Subset selection is a key task in evolutionary multi-objective optimization. Addressing this combinatorial challenge is key to generating finite μ-point Pareto front approximations (PFAs) that accurately represent the Pareto front, regardless of its geometric shape and dimension. For instance, subset selection is used both in bounded-size archiving and in the environmental selection of an EMO algorithm. A significant challenge remains in designing algorithms for medium-and large-scale subset selection instances such as those that involve unbounded external archives (UEA). In this article, we propose two efficient algorithms based on Riesz s-energy (RSE). These algorithms employ two main strategies: greedy inclusion and iterative replacement, exhibiting linear time and space complexity for medium-and large-scale subset selection instances. Our experimental results show that our RSE-based algorithms produce target subsets with high diversity at a low processing time, outperforming six state-of-the-art subset selection algorithms. Additionally, we validated their effectiveness in extracting a representative PFA from a UEA connected to a multi-objective evolutionary algorithm, producing well-diversified subsets regardless of the Pareto front geometry and its dimension.

The development of specialized, domain-specific operators has significantly enhanced the performance of evolutionary algorithms for solving optimization problems. However, creating such operators often requires substantial effort from human experts, making the process slow, resource-intensive, and heavily reliant on domain knowledge. To overcome these limitations, generation hyper-heuristics provide a framework for automating the design of variation operators by evolving combinations of heuristic components without direct expert input. In this work, we propose a generation hyper-heuristic method based on grammatical evolution to automatically design variation operators (crossover and mutation) tailored to the multi-objective quadratic assignment problem (mQAP)—a challenging combinatorial optimization problem with many real-world applications. Using the proposed method, variation operators were generated considering six mQAP instances with two and three objectives, leveraging MOEA/D as a multi-objective optimizer. For validation, the generated operators were evaluated on unseen instances. Our experimental results indicate that the evolved operators enhance the performance of MOEA/D compared to standard crossover operators. Furthermore, the top-performing operator in training did not always generalize best to larger instances, while some lower-ranked operators showed better adaptability. These results highlight the potential of automated operator design in effectively tackling complex optimization problems like the mQAP.

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.

Metaheuristics are optimization techniques that can be adapted to different problem domains. They are considered general algorithms that have been successful in solving various problems. However, the increase in redundant ‘novel’ metaheuristics lacking solid validation has become a criticized trend in the research area. In this work, we present a software tool offering interactive development and metaheuristics customization. This software is based on a standard and theoretical framework for analyzing, modifying, and generating metaheuristics. We take advantage of this framework to examine some algorithms recently reported as innovative in the literature and reveal their true nature. Plus, these tools enable rapid prototyping of metaheuristics and allow for static validation experimentation and comparison of algorithms. It helps in the refinement and fine-tuning of the metaheuristics. We present a case study demonstrating the proposed tool's effectiveness, which promises influential scientific contributions and unlocks the potential of metaheuristics.

The distribution of objective vectors in a Pareto Front Approximation (PFA) is crucial for representing the associated manifold accurately. Distribution Indicators (DIs) assess the distribution of a PFA numerically, utilizing concepts like distance calculation, Biodiversity, Entropy, Potential Energy, or Clustering. Despite the diversity of DIs, their strengths and weaknesses across assessment scenarios are not well-understood. This paper introduces a taxonomy for classifying DIs, followed by a preference analysis of nine DIs, each representing a category in the taxonomy. Experimental results, considering various PFAs under controlled scenarios (loss of coverage, loss of uniformity, pathological distributions), reveal that some DIs can be misleading and need cautious use. Additionally, DIs based on Biodiversity and Potential Energy are promising for PFA evaluation and comparison of Multi-Objective Evolutionary Algorithms.

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.

Reference sets are widely used by many multi-objective evolutionary algorithms (MOEAs) to decompose the objective space, define search directions, or calculate quality indicators (QIs) embedded into the selection mechanisms. Well-known MOEAs adopt the generation of uniformly distributed points on a unit simplex to construct such reference sets. Although these mechanisms are useful for approximating Pareto fronts with regular shapes, i.e., simplex-like shapes, they have difficulties representing Pareto fronts with irregular geometries. To overcome this drawback, many reference set adaptation methods have been proposed so far. However, some adaptation methods present a degraded performance on regular Pareto front shapes, while others promote a balanced performance. Nevertheless, an extensive assessment has not been made. In this paper, a MOEA based on the inverted generational distance plus indicator, using an adaptive reference set, is used to study the performance of well-known adaptation methods. Although an adaptation method promotes a balanced performance on both regular and irregular Pareto front shapes, results show some difficulties related to the distribution of solutions in complex Pareto front shapes. The results of this study allow detecting the main drawbacks of adaptation methods, which can be addressed by using diversity-oriented selection mechanisms in the generation of reference sets. Hence, these could impact the generation of reference set-based MOEAs achieving good coverage, convergence, and diversity regardless of the Pareto front shape.

