Reference Point Specification in Inverted Generational Distance for Triangular Linear Pareto Front

The hypervolume indicator has frequently been used for comparing evolutionary multi-objective optimization (EMO) algorithms. A reference point is needed for hypervolume calculation. However, its specification has not been discussed in detail from a viewpoint of fair performance comparison. A slightly worse point than the nadir point is usually used for hypervolume calculation in the EMO community. In this paper, we propose a reference point specification method for fair performance comparison of EMO algorithms. First, we discuss the relation between the reference point specification and the optimal distribution of solutions for hypervolume maximization. It is demonstrated that the optimal distribution of solutions strongly depends on the location of the reference point when a multi-objective problem has an inverted triangular Pareto front. Next, we propose a reference point specification method based on theoretical discussions on the optimal distribution of solutions. The basic idea is to specify the reference point so that a set of well-distributed solutions over the entire linear Pareto front has a large hypervolume and all solutions in such a solution set have similar hypervolume contributions. Then, we examine whether the proposed method can appropriately specify the reference point through computational experiments on various test problems. Finally, we examine the usefulness of the proposed method in a hypervolume-based EMO algorithm. Our discussions and experimental results clearly show that a slightly worse point than the nadir point is not always appropriate for performance comparison of EMO algorithms.

So far, many performance indicators have been proposed to compare different evolutionary multiobjective optimization algorithms (MOEAs). Among them, the inverted generational distance (IGD) is one of the most commonly used, mainly because it can measure a population’s convergence, diversity, and evenness. However, the effectiveness of IGD highly depends on the quality of the reference set. That is to say, all the reference points should be as close to the Pareto front (PF) as possible and evenly distributed to become ready for a fair performance evaluation. Currently, it is still challenging to generate well-configured reference sets, even if the PF can be given analytically. Therefore, biased reference sets might be a significant source of systematic error. However, in most MOEA literature, biased reference sets are utilized in experiments without an error estimation, which may make the experimental results unconvincing. In this paper, we propose an exact IGD (eIGD) for continuous PF, which is derived from the original IGD under an additional assumption that the reference set is perfect, i.e., the PF itself is directly utilized as an infinite-sized reference set. Therefore, the IGD values produced by biased reference sets can be compared with eIGD so that systematic error can be quantitatively evaluated and analyzed.KeywordsMultiobjective optimizationEvolutionary computationPerformance indicatorReference setDifferential geometry

Assessing the performance of Pareto front (PF) approximations is a key issue in the field of evolutionary multi/many-objective optimization. Inverted generational distance (IGD) has been widely accepted as a performance indicator for evaluating the comprehensive quality for a PF approximation. However, IGD usually becomes infeasible when facing a real-world optimization problem as it needs to know the true PF a priori. In addition, the time complexity of IGD grows quadratically with the size of the solution/reference set. To address the aforementioned issues, a grid-based IGD (Grid-IGD) is proposed to estimate both convergence and diversity of PF approximations for multi/many-objective optimization. In Grid-IGD, a set of reference points is generated by estimating PFs of the problem in question, based on the representative nondominated solutions of all the approximations in a grid environment. To reduce the time complexity, Grid-IGD only considers the closest solution within the grid neighborhood in the approximation for every reference point. Grid-IGD also possesses other desirable properties, such as Pareto compliance, immunity to dominated/duplicate solutions, and no need of normalization. In the experimental studies, Grid-IGD is verified on both the artificial and real PF approximations obtained by five many-objective optimizers. Effects of the grid specification on the behavior of Grid-IGD are also discussed in detail theoretically and experimentally.

The Hausdorff distance d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> is a widely used tool to measure the distance between different objects in several research fields. Possible reasons for this might be that it is a natural extension of the well-known and intuitive distance between points and/or the fact that d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> defines in certain cases a metric in the mathematical sense. In evolutionary multiobjective optimization (EMO) the task is typically to compute the entire solution set-the so-called Pareto set-respectively its image, the Pareto front. Hence, d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> should, at least at first sight, be a natural choice to measure the performance of the outcome set in particular since it is related to the terms spread and convergence as used in EMO literature. However, so far, d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> does not find the general approval in the EMO community. The main reason for this is that d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> penalizes single outliers of the candidate set which does not comply with the use of stochastic search algorithms such as evolutionary strategies. In this paper, we define a new performance indicator, Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> , which can be viewed as an “averaged Hausdorff distance” between the outcome set and the Pareto front and which is composed of (slight modifications of) the well-known indicators generational distance (GD) and inverted generational distance (IGD). We will discuss theoretical properties of Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (as well as for GD and IGD) such as the metric properties and the compliance with state-of-theart multiobjective evolutionary algorithms (MOEAs), and will further on demonstrate by empirical results the potential of Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> as a new performance indicator for the evaluation of MOEAs.

