Crossover Invariant Subsets of the Search Space for Evolutionary Algorithms
本文建立了一个数学框架，研究交叉算子与搜索空间不变子集的关系，证明了掩码交叉是使Antonisse模式族不变的最大变换族，并给出了传统交叉下子集完整动态跨度所需的迭代次数。
This paper addresses the relationship between schemata and crossover operators. In Appendix A a general mathematical framework is developed which reveals an interesting correspondence between the families of reproduction transformations and the corresponding collections of invariant subsets of the search space. On the basis of this mathematical apparatus it is proved that the family of masked crossovers is, for all practical purposes, the largest family of transformations whose corresponding collection of invariant subsets is the family of Antonisse's schemata. In the process, a number of other interesting facts are shown. It is proved that the full dynastic span of a given subset of the search space under either one of the traditional families of crossover transformations (one-point crossovers or masked crossovers) is obtained after [log2n] iterations where n is the dimension of the search space. The generalized notion of invariance introduced in the current paper unifies Radcliffe's notions of firespectfl and figene transmissionfl. Besides providing basic tools for the theoretical analysis carried out in the current paper, the general facts established in Appendix A provide a way to extend Radcliffe's notion of figenetic representation functionfl to compare various evolutionary computation techniques via their representation.