Comparative Analysis of Indicators for Multiobjective Diversity Optimization

TL;DR

Different diversity indicators from the perspective of indicator-based evolutionary algorithms (IBEA) with multiple objectives are discussed, 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.

Abstract

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.

Figures (5)

• Figure 1: Max-min is identical for both (A) and (B), although all pairwise distances increase from (A) to (B), except for nearest neighbors, which remain unchanged. The Sum indicator (aka Total Distance Indicator, sum of all pairwise distances in the set) for (A) is larger than for (C) though intuitively the dissimilarity is much larger for (C). Also the spread and evenness are much better in (C).
• Figure 3: Sample of 20 points distributed in the box $\{(x,y), x \in [0, 10], y \in [0, 10]\}$ (left). Obtained efficient set and contours of shifted Himmelblau and Paraboloid objective functions for the biobjective problem with $\epsilon$-dominance, $\epsilon = 1$ (right).
• Figure 4: Typical run. Iterations 6 and 7 of the NOAH algorithm with the Max-min diversity indicator. From left to right: normal optimization and Max-min diversity optimization for iteration 6, followed by normal optimization and Max-min diversity optimization for iteration 7.
• Figure 5: Diversity measures averaged over 30 runs for each algorithm. From left to right: NOAH with Max-min optimization, s-Energy optimization and SP optimization. Solid lines represent the mean over 30 runs, with dashed lines indicating means $\pm$ the standard error of the mean.
• Figure 6: Diversity measures calculated during the execution of the three algorithms, and the Hausdorff distance calculated at two stages of each iteration: after objective optimization and after diversity optimization.

Theorems & Definitions (14)

• Definition 1
• Remark 1
• Definition 2
• Remark 2
• Lemma 1
• Definition 3: Theoretical Properties of Diversity Indicators
• Proposition 2
• Corollary 5
• Remark 3
• Definition 4