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.

翻译：进化算法的数学运行时分析传统上关注算法在随机初始化条件下找到特定质量解所需的时间。在实际应用中，可能存在优于随机解的猜测解。我们针对此类场景展开数学运行时分析。研究发现，不同算法从更优初始化中获益的程度差异显著，且算法的最优参数配置可能强烈依赖于初始解的质量。为克服这一困难，自适应调整和随机重尾参数选择策略可能更具优势。最后，我们发现所找到的最佳进化算法性能与对应黑箱复杂度之间存在较大差距，这暗示着能够更好利用优质初始解的进化算法仍有待发掘。这些初步发现源于对$(1+1)$进化算法以及静态、自适应调整和重尾$(1 + (\lambda,\lambda))$遗传算法在OneMax基准测试中性能的分析。我们乐观地认为，如何从优质初始解中获益这一问题的研究意义将超越这些初步案例。