The figures below are the result of R-NSGA-II algorithm in solving DTLZ2 problem with 3 objectives. The green rhomboids represent the reference points, (0.2 0.5 0.6) and (0.7 0.8 0.5).
the result in Figure 1 looks good. However, I do not know what is wrong with the Inverted generational distance (IGD) curve.
To calculate the IGD values I have used this Matlab code.
maxGen = 200;
for gen =1:maxGen
Distance = min(pdist2(PF,popObj),[],2);
IGD_values(gen) = mean(Distance);
end
I think you cannot use the original IGD to evaluate the performance of a preference-based EMO algorithm. We have developed systematic way for this purpose. Please find more details from the following reference:
K. Li, K. Deb, X. Yao, “R-Metric: Evaluating the Performance of Preference-Based Evolutionary Multi-Objective Optimization Using Reference Points”, IEEE Trans. Evolutionary Computation (TEVC), in press.
Figure 1 is not clear for me to comment on the results; and accordingly the core of the question. But in general I can say the following:
In multi-objective optimization, there are quality indicators that are i. Pareto-Compliant; and ii. not Pareto-compliant. The quality of an approximation set is measured by a so-called (unary) quality indicator I : Ω → R, assessing a specific property of the approximation set. Likewise, an l-ary quality indicator I : Ω^l → R quantifies quality differences between l approximation sets. A quality indicator is not Pareto-compliant if it contradicts the order induced by the Pareto-dominance relations [1,2]. IGD is not Pareto-compliant, which could be the reason behind your question.
See:
[1] E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. G. Da Fonseca, Performance assessment of multiobjective optimizers: an analysis and review, IEEE Transactions on Evolutionary Computation, 7 (2003), pp. 117–132.
[2] Al-Dujaili, Abdullah, and Sundaram Suresh. "Multi-objective simultaneous optimistic optimization." Information Sciences 424 (2018): 159-174.
[3] Bezerra LC, López-Ibáñez M, Stützle T. An empirical assessment of the properties of inverted generational distance on multi-and many-objective optimization. InInternational Conference on Evolutionary Multi-Criterion Optimization 2017
Thank you for the answers.
In this case why the IGD Indicator is used by several studies to compare the performance of multiobjective optimization algorithms, such as multiobjective PSO, NSGA-II, R-NSGA-II and others?
Hi Abdullah Al-Dujaili,
In Figure 1, the dark blue dots represent the solutions that are clustered near the reference points. While the other graph represents the Pareto front of DTLZ2 problem with 3 objectives. I have added another figure from the original paper: "Reference Point Based Multi-Objective Optimization Using Evolutionary Algorithms"
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