First steps towards a runtime analysis when starting with a good solution

The mathematical runtime analysis of evolutionary algorithms traditionally regards the time an algorithm needs to find a solution of a certain quality when initialized with a random population. In practical applications it may be possible to guess solutions that are better than random ones. We start a mathematical runtime analysis for such situations. We observe that different algorithms profit to a very different degree from a better initialization. We also show that the optimal parameterization of the algorithm can depend strongly on the quality of the initial solutions. To overcome this difficulty, self-adjusting and randomized heavy-tailed parameter choices can be profitable. Finally, we observe a larger gap between the performance of the best evolutionary algorithm we found and the corresponding black-box complexity. This could suggest that evolutionary algorithms better exploiting good initial solutions are still to be found. These first findings stem from analyzing the performance of the $$(1+1)$$ evolutionary algorithm and the static, self-adjusting, and heavy-tailed $$(1 + (\lambda,\lambda ))$$ GA on the OneMax benchmark, but we are optimistic that the question how to profit from good initial solutions is interesting beyond these first examples.