The diversity optimization is the class of optimization problems, in which we aim at finding a diverse set of good solutions. One of the frequently used approaches to solve such problems is to use evolutionary algorithms which evolve a desired diverse population. This approach is called evolutionary diversity optimization (EDO). In this paper, we analyse EDO on a 3-objective function LOTZ$_k$, which is a modification of the 2-objective benchmark function (LeadingOnes, TrailingZeros). We prove that the GSEMO computes a set of all Pareto-optimal solutions in $O(kn^3)$ expected iterations. We also analyze the runtime of the GSEMO$_D$ (a modification of the GSEMO for diversity optimization) until it finds a population with the best possible diversity for two different diversity measures, the total imbalance and the sorted imbalances vector. For the first measure we show that the GSEMO$_D$ optimizes it asymptotically faster than it finds a Pareto-optimal population, in $O(kn^2\log(n))$ expected iterations, and for the second measure we show an upper bound of $O(k^2n^3\log(n))$ expected iterations. We complement our theoretical analysis with an empirical study, which shows a very similar behavior for both diversity measures that is close to the theory predictions.

翻译：多样性优化是一类旨在寻找多样化优质解集的优化问题。解决此类问题的常用方法之一是使用进化算法来演化出所需的多样化种群，这种方法被称为进化多样性优化（EDO）。本文针对三目标函数LOTZ$_k$（即对二目标基准函数(LeadingOnes, TrailingZeros)的改进）分析了EDO算法。我们证明了GSEMO算法能够在$O(kn^3)$期望迭代次数内计算出所有帕累托最优解集。同时，针对两种不同的多样性度量指标——总失衡度和排序失衡向量，我们分析了GSEMO$_D$（GSEMO的多样性优化改进版本）在找到具有最优多样性的种群前的运行时间。对于第一种度量指标，我们证明了GSEMO$_D$在$O(kn^2\log(n))$期望迭代次数内即可渐进快地完成优化（快于其找到帕累托最优种群的时间）；对于第二种度量指标，我们给出了$O(k^2n^3\log(n))$期望迭代次数的上界。我们通过实证研究对理论分析进行了补充，结果表明两种多样性度量指标均展现出与理论预测高度相似的行为特征。