index is designed to estimate the diversity (power) of solutions in the same rank. Comprehensive experimental results based on 22 well-known single-objective and 25 multi-objective benchmark instances clearly show that our proposed approach has better solution quality compared with other popular metaheuristics and state-of-the-art ICA variants. The proposed approach can be used as an optimization module in any intelligent decision-making systems to enhance efficiency and accuracy. The designed fuzzy inferencing mechanisms can also be incorporated into any single- or multi-objective optimizers for parameter tuning purposes, to make the optimizers more adaptive to new problems or environments.

... There are a number of potential pitfalls that can inhibit an optimisation task (Weise et al., 2009). Firstly, it must be ensured that the global maximum of the function under consideration is found and not just a local maximum. ...

Dean Markwick

Many statistical problems involve modelling the times at which events occur. There are cases where events can occur in clusters with sudden jumps in the total number of occurrences. To model such data an intensity function can be constructed which describes the probability of an event occurring at a specific time. The Hawkes process is a point process model with a conditional intensity function that provides a change in intensity for each event occurrence and as such the Hawkes process can be used to explain event clustering. The flexibility and extendability of the Hawkes process will be highlighted in this thesis. I extend the Hawkes process by using nonparametric Bayesian methods where different components of the Hawkes process are constructed using a Dirichlet process which is a Bayesian prior for distributions. This allows for a data driven approach and removes the need for parametric assumptions. This Bayesian approach also allows for a hierarchical structure to be integrated in the models where appropriate. These extended Hawkes process are applied to different application domains including: extreme value theory, financial trading and soccer goal occurrence modelling. Each new application introduces a different extension to the Hawkes process and illustrates how it improves on existing methodology. From this research I also wrote a new software package for using Dirichlet processes. This software enables users to easily construct Dirichlet process objects that can incorporated into existing inference workflows. This allows users to introduce nonparametric methods without needing to program their own inference methods.

... Given the fact that as many dimensions the problem has, as much time would be needed to check all possible solutions, we conclude that exhaustive search is not an option. Thus, optimization is a difficult process (Weise et al. 2009), because the goal is to find one of the possible sub-optimal solutions existing in the search space, without knowing either the global optimum, or if there is already a better sub-optimal solution that has not been discovered yet. ...

In the last decade, we observe an increasing number of nature-inspired optimization algorithms, with authors often claiming their novelty and their capabilities of acting as powerful optimization techniques. However, a considerable number of these algorithms do not seem to draw inspiration from nature or to incorporate successful tactics, laws, or practices existing in natural systems, while also some of them have never been applied in any optimization field, since their first appearance in literature. This paper presents some interesting findings that have emerged after the extensive study of most of the existing nature-inspired algorithms. The need for irrationally introducing new nature inspired intelligent (NII) algorithms in literature is also questioned and possible drawbacks of NII algorithms met in literature are discussed. In addition, guidelines for the development of new nature-inspired algorithms are proposed, in an attempt to limit the misleading appearance of variation of metaheuristics as nature inspired optimization algorithms.

Loris Serafino

The chapter considers the recent but already classic theoretical result called No Free Lunch Theorem in the context of optimization practice. The No Free Lunch Theorem is probably the fundamental theoretical result of the Machine Learning field but its practical meaning and implication for practitioners facing "real life" industrial and design optimization problems are rarely addressed in the technical literature. This discussion is intended for a broad audience of mathematicians, engineers, and computer scientists and presents a probabilistic understanding of the theorem that can shed light to its meaning and impact in the industrial optimization practice.

jMetal is a Java-based framework for multi-objective optimization with metaheuristics that is widely used in the field. The jMetal project started in 2006, and it is continuously evolving. In this chapter, we describe a set of new features that are in current development, aimed at facilitating the analysis of the results of executing multi-objective algorithms and at to improve the usability of jMetal to solve real-world problems. Concretely, we focus on the new features related to algorithm templates, visualization, analysis of results, auto-configuration of multi-objective metaheuristics, and parallelism.

In the field of complex problem optimization with metaheuristics, semantics has been used for modeling different aspects, such as: problem characterization, parameters, decision-maker's preferences, or algorithms. However, there is a lack of approaches where ontologies are applied in a direct way into the optimization process, with the aim of enhancing it by allowing the systematic incorporation of additional domain knowledge. This is due to the high level of abstraction of ontologies, which makes them difficult to be mapped into the code implementing the problems and/or the specific operators of metaheuristics. In this paper, we present a strategy to inject domain knowledge (by reusing existing ontologies or creating a new one) into a problem implementation that will be optimized using a metaheuristic. Thus, this approach based on accepted ontologies enables building and exploiting complex computing systems in optimization problems. We describe a methodology to automatically induce user choices (taken from the ontology) into the problem implementations provided by the jMetal optimization framework. With the aim of illustrating our proposal, we focus on the urban domain. Concretely, we start from defining an ontology representing the domain semantics for a city (e.g., building, bridges, point of interest, routes, etc.) that allows defining a-priori preferences by a decision maker in a standard, reusable, and formal (logic-based) way. We validate our proposal with several instances of two use cases, consisting in bi-objective formulations of the Traveling Salesman Problem (TSP) and the Radio Network Design problem (RND), both in the context of an urban scenario. The results of the experiments conducted show how the semantic specification of domain constraints are effectively mapped into feasible solutions of the tackled TSP and RND scenarios. This proposal aims at representing a step forward towards the automatic modeling and adaptation of optimization problems guided by semantics, where the annotation of a human expert can be now considered during the optimization process.

