Indicator-based (multiobjective) diversity optimization aims at finding a set of near (Pareto-)optimal solutions that maximizes a diversity indicator, where diversity is typically interpreted as the number of essentially different solutions. Whereas, in the first diversity-oriented evolutionary multiobjective optimization algorithm, the NOAH algorithm by Ulrich and Thiele, the Solow Polasky Diversity (also related to Magnitude) served as a metric, other diversity indicators might be considered, such as the parameter-free Max-Min Diversity, and the Riesz s-Energy, which features uniformly distributed solution sets. In this paper, focusing on multiobjective diversity optimization, we discuss different diversity indicators from the perspective of indicator-based evolutionary algorithms (IBEA) with multiple objectives. We examine theoretical, computational, and practical properties of these indicators, such as monotonicity in species, twinning, monotonicity in distance, strict monotonicity in distance, uniformity of maximizing point sets, computational effort for a set of size~n, single-point contributions, subset selection, and submodularity. We present new theorems -- including a proof of the NP-hardness of the Riesz s-Energy Subset Selection Problem -- and consolidate existing results from the literature. In the second part, we apply these indicators in the NOAH algorithm and analyze search dynamics through an example. We examine how optimizing with one indicator affects the performance of others and propose NOAH adaptations specific to the Max-Min indicator.

翻译：基于指标的（多目标）多样性优化旨在寻找一组近似（帕累托）最优解，以最大化多样性指标，其中多样性通常被解释为本质不同解的数量。在首个面向多样性的进化多目标优化算法——Ulrich和Thiele提出的NOAH算法中，Solow Polasky多样性（亦与Magnitude相关）被用作度量指标，但其他多样性指标也可能被考虑，例如无参数的Max-Min多样性以及以均匀分布解集为特征的Riesz s-Energy。本文聚焦于多目标多样性优化，从基于指标的进化多目标算法（IBEA）的视角探讨不同的多样性指标。我们研究了这些指标的理论、计算和实践特性，包括物种单调性、孪生性、距离单调性、距离严格单调性、最大化点集的均匀性、规模为n的集合计算复杂度、单点贡献度、子集选择以及子模性。我们提出了新的定理——包括对Riesz s-Energy子集选择问题NP难度的证明——并整合了文献中的现有结果。在第二部分中，我们将这些指标应用于NOAH算法，并通过实例分析搜索动态。我们研究了使用某一指标进行优化如何影响其他指标的性能，并提出了针对Max-Min指标的NOAH算法改进方案。