The role of completeness in convergence of evolutionary algorithms

It is interesting that typically in the proof of convergence of evolutionary algorithms only elitist selection is considered. In this paper, we stress out that truly in reaching optimum of the fitness function completeness of search plays probably even more important role. The elitist selection helps in reaching the optimum. It allows to converge faster and not to lose the optimum in cases when we are uncertain that the optimum has been reached. An evolutionary search using elitist selection, but being incomplete, will not reach the optimum. This paper provides sufficient conditions for finding the best (optimal) solutions, or the best (totally optimal) solutions with minimal search costs. To do that we utilized a new formal model of evolutionary computation the evolutionary Turing machine. The results are applicable both to genetic algorithms, genetic programming, evolution strategies and evolutionary programming. The problem of finding total optimum, optimizing together the quality of solution together with an evolutionary algorithm is considered as a multiobjective optimization allowing to tackle real-world problems where the complexity of evolutionary search becomes an issue. A new classification of evolutionary procedures into easy, hard, and solvable in the limit classes, is proposed.