  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. We are optimistic that the question how to profit from good initial
solutions is interesting beyond these first examples.
