Static and Self-Adjusting Mutation Strengths for Multi-valued Decision Variables

Abstract

The most common representation in evolutionary computation are bit strings. With very little theoretical work existing on how to use evolutionary algorithms for decision variables taking more than two values, we study the run time of simple evolutionary algorithms on some OneMax-like functions defined over $$\varOmega = \{0, 1, \ldots , r-1\}^n$$ . We observe a crucial difference in how we extend the one-bit-flip and standard-bit mutation operators to the multi-valued domain. While it is natural to modify a random position of the string or select each position of the solution vector for modification independently with probability 1/n, there are various ways to then change such a position. If we change each selected position to a random value different from the original one, we obtain an expected run time of $$\varTheta (nr \log n)$$ . If we change each selected position by $$+1$$ or $$-1$$ (random choice), the optimization time reduces to $$\varTheta (nr + n\log n)$$ . If we use a random mutation strength $$i \in \{0,1,\ldots ,r-1\}$$ with probability inversely proportional to i and change the selected position by $$+i$$ or $$-i$$ (random choice), then the optimization time becomes $$\varTheta (n \log (r)(\log n +\log r))$$ , which is asymptotically faster than the previous if $$r = \omega (\log (n) \log \log (n))$$ . Interestingly, a better expected performance can be achieved with a self-adjusting mutation strength that is based on the success of previous iterations. For the mutation operator that modifies a randomly chosen position, we show that the self-adjusting mutation strength yields an expected optimization time of $$\varTheta (n (\log n + \log r))$$ , which is best possible among all dynamic mutation strengths. In our proofs, we use a new multiplicative drift theorem for computing lower bounds, which is not restricted to processes that move only towards the target.