A Practical Approach to Metaheuristics using LabVIEW and MATLAB®

Pedro Ponce-Cruz, Arturo Molina Gutiérrez, Ricardo A. Ramírez-Mendoza, Efraín Méndez Flores, Alexandro Antonio Ortiz Espinoza, David Christopher Balderas Silva

• Publisher: Chapman and Hall/CRC
• Year: 2020
• ISBN:
  9781000073478
• Edition: 1
• Topic: Computer Science
• Subtopic: Computer Science General
• Index: Computer Science

Metaheuristic optimization has become a prime alternative for solving complex optimization problems in several areas. Hence, practitioners and researchers have been paying extensive attention to those metaheuristic algorithms that are mainly based on natural phenomena. However, when those algorithms are implemented, there are not enough books that deal with theoretical and experimental problems in a friendly manner so this book presents a novel structure that includes a complete description of the most important metaheuristic optimization algorithms as well as a new proposal of a new metaheuristic optimization named earthquake optimization. This book also has several practical exercises and a toolbox for MATLAB® and a toolkit for LabVIEW are integrated as complementary material for this book. These toolkits allow readers to move from a simulation environment to an experimentation one very fast. This book is suitable for researchers, students, and professionals in several areas, such as economics, architecture, computer science, electrical engineering, and control systems.

The unique features of this book are as follows:

• Developed for researchers, undergraduate and graduate students, and practitioners

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• A friendly description of the main metaheuristic optimization algorithms

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• Theoretical and practical optimization examples

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• A new earthquake optimization algorithm

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• Updated state-of-the-art and research optimization projects

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The authors are multidisciplinary/interdisciplinary lecturers and researchers who have written a structure-friendly learning methodology to understand each metaheuristic optimization algorithm presented in this book.

II

Metaheuristic Optimization

CHAPTER 3

Basic Metaheuristic Optimization Algorithms

3.1 Exhaustive Search

The most basic form of search is to look at all possible solutions and to decide which one is the best. This method is called exhaustive search or brute force method. Although this search ideally guarantees the finding of an optimal solution, the time it requires is high, and in some cases, it is inefficient. As an alternative, it is possible to use an algorithm that limits the search using some kind of heuristics.

3.2 Random Optimization

A simple way of searching the space is random optimization. The method basically searches the space using a random number over certain number of iterations, and if at any step a better solution is found, then it replaces the current one. Although this method does not guarantee the finding of the optimal solution, with enough iterations it may find a solution that is good enough. The pseudocode for random search is shown in Figure 3.1.

Figure 3.1 Random optimization pseudocode.

3.3 Nelder-Mead Algorithm

The Nelder-Mead algorithm [51,75] is a blackbox optimization method that operates similar to some evolutionary algorithms. Differently than some other algorithms, the Nelder-Mead algorithm does not solely resort to selection for improving the average solution, and it does not contain stochastic decision.

The method determines from the three vertices what is the worst vertex, which would be the one with the largest function value, and this worst vertex will be replaced with a new vertex to form a new triangle. This process will be repeated, thus having different shapes of triangle, and in general, this method is trying to make the triangle smaller until the minimum is found. The algorithm has mainly 4 steps:

• Reflection
• Expansion
• Contraction
• Shrinking.

The Nelder-Mead method starts with a triangle with three vertices x₁, x₂, and x₃. These vertices are evaluated in f (x), and the order of selecting the smaller is the best, the second smaller is good and the largest is the worst, which are defined as B, G, and W, respectively.

After the vertices are ordered, a midpoint between B and G is calculated in order to find a new point that will replace W. The midpoint is the average of the coordinates:

$$M=\frac{{\displaystyle B+G}}{{\displaystyle 2}}(3.1)$$

Then, since the function decreases from W to either B or G, it is possible that a smaller value can be found at the opposite side of W that passes through the midpoint M. Hence, a reflection point can be calculated by getting the distance d from W to M and using this distance on the other side. The formula for R is

$$R=M+(M-W)(3.2)$$

If f (R) is smaller than f (W), then the movement is assumed as correct. At this point, it can be supposed that the minimum is farther from R, so the segment is expanded to the point E by moving an extra distance d. The expansion point E can be calculated as follows:

$$E=R+(R-M)(3.3)$$

Assuming that f(E) is smaller than f(R), E will be a better vertex.

If the function f(R) is larger than or the same as f(W), then another point must be calculated. At this point, two contraction midpoints C₁ and C₂ are selected in betwee...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• List of Figures
• List of Tables
• Foreword
• Preface
• Authors
• Section I Basis
• Section II Metaheuristic Optimization
• Appendix
• Bibliography
• Index