Xin-She Yang - Nature-Inspired Optimization Algorithms

Algorithms are important tools for solving problems computationally. All computation involves algorithms, and the efficiency of an algorithm largely determines its usefulness. This chapter provides an overview of the fundamentals of algorithms and their links to self-organization, exploration, and exploitation. A brief history of recent nature-inspired algorithms for optimization is outlined in this chapter.

Keywords

Algorithm

Optimization

Nature-inspired

Metaheuristic

Self-organization

Optimization is paramount in many applications, such as engineering, business activities, and industrial designs. Obviously, the aims of optimization can be anythingto minimize the energy consumption and costs, to maximize the profit, output, performance, and efficiency. It is no exaggeration to say that optimization is needed everywhere, from engineering design to business planning and from Internet routing to holiday planning. Because resources, time, and money are always limited in real-world applications, we have to find solutions to optimally use these valuable resources under various constraints. Mathematical optimization or programming is the study of such planning and design problems using mathematical tools. Since most real-world applications are often highly nonlinear, they require sophisticated optimization tools to tackle. Nowadays, computer simulations become an indispensable tool for solving such optimization problems with various efficient search algorithms.

Behind any computer simulation and computational methods, there are always some algorithms at work. The basic components and the ways they interact determine how an algorithm works and the efficiency and performance of the algorithm.

This chapter introduces algorithms and analyzes the essence of the algorithm. Then we discuss the general formulation of an optimization problem and describe modern approaches in terms of swarm intelligence and bio-inspired computation. A brief history of nature-inspired algorithms is reviewed.

In essence, an algorithm is a step-by-step procedure of providing calculations or instructions. Many algorithms are iterative. The actual steps and procedures depend on the algorithm used and the context of interest. However, in this book, we mainly concern ourselves with the algorithms for optimization, and thus we place more emphasis on iterative procedures for constructing algorithms.

For example, a simple algorithm of finding the square root of any positive number or , can be written as

(1.1)

starting from a guess solution , say, . Here, is the iteration counter or index, also called the pseudo-time or generation counter .

This iterative equation comes from the rearrangement of in the following form:

(1.2)

For example, for with , we have

(1.3)

(1.4)

(1.5)

We can see that after just five iterations (or generations) is very close to the true value of which shows that this iteration method is very efficient.

The reason that this iterative process works is that the series converges to the true value due to the fact that

(1.6)

as . However, a good choice of the initial value will speed up the convergence. A wrong choice of could make the iteration fail; for example, we cannot use as the initial guess, and we cannot use either since (in this case, the iterations will approach another root: ).

So a sensible choice should be an educated guess. At the initial step, if is the lower bound and is upper bound. If , then