Pareto compliance is a critical property of quality indicators (QIs) focused on measuring convergence to the Pareto front. This property allows a QI not to contradict the order imposed by the Pareto dominance relation. Hence, Pareto-compliant QIs are remarkable when comparing approximation sets of multi-objective evolutionary algorithms (MOEAs) since they do not produce misleading conclusions. However, the practical effect of Pareto-compliant QIs as the backbone of MOEAs' survival mechanisms is not entirely understood. In this paper, we study the practical effect of IGD++ (which is a combination of IGD+ and the hypervolume indicator), IGD+, and IGD, which are Pareto-compliant, weakly Pareto-compliant, and not Pareto-compliant QIs, respectively. To this aim, we analyze the convergence and diversity properties of steady-state MOEAs based on the previously mentioned QIs throughout the whole evolutionary process. Our experimental results showed that, in general, the practical effect of a Pareto-compliant QI in a selection mechanism is not very significant concerning weaker QIs, taking into account the whole evolutionary process.

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.

Currently, reference sets, which are a collection of feasible or infeasible points in objective space, are the backbone of several multi-objective evolutionary algorithms (MOEAs) and quality indicators (QIs). For both MOEAs and QIs, an important question is how to construct the reference set regardless of the dimensionality of the objective space, preserving well-diversified solutions. The Simplex-Lattice-Design method (SLD) that constructs a set of convex weights in a simplex, has been usually used to define reference sets. However, it is not a good option since Pareto fronts with irregular geometries cannot be completely intersected by the weight vectors. In this paper, we propose a tool based on the Riesz s-energy to generate reference sets exhibiting good diversity properties. Our experimental results support the Riesz s-energy-based reference sets as a better option due to their invariance to the Pareto front shape and the objective space dimensionality.

In recent years, several indicator-based multi-objective evolutionary algorithms (IB-MOEAs) have been proposed. Each IB-MOEA presents different search preferences depending on the quality indicator (QI) that it uses in its selection mechanism. However, due to these search biases, IB-MOEAs behave differently on each multi-objective optimization problem, producing Pareto front approximations whose characteristics are related to the QI on which they are based. In this paper, we propose a novel algorithm based on the island model that aims to take advantage of the cooperation of individual IB-MOEAs based on the indicators hypervolume, R2, IGD+,+, and Δpwith the aim of improving both convergence and distribution of the Pareto fronts produced. Our experimental results, taking into account seven quality indicators, empirically show that the cooperation of several IB-MOEAs is better than using panmictic versions of them. Additionally, we also show that the performance of our proposal does not depend on the Pareto front shape of the problem being solved.

In recent years, quality indicators (QIs) have been employed to design selection mechanisms for multi-objective evolutionary algorithms (MOEAs). These indicator-based MOEAs (IB-MOEAs) generate Pareto front approximations that present convergence and diversity characteristics strongly related to the QI that guides the selection mechanism. However, on complex multi-objective optimization problems, the performance of IB-MOEAs is far from being completely understood. In this paper, we empirically analyze the convergence and diversity properties of five steady-state IB-MOEAs based on the hypervolume, R2, IGD+, ∈+, and Δp. Regarding convergence, we analyze their speed of convergence and the final closeness to the true Pareto front. The IB-MOEAs adopted in our study were tested on problems having different Pareto front shapes, and were taken from six test suites. Our experimental results show general and particular strengths and weaknesses of the adopted IB-MOEAs. We believe that these results are the first step towards a deeper understanding of the behavior of IB-MOEAs.

In recent years, Indicator-based Multi-Objective Evolutionary Algorithms (IB-MOEAs) have become a relatively popular alternative for solving multi-objective optimization problems. IB-MOEAs are normally based on the use of a single performance indicator. However, the effect of the combination of multiple performance indicators for selecting solutions is a topic that has rarely been explored. In this paper, we propose a hyper-heuristic which combines the strengths and compensates for the weaknesses of four density estimators based on R2, IGD+, ϵ+ and Δp. The selection of the indicator to be used at a particular moment during the search is done using online learning and a Markov chain. Additionally, we propose a novel framework that aims to reduce the computational cost involved in the calculation of the indicator contributions. Our experimental results indicate that our proposed approach can outperform state-of-the-art MOEAs based on decomposition (MOEA/D) reference points (NSGA-III) and the R2 indicator (R2-EMOA) for problems with both few and many objectives.