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：多样性优化是一类优化问题，其目标是在寻找一组多样性良好的优质解。解决此类问题的常用方法之一是使用进化算法来进化出具有所需多样性的种群，这种方法被称为进化多样性优化（EDO）。本文分析了在3目标函数LOTZ$_k$上的EDO，该函数是对2目标基准函数（LeadingOnes, TrailingZeros）的改进。我们证明了GSEMO能在$O(kn^3)$期望迭代次数内计算出所有帕累托最优解的集合。此外，我们分析了GSEMO$_D$（GSEMO用于多样性优化的改进版本）在两种不同多样性度量——总失衡度和排序失衡向量——下，找到具有最佳可能多样性的种群所需的运行时。对于第一种度量，我们证明GSEMO$_D$能在$O(kn^2\log(n))$期望迭代次数内渐近更快地完成优化，找到帕累托最优种群；对于第二种度量，我们给出了$O(k^2n^3\log(n))$期望迭代次数的上界。我们通过实证研究补充了理论分析，结果表明两种多样性度量下的行为非常相似，且接近理论预测。