Comparison of Hypervolume, IGD and IGD+ from the Viewpoint of Optimal Distributions of Solutions

Ishibuchi, Hisao; Imada, Ryo; Masuyama, Naoki; Nojima, Yusuke

doi:10.1007/978-3-030-12598-1_27

Cited by 70 publications

(28 citation statements)

References 26 publications

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“…The τ value of IGD and IGD + in Table III is also −0.60 on F concave . These results of IGD and IGD + are consistent with [40], [31].…”

Section: B Ranking Information Based Analysissupporting

confidence: 87%

“…The HV contribution-based selection in SMS-EMOA was replaced by the IGD contribution-based selection. The optimal µdistributions for HV, IGD, and IGD + approximated by SMS-EMOA in a similar manner were analyzed in [31]. The results show that the optimal µ-distributions for IGD are almost uniform in all problems.…”

Section: B Optimal µ-Distribution Based Approachesmentioning

confidence: 95%

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An Analysis of Quality Indicators Using Approximated Optimal Distributions in a 3-D Objective Space

Tanabe

Ishibuchi

2020

IEEE Trans. Evol. Computat.

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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.

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“…The τ value of IGD and IGD + in Table III is also −0.60 on F concave . These results of IGD and IGD + are consistent with [40], [31].…”

Section: B Ranking Information Based Analysissupporting

confidence: 87%

Section: B Optimal µ-Distribution Based Approachesmentioning

confidence: 95%

An Analysis of Quality Indicators Using Approximated Optimal Distributions in a 3-D Objective Space

Tanabe

Ishibuchi

2020

IEEE Trans. Evol. Computat.

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“…6. It is known that the obtained solution sets by SMS-EMOA depend on the reference point specification for hypervolume calculation [26], [27]. Especially when the Pareto front is inverted triangular, totally different solution sets can be obtained depending on the reference point specification.…”

Section: Elitist Emo Algorithmsmentioning

confidence: 99%

“…For example, by using the IGD (inverted generational distance) indicator [40], we may be able to obtain more uniformly distributed solutions than distance-based solution selection. This is because IGD minimization leads to higher uniformity of solutions than the use of other performance indicators as objective functions such as hypervolume maximization [26], [41]. Proposal of new solution selection methods is a promising future research direction.…”

Section: Solution Selectionmentioning

confidence: 99%

A New Framework of Evolutionary Multi-Objective Algorithms with an Unbounded External Archive

Ishibuchi¹,

Pang²,

Shang³

2020

Preprint

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This paper proposes a new framework for the design of evolutionary multi-objective optimization (EMO) algorithms. The main characteristic feature of the proposed framework is that the optimization result of an EMO algorithm is not the final population but a subset of the examined solutions during its execution. As a post-processing procedure, a pre-specified number of solutions are selected from an unbounded external archive where all the examined solutions are stored. In the proposed framework, the final population does not have to be a good solution set. The point of the algorithm design is to examine a wide variety of solutions over the entire Pareto front and to select well-distributed solutions from the archive. In this paper, first we explain difficulties in the design of EMO algorithms in the existing two frameworks: non-elitist and elitist. Next, we propose the new framework of EMO algorithms. Then we demonstrate advantages of the proposed framework over the existing ones through computational experiments. Finally we suggest some interesting and promising future research topics.

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“…In the last decades, various performance indicators have been proposed to assess the performance of EMO algorithms, see [27]. The most popular performance indicator amongst the EMO society in recent years has been the Hypervolume (HV) indicator [42] and the Inverted Generational Distance+ (IGD + ), see [27,29,51].…”

Section: Comparison Methodsmentioning

confidence: 99%

On Algorithmic Descriptions and Software Implementations for Multi-objective Optimisation: A Comparative Study

2020

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Multi-objective optimisation is a prominent subfield of optimisation with high relevance in real-world problems, such as engineering design. Over the past 2 decades, a multitude of heuristic algorithms for multi-objective optimisation have been introduced and some of them have become extremely popular. Some of the most promising and versatile algorithms have been implemented in software platforms. This article experimentally investigates the process of interpreting and implementing algorithms by examining multiple popular implementations of three well-known algorithms for multi-objective optimisation. We observed that official and broadly employed software platforms interpreted and thus implemented the same heuristic search algorithm differently. These different interpretations affect the algorithmic structure as well as the software implementation. Numerical results show that these differences cause statistically significant differences in performance.

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Comparison of Hypervolume, IGD and IGD+ from the Viewpoint of Optimal Distributions of Solutions

Cited by 70 publications

References 26 publications

An Analysis of Quality Indicators Using Approximated Optimal Distributions in a 3-D Objective Space

An Analysis of Quality Indicators Using Approximated Optimal Distributions in a 3-D Objective Space

A New Framework of Evolutionary Multi-Objective Algorithms with an Unbounded External Archive

On Algorithmic Descriptions and Software Implementations for Multi-objective Optimisation: A Comparative Study

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