Table Of ContentApplied Evolutionary
Algorithms for
Engineers Python
Using
Leonardo Azevedo Scardua
Federal Institute of Technology of Espírito Santo
Vitoria, Brazil
p,
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Preface
This book has been written for those who seek to apply evolutionary algorithms
to problems in engineering and science. To this end, it provides the theoretical
background necessary for the understanding of the presented evolutionary algorithms
and their shortcomings, while also discussing themes that are pivotal to the successful
application of evolutionary algorithms to real-world problems.
The theoretical descriptions are illustrated with Python implementations of the
algorithms, which not only allow readers to consolidate their understanding but also
provide a sound starting point for those intending to apply evolutionary algorithms
to optimization problems in their fields of work. Python has been chosen due to its
widespread adoption in the Artificial Intelligence community. Those familiar with
high level languages such as MATLABTM would find no difficulty in reading the
provided Python implementations of the evolutionary algorithms.
Instead of attempting to encompass most of the existing evolutionary
algorithms, this book focuses on those algorithms that researchers have recently
applied to difficult engineering optimization problems, such as control problems
with continuous action spaces and the training of high-dimensional convolutional
neural networks. The basic characteristics of real-world optimization problems are
presented, together with advice on how to properly apply evolutionary algorithms
to them. The applied nature of this book is enforced by the presentation of cases of
successful application of evolutionary algorithms to optimization problems which
are closely related to real-world problems. The presentation is complemented by
Python source code, allowing the user to gain insight into the idiosyncrasies of the
practical application of evolutionary algorithms.
All source code presented in this book was developed in Python 3.7.6. The
implementations are meant to be didactic instead of computationally efficient. For
this reason, vectorization and other Python implementation tricks which specifically
aim at reducing execution times, but add extra complexity to the source code, have
been avoided.
iv * Applied Evolutionary Algorithms for Engineers Using Python
This book is organized as follows. Chapter 1 provides an introduction to what
makes an optimization problem harder to solve and why evolutionary algorithms are a
suitable approach to the solution of those problems. Chapter 2 presents an introduction
to the field of optimization. Those familiar with the core concepts of this field can
safely skip the chapter. Chapter 3 presents an introduction to evolutionary algorithms,
providing the theoretical basis necessary for understanding of the remainder of the
book. Chapters 4, 5, 6 and 7 describe evolutionary algorithms designed to deal with
single-objective optimization problems. Evolutionary algorithms designed to handle
multi-objective optimization problems are presented in Chapters 8 and 9. Chapter 10
describes how to apply evolutionary algorithms to difficult optimization problems,
while Chapter 11 describes how to assess the solutions produced by evolutionary
algorithms. Chapters 12 and 13 provide detailed descriptions of the application of
evolutionary algorithms to two difficult optimization problems.
Contents
Preface iii
Glossary viii
SECTION I: INTRODUCTION
1. EbolutionaryAlgorithmsandDifficultOptimizationProblems 2
1.1 What Makes an Optimization Problem Harder to Solve 3
1.2 Why Evolutionary Algorithms 6
2. Introduction toOptimization 7
2.1 What is Optimization 7
2.2 Solutions of an Optimization Problem 8
2.3 Maximization or Minimization 8
2.4 Basic Mathematical Formulation 8
2.5 Constraints and Feasible Regions 9
2.6 Local Solutions and Global Solutions 11
2.7 Multimodality 11
2.8 Multi-Objective Optimization 11
2.9 Combinatorial Optimization 15
3. Introduction to Evolutionary Algorithms 18
3.1 Representing Candidate Solutions 19
3.1.1 Discrete Representations 20
3.1.2 Integer Representation 24
3.1.3 Real-valued Representation 24
3.2 Comparing Representations on a Benchmark Problem 24
3.3 The Fitness Function 25
vi * Applied Evolutionary Algorithms for Engineers Using Python
3.4 Population 32
3.5 Selecting Parents 32
3.5.1 Selection Probabilities 33
3.5.2 Sampling 38
3.5.3 Selection of Individuals 43
3.6 Crossover (Recombination) 46
3.6.1 Recombination for Discrete Representations 47
3.6.2 Recombination for Real-valued Representations 52
3.7 Mutation 57
3.7.1 Mutation for Binary Representations 58
3.7.2 Mutation for Real-valued Representations 60
3.7.3 Mutation for Integer Representations 65
3.8 Elitism 66
SECTION II: SINGLE-OBJECTIVE EVOLUTIONARY
ALGORITHMS
4. SwarmOptimization 70
4.1 Ant Colony Optimization 71
4.2 Particle Swarm Optimization 80
5. Evolution Strategies 86
5.1 Recombination Operators 87
5.2 Mutation Operators 89
5.3 The (1 + 1) ES 91
5.4 The (u + y) ES 98
5.5 Natural Evolution Strategies 104
5.6 Covariance Matrix Adaptation Evolution Strategies 108
6. GeneticAlgorithms 117
6.1 Real-Valued Genetic Algorithm 118
6.2 Binary Genetic Algorithm 122
7. Differential Ebolution 127
SECTION III: MULTI-OBJECTIVE EVOLUTIONARY
ALGORITHMS
8. Non-DominatedSortedGeneticAlgorithmII 137
9. MultiobjectibeEbolutionaryAlgorithmBasedonDecomposition 149
Contents * vii
SECTION IV: APPLYING EVOLUTIONARY
ALGORITHMS
10. SolbingOptimizationProblems withEbolutionaryAlgorithms 160
10.1 Benchmark Problems 160
10.1.1 Single-Objective 160
10.1.2 Multi-Objective 162
10.1.3 Noisy 163
10.2 Dealing with Constraints 164
10.3 Dealing with Costly Objective Functions 168
10.4 Dealing with Noise 169
10.5 Evolutionary Multi-Objective Optimization 180
10.6 Some Auxiliary Functions 184
11. AssessingthePerformanceofEbolutionaryAlgorithms 186
11.1 A Cautionary Note 186
11.2 Performance Metric 187
11.3 Confidence Intervals 187
11.4 Assessing the Performance of Single-Objective 195
Evolutionary Algorithms
11.5 Assessing the Performance of Multi-Objective 198
Evolutionary Algorithms
11.6 Benchmark Functions 203
12. CaseStudy;OptimalDesignofaGear Train System 207
13. CaseStudy;Teaching aLeggedRobotHowtoWalk 217
References 233
Index 241
Glossary
ACO Ant Colony Optimization
Adam Adaptive Moment Estimation
AI Artificial Intelligence
AS Ant System
CMAES Covariance Matrix Adaptation Evolution Strategies
COP Constrained Optimization Problem
CSP Chebyshev Scalarization Problem
DE Differential Evolution
EA Evolutionary Algorithm
Elitist AS Elitist Ant System
EMO Evolutionary Multi-Objective Optimization
ES Evolution Strategies
FFNN Feed-Forward Neural Network
GA Genetic Algorithm
IGD Inverted Generational Distance
MOEA Multi-Objective Evolutionary Algorithm
MOEA/D Multi-Objective Evolutionary Algorithm Based on Decomposition
MOP Multi-Objective Optimization Problem
NES Natural Evolution Strategies
NFL No Free Lunch
NN Neural Network
NSGA Non-dominated Sorting Genetic Algorithm
NSGA-II Non-dominated Sorting Genetic Algorithm II
PSO Particle Swarm Optimization
RMSE Root Mean Squared Error
SBX Simulated Binary Crossover
SGD Stochastic Gradient Descent
SUS Stochastic Universal Sampling
TSP Traveling Salesman Problem
I
INTRODUCTION
