For over 25 years, most multi-objective evolutionary algorithms (MOEAs) have adopted selection criteria based on Pareto dominance. However, the performance of Pareto-based MOEAs quickly degrades when solving multi-objective optimization problems (MOPs) having four or more objective functions (the so-called many-objective optimization problems), mainly because of the loss of selection pressure. Consequently, in recent years, MOEAs have been coupled with indicator-based selection mechanisms in furtherance of increasing the selection pressure so that they can properly solve many-objective optimization problems. Several research efforts have been conducted since 2003 regarding the design of the so-called indicator-based (IB) MOEAs. In this article, we present a comprehensive survey of IB-MOEAs for continuous search spaces since their origins up to the current state-of-the-art approaches. We propose a taxonomy that classifies IB-mechanisms into two main categories: (1) IB-Selection (which is divided into IB-Environmental Selection, IB-Density Estimation, and IB-Archiving) and (2) IB-Mating Selection. Each of these classes is discussed in detail in this article, emphasizing the advantages and drawbacks of the selection mechanisms. In the final part, we provide some possible paths for future research.

For almost 20 years, quality indicators (QIs) have promoted the design of new selection mechanisms of multiobjective evolutionary algorithms (MOEAs). Each indicator-based MOEA (IB-MOEA) has specific search preferences related to its baseline QI, producing Pareto front approximations with different properties. In consequence, an IB-MOEA based on a single QI has a limited scope of multiobjective optimization problems (MOPs) in which it is expected to have a good performance. This issue is emphasized when the associated Pareto front geometries are highly irregular. In order to overcome these issues, we propose here an island-based multiindicator algorithm (IMIA) that takes advantage of the search biases of multiple IB-MOEAs through a cooperative scheme. Our experimental results show that the cooperation of multiple IB-MOEAs allows IMIA to perform more robustly (considering several QIs) than the panmictic versions of its baseline IB-MOEAs as well as several state-of-the-art MOEAs. Additionally, IMIA shows a Pareto-front-shape invariance property, which makes it a remarkable optimizer when tackling MOPs with complex Pareto front geometries.

The most relevant property that a quality indicator (QI) is expected to have is Pareto compliance, which means that every time an approximation set strictly dominates another in a Pareto sense, the indicator must reflect this. The hypervolume indicator and its variants are the only unary QIs known to be Pareto-compliant but there are many commonly used weakly Pareto-compliant indicators such as R2, IGD+, and ε+. Currently, an open research area is related to finding new Pareto-compliant indicators whose preferences are different from those of the hypervolume indicator. In this article, we propose a theoretical basis to combine existing weakly Pareto-compliant indicators with at least one being Pareto-compliant, such that the resulting combined indicator is Pareto-compliant as well. Most importantly, we show that the combination of Pareto-compliant QIs with weakly Pareto-compliant indicators leads to indicators that inherit properties of the weakly compliant indicators in terms of optimal point distributions. The consequences of these new combined indicators are threefold: (1) to increase the variety of available Pareto-compliant QIs by correcting weakly Pareto-compliant indicators, (2) to introduce a general framework for the combination of QIs, and (3) to generate new selection mechanisms for multiobjective evolutionary algorithms where it is possible to achieve/adjust desired distributions on the Pareto front.

Measuring the diversity of a Pareto Front Approximation (PFA) is critical when comparing the performance of Multi-Objective Evolutionary Algorithms (MOEAs). In the literature, some Quality Indicators (QIs) measure diversity according to their specific preferences. However, just a few QIs have mathematical properties proven. In this paper, we propose the Riesz s-energy (Es) as a QI to evaluate the diversity and spread of PFAs. Theoretical results show that Es holds (1) some of the Weitzman properties of a desirable diversity QI, (2) monotonicity, (3) the submodularity property (for -Es), and (4) that it is invariant under rotations. We provide numerical evidence on the behavior of Es in both artificial PFAs and PFAs generated by state-of-the-art MOEAs. The mathematical properties that Es satisfies show its usefulness when it is utilized as a diversity QI in Evolutionary Multi-Objective Optimization.