Linear Evolutionary Algorithm

Abstract

During the past three decades, global optimization problems (including single-objective optimization problems (SOP) and multi-objective optimization problems (MOP)) have been intensively studied not only in Computer Science, but also in Engineering. There are many solutions in literature, such as gradient projection method [1-3], Lagrangian and augmented Lagrangian penalty methods [4-6], and aggregate constraint method [7-9]. Among these methods, penalty function method is an important approach to solve global optimization problems.. To obtain the optimal solution of the original problem, the first step is to convert the optimization problem into an unconstrained optimization problem with a certain penalty function (such as Lagrangian multiplier). As the penalty multiplier approaches zero or infinite, the iteration point might approach optimal too. However, at the same time, the objective function of the unconstrained optimization problem might gradually become worse. This leads to increased computational complexity and long computational time in implementing the penalty function method to solve the complex optimization problems. In most of the research, both the original constraints and objective function are required to be smooth (or differentiable). However, in real-world problem, it is seldom to be able to guarantee a derivative for of the specific complex optimization problem. Hence, the development of efficient algorithms for handling complex optimization problems is of great importance. In this chapter, we present a new framework and algorithm that can solve problems belong to the family of stochastic search algorithms, often referred to as evolutionary algorithms. Evolutionary algorithms (EAs) are stochastic optimization techniques based on natural evolution and survival of the fittest strategy found in biological organisms. Evolutionary algorithms have been successfully applied to solve complex optimization problems in business [10,11], engineering [12,13], and science [14,15]. Some commonly used EAs are Genetic algorithms (GAs)[16], Evolutionary Programming (EP)[17], Evolutionary Strategy (ES)[18] and Differential Evolution (DE)[19]. Each of these methods has its own characteristics, strengths and weaknesses. In general, a EA algorithm generate a set of initial solutions randomly based on the given seed and population size. Afterwards, it will go through evolution operations such as cross-over and mutation before evaluated by the