Dynamics of information and optimal control of mutation in evolutionary systems

Evolutionary systems are used for search and optimization in complex problems and for modelling population dynamics in nature. Individuals in populations reproduce by simple mechanisms, such as mutation or recombination of their genetic sequences, and selection ensures they evolve in the direction of increasing fitness. Although successful in many applications, evolution towards an optimum or high fitness can be extremely slow, and the problem of controlling parameters of reproduction to speed up this process has been investigated by many researchers. Here, we approach the problem from two points of view: 1) as optimization of evolution in time; 2) as optimization of evolution in information. The former problem is often intractable, because analytical solutions are not available. The latter problem, on the other hand, can be solved using convex analysis, and the resulting control, optimal in the sense of information dynamics, can achieve good results also in the sense of time evolution. The principle is demonstrated on the problem of optimal mutation rate control in Hamming spaces of sequences. To facilitate the analysis, we introduce the notion of a relatively monotonic fitness landscape, and obtain general formula for transition probability by simple mutation in a Hamming space. Several rules for optimal control of mutation are presented, and the resulting dynamics are compared and discussed.