Most evolutionary multiobjective optimization algorithms are designed to approximate the entire Pareto front. During the last decade, a series of preference-based evolutionary algorithms have been developed, where a part of the Pareto front is approximated by incorporating the preferences of a Decision Maker. However, only a few such algorithms are able to obtain well-distributed solutions covering the complete “region of interest” that is determined by a reference point. In this paper, a preference-based evolutionary algorithm for approximating the region of interest is proposed. It is based on the state-of-the-art genetic algorithm NSGA-II and the CHIM approach introduced in the NBI method which is used to obtain uniformly distributed solutions in the region of interest. The efficiency of the proposed algorithm has been experimentally evaluated and compared to other state-of-the-art multiobjective preference-based evolutionary algorithms by solving a set of multiobjective optimization benchmark problems. It has been shown that the incorporation of the Decision Maker’s preferences and the CHIM approach into the NSGA-II algorithm allows approximating the whole region of interest accurately while maintaining a good distribution of the obtained solutions.

Although quality indicators play a crucial role in benchmarking evolutionary multi-objective optimization algorithms, their properties are still unclear. One promising approach for understanding quality indicators is the use of the optimal distribution of objective vectors that optimizes each quality indicator. However, it is difficult to obtain the optimal distribution for each quality indicator, especially when its theoretical property is unknown. Thus, optimal distributions for most quality indicators have not been well investigated. To address these issues, first, we propose a problem formulation of finding the optimal distribution for each quality indicator on an arbitrary Pareto front. Then, we approximate the optimal distributions for nine quality indicators using the proposed problem formulation. We analyze the nine quality indicators using their approximated optimal distributions on eight types of Pareto fronts of three-objective problems. Our analysis demonstrates that uniformly-distributed objective vectors over the entire Pareto front are not optimal in many cases. Each quality indicator has its own optimal distribution for each Pareto front. We also examine the consistency among the nine quality indicators.

Testing and assessing the performance of proposed algorithms in a right way is a very important part in evolutionary optimization. If the number of optimization objectives is more than three, it's commonly referred as many-objective optimization problem(MaOP). Despite the fact there exists many test functions and assessment metrics, all of which are designed for testing and assessing the performance of evolutionary algorithms, the test functions DTLZ proposed by Deb(2005) and the assessment metrics Inverted Generational Distance(IGD) proposed by Bosman and Thierens(2003) are both widely recognized and used in most of evolutionary optimizations, especially in many-objective evolutionary optimization. When using IGD as a assessment tool, the vital task is to distribute reference points uniformly along the Pareto front of test functions. To overcome it, however, is really difficult with the number of objectives increase. For dealing with such difficulty, this paper introduce an approach called Inverse Transform Technique(ITT) suggested by Fang and Wang(1999) and successfully distributes reference points uniformly along the Pareto front of DTLZ test functions. In the final, three representative evolutionary algorithms: NSGA-II, MOEA/D and MOEA/D-M2M are selected to test their performance in dealing with many-objective optimization. As a result, extensive experiments and analysis are conducted on four DTLZ test functions from five to ten objectives.

The problem of atmospheric duct inversion is usually solved as a single objective optimization problem. Based on ground-based Global Positioning System (GPS) phase delay and propagation loss, this paper develops a multi-objective method including the effect of source frequency and receiving antenna height. The diversity and convergence of solution sets are evaluated for seven multi-objective evolutionary algorithms with three performance metrics: Hypervolume (HV), Inverted Generational Distance (IGD), and the averaged Hausdorff distance (). The inversion results are compared with the simulation results, and the experimental comparison is conducted on three groups of test situations. The results demonstrate that the ranking of algorithm performance varies because of the different methods used to calculate performance metrics. Moreover, when the algorithms show overwhelming performance using performance metrics, the inversion result is not more close to the real value. In the comparison of computational experiments, it was found that, as the retrieved parameter dimension increases, the inversion result becomes more unstable. When the observed data are sufficient, the inversion result seems to be improved.

