Schema theorem for computational gene transposition and performance analysis

Abstract

Since the introduction of Jumping Gene Genetic Algorithm (JGGA), the gene transposition in computational evolutionary algorithm has re-gained its power and attentions. In the last few years, many successful applications of JGGA have been reported in various areas, such as antenna designs, resource management in networking systems, control, and so on. The remarkable performance of JGGA is mainly due to the two newly designed genetic transposition mechanisms, namely the copy-and-paste and the cut-and-paste operations. However, though extensive simulations and applications have been carried out to provide a statistical justification on their effectiveness, rigorous theoretical analysis is still in vain. The objective of this thesis is hence to resolve this problem, aiming to provide an explanation on why these two JG operations can enhance the optimization process. Our approach relies on the concept of schema, so that schema theorem for these two JG operations can be devised. Unlike the Holland's or other approximate schema approach, both the destruction and reconstruction of schemata are taken into account and hence an exact schema evolution equation for each operation is to be obtained, serving as a solid foundation for the theoretical studies. The schema evolution equations reflect an important nature of these JG operations from which a theorem, called the theorem of equilibrium, is established. It is mathematically proved that either the copy-and-paste or the cut-and-paste operation will force a uniform distribution of schemata in every primary schemata competition set, despite its initial distribution. This phenomenon implies a global search on the domain of solutions when the population size is sufficiently large. For a finite population, diversity can still be well maintained in the population, which is supported by simulation results. To further analyze the effect of the length of JG, the convergence speed of equilibrium in primary schemata competition sets has been studied. It is also concluded that a low-order schema, with all its definite bits locating away from the edges, will have a fast convergence speed to the equilibrium, implying a high degree of diversity in the population. Finally, some practical applications are presented, not only to supplement the list of JGGA applications, but also to demonstrate the outperforming searching ability of JGGA.