Mixed-discrete optimization deals with mathematical optimization problems with multiple types of variables: discrete (nominal) taking values from a not-sortable set of possible elements, integer variables and variables taking values in a continuous domain. Mixed-discrete problems appear naturally in many contexts such as in the real world in the engineering domain, bioinformatics and data sciences, and this has led to an increased interest in the design of strong algorithms for different variants of the problem. Much effort has been spent over the last decades in studying and developing new methodologies, but unfortunately mixed-discrete optimization problems are much less understood then their "non-mixed" counterparts. In this chapter we will focus on the rather new approaches to handle mixed-discrete problems by means of surrogate methods.

In this paper, we investigate how systemic errors due to random sampling impact on automated algorithm selection for bound-constrained, single-objective, continuous black-box optimization. We construct a machine learning based algorithm selector, which uses exploratory landscape analysis features as inputs. We test the accuracy of the recommendations experimentally using resampling techniques and the hold-one-instance-out and hold-one-problem-out validation methods. The results demonstrate that the selector remains accurate even with sampling noise, although not without trade-offs.

The Transit Network Design and Frequency Setting Problem (TNDFSP) can be defined as the creation of effective routes in a public transport network and the determination of relevant frequencies. Generally, the TNDFSP problem is in the same category as the Traveling Salesman Problem (TSP), which is known to be a non-deterministic polynomial-period (NP-hard) difficult problem. This study consists of two stages: first, the design of a public transport network with an evolutionary optimization technique – the Intelligent Water Drops (IWD) algorithm – based on the TSP and the determination of relevant frequencies; and second, the assignment of passengers to designated routes. All decisions related to public transport network design may be evaluated by considering environmental costs in relation to passengers, operators, and the environment. This study presents an acceptable, constructive, and original algorithm.

Karsten Weicker

Evolutionäre Algorithmen sind relativ neue Methoden zur Lösung von Optimierungsproblemen in Industrie, Wirtschaft und Forschung. Inspiriert durch die biologische Evolution imitieren sie das Wechselspiel zwischen Variation von Individuen und Selektion. In diesem Lehrbuch wird neben der Darstellung der Standardalgorithmen vor allem das gängige Verständnis für die Arbeitsweise und die zu Grunde liegenden Prinzipien vermittelt. Darüber hinaus werden spezielle Anforderungen aus der Praxis, wie z. B. die Beachtung von Randbedingungen, Mehrzieloptimierung und verrauschte oder zeitabhängige Probleme, diskutiert. In der nun vorliegenden zweiten Auflage wurde neben einer anschaulichen Darstellung der Grundlagen anhand vieler Beispiele insbesondere die praktische Anwendung, z. B. durch Fallbeispiele aus verschiedenen Themenbereichen, stärker berücksichtigt.

Stuart A Kauffman

Stuart Kauffman here presents a brilliant new paradigm for evolutionary biology, one that extends the basic concepts of Darwinian evolution to accommodate recent findings and perspectives from the fields of biology, physics, chemistry and mathematics. The book drives to the heart of the exciting debate on the origins of life and maintenance of order in complex biological systems. It focuses on the concept of self-organization: the spontaneous emergence of order widely observed throughout nature. Kauffman here argues that self-organization plays an important role in the emergence of life itself and may play as fundamental a role in shaping life's subsequent evolution as does the Darwinian process of natural selection. Yet until now no systematic effort has been made to incorporate the concept of self-organization into evolutionary theory. The construction requirements which permit complex systems to adapt remain poorly understood, as is the extent to which selection itself can yield systems able to adapt more successfully. This book explores these themes. It shows how complex systems, contrary to expectations, can spontaneously exhibit stunning degrees of order, and how this order, in turn, is essential for understanding the emergence and development of life on Earth. Topics include the new biotechnology of applied molecular evolution, with its important implications for developing new drugs and vaccines; the balance between order and chaos observed in many naturally occurring systems; new insights concerning the predictive power of statistical mechanics in biology; and other major issues. Indeed, the approaches investigated here may prove to be the new center around which biological science itself will evolve. The work is written for all those interested in the cutting edge of research in the life sciences.

Martijn Huynen

RNA secondary-structure folding algorithms predict the existence of connected networks of RNA sequences with identical secondary structures. Fitness landscapes that are based on the mapping between RNA sequence and RNA secondary structure hence have many neutral paths. A neutral walk on these fitness landscapes gives access to a virtually unlimited number of secondary structures that are a single point mutation from the neutral path. This shows that neutral evolution explores phenotype space and can play a role in adaptation.

Gunar E. Liepins
Michael D. Vose

We address deceptiveness, one of at least four reasons genetic algorithms can fail to converge to function optima. We construct fully deceptive functions and other functions of intermediate deceptiveness. For the fully deceptive functions of our construction, we specify linear transformations that induce changes of representation to render the functions fully easy. We further model genetic algorithm selection and recombination as the interleaving of linear and quadratic operators. Spectral analysis of the underlying matrices allows us to draw preliminary conclusions about fixed points and their stability. We also obtain an explicit formula relating the nonuniform Walsh transform to the dynamics of genetic search.

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