A parallel multiple reference point approach for multi-objective optimization

Over the past decades, evolutionary multi-objective optimization algorithms have shown their strength on solving the multi-objective optimization problems. The S-Metric Selection Evolutionary Multiobjective Optimization Algorithm (SMS-EMOA) is a state-of-the-art algorithm which uses the hypervolume indicator as selection criterion and performs well in finding well distributed solutions to approximate the Pareto front. In this paper, the concept of multiple dynamic reference points is integrated to the SMS-EMOA which can balance the trade-off of exploration and exploitation by changing the number of reference points. This way it combines concepts of indicator based and decomposition based EMOA design in a promising manner. The proposed algorithm is compared with other well established EMOA on the classical ZDT benchmark which have 5 different test problems with two objectives. The results show that SMS-EMOA with multiple dynamic reference points outperforms other state-of-the-art algorithms, including SMS-EMOA, by covering more hypervolume which means it has a better approximation of the Pareto front.

How to evaluate Pareto front approximations generated by multi/many-objective optimizers is a critical issue in the field of multiobjective optimization. Currently, there exist two types of comprehensive quality indicators (i.e., volume-based and distance-based indicators). Distance-based indicators, such as inverted generational distance (IGD), are usually computed by summing up the distance of each reference point to its nearest solution. Their high computational efficiency leads to their prevalence in many-objective optimization. However, in the existing distance-based indicators, the distributions of the solution sets are usually neglected, leading to their lack of ability to well distinguish between different solution sets. This phenomenon may become even more severe in high-dimensional space. To address such an issue, a kernel-based indicator (KBI) is proposed as a comprehensive indicator. Different from other distance-based indicators, a kernel-based maximum mean discrepancy is adopted in KBI for directly measuring the difference that can characterize the convergence, spread, and uniformity of two sets, i.e., the solution set and reference set, by embedding them in reproducing kernel Hilbert space (RKHS). As a result, KBI not only reflects the distance between the solution set and the reference set but also can reflect the distribution of the solution set itself. In addition, to maintain the desirable weak Pareto compliance property of KBI, a nondominated set reconstruction approach is also proposed to shift the original solution set. The detailed theoretical and experimental analysis of KBI is provided in this article. The properties of KBI have also been analyzed by the optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> -distribution.

Characterization of graph properties for improved Pareto fronts using heuristics and EA for bi-objective graph coloring problem

An Area Partitioning and Allocation (APA) approach was presented in[1]. The approach focused on optimizing the coverage performance of Autonomous Industrial Robots (AIRs) using multiple conflicting objectives and Voronoi partitioning. However, questions related to the optimality, convergence, and consistency of the Pareto solutions were not studied in details. In this paper, Inverted Generational Distance (IGD) metric is used to verify the convergence of the Pareto front towards Pareto optimal front (PF*). The consistency in obtaining similar Pareto fronts for independent optimization runs is studied. The computational complexity of the approach with respect to the size of the coverage area and the number of AIRs is also discussed. Two application scenarios are used in this research.

Selecting the best multi-objective algorithms for photovoltaic energy storage system (PV-ESS) design remains challenging due to limited benchmarking across renewable energy studies. This study addresses this gap through a systematic evaluation of four widely used multi-objective optimization algorithms: NSGA-II, Multi-Objective Particle Swarm Optimization (MOPSO), weighted-sum scalarization, and ε-constraint methods. Performance assessment utilized three Pareto front quality metrics: Inverted Generational Distance (IGD) for convergence quality, hypervolume (HV) for objective-space coverage, and spacing for solution distribution uniformity. The algorithms were tested on PV-ESS design problems in three developing economies (Nigeria, South Africa, India) under identical problem formulations and computational resources. NSGA-II achieved superior performance across all metrics in all three case studies. For convergence quality, NSGA-II attained a mean IGD of 0.0083, outperforming MOPSO by 29%, ε-constraint by 64%, and weighted-sum by 131%. For objective-space coverage, NSGA-II achieved a mean HV of 0. 700, representing 10–16% better coverage than other methods. For solution distribution, NSGA-II showed a mean spacing of 0.076, indicating 30–117% more uniform Pareto fronts. Computational efficiency analysis revealed that NSGA-II’s runtime is between 5.5 and 7.8 h per case, providing better quality–time ratios compared to ε-constraint methods (which are 18 times slower), while avoiding MOPSO’s premature convergence. Statistical validation confirmed NSGA-II’s superiority, with p &lt; 0.01 across all quality metrics. These results establish NSGA-II as the best algorithm for lifecycle-aware PV-ESS optimization, offering quantitative, evidence-based guidance for practitioners selecting optimization tools for renewable energy system design. The demonstrated performance leads to $ 45,000–$ 60,000 lifecycle cost savings per MW/MWh of system capacity through improved Pareto front identification.

A fuzzy-guided adaptive algorithm with hierarchy mechanism for solving dynamic multi-objective optimization problems