Algorithms for Variable-Size Optimization

Applications in Space Systems and Renewable Energy

Ossama Abdelkhalik

• Publisher: CRC Press
• Year: 2021
• ISBN:
  9781351119085
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Many systems architecture optimization problems are characterized by a variable number of optimization variables. Many classical optimization algorithms are not suitable for such problems. The book presents recently developed optimization concepts that are designed to solve such problems. These new concepts are implemented using genetic algorithms and differential evolution. The examples and applications presented show the effectiveness of the use of these new algorithms in optimizing systems architectures.

The book focuses on systems architecture optimization. It covers new algorithms and its applications, besides reviewing fundamental mathematical concepts and classical optimization methods. It also provides detailed modeling of sample engineering problems. The book is suitable for graduate engineering students and engineers. The second part of the book includes numerical examples on classical optimization algorithms, which are useful for undergraduate engineering students.

While focusing on the algorithms and their implementation, the applications in this book cover the space trajectory optimization problem, the optimization of earth orbiting satellites orbits, and the optimization of the wave energy converter dynamic system: architecture and control. These applications are illustrated in the starting of the book, and are used as case studies in later chapters for the optimization methods presented in the book.

II
CLASSICAL OPTIMIZATION ALGORITHMS

Chapter 3
Fundamentals and Core Algorithms

Part I presented the necessary conditions of optimality. It was also shown that for relatively simple optimization problems, the necessary conditions for optimality can be used to find the optimal solution. For most practical optimization problems, numerical methods are needed to search for the optimal solution. This chapter presents basic numerical methods that are designed to solve relatively simple problems numerically. These algorithms however are building blocks that are usually used in more complex numerical algorithms that will be presented in subsequent chapters in Part II.

Consider the case of an unconstrained optimization problem that has only one optimization variable x, and the objective function is f(x). The necessary condition for optimality in this case is:

$$\varphi (x)\equiv \frac{df}{dx}=0$$ (3.1)

Solving the necessary condition for optimality can be thought of as a root finding problem to find the roots of φ(x) = 0. Numerical algorithms for root finding include the Newton-Raphson and the bisection algorithms. The Newton-Raphson algorithm starts with an initial guess x₀ and updates this guess over subsequent iterations using the gradient dφ(x)/dx. The bisection algorithm does not need the gradient information. It starts with two points x₁ and x₂ selected such that φ(x₁)φ(x₂) < 0 (hence x₁ and x₂ are on two opposite sides of the root), and repeatedly halves the i...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication Page
• Preface
• Contents
• Symbol Description
• Section I: Background and Motivation
• Section II: Classical Optimization Algorithms
• Section III: Variable-Size Design Space Optimization
• Section IV: Applications
• Bibliography
• Index