Progressively Interactive Evolutionary Multi-Objective Optimization ...

F2
0
−0.2
−0.4
−0.6
−0.8
−1
−2 −1.8 −1.6 −1.4
F1
−1.2 −1 −0.8
Figure 11: Final archive solutions for
problemTP1.
F2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−2 −1.8 −1.6 −1.4
F1
−1.2 −1 −0.8
Figure 12: Attainment surfaces(0%, 50%
and100%)forproblemTP1from21runs.
Theattainment surfacesobtainedforthe archivesolutions over21runs areshown
in Figure 12. All three surfaces are so close to each other that they are difficult to be
distinguished from one another. This indicates the robustness of the procedure. The
hypervolume values are computed after normalizing the upper level objective values
bytheirminimumandmaximumvalues. Thehypervolumesforthe0%,50%and100%
attainment surfaces are 0.3583, 0.3678 and 0.3700, respectively. The difference in the
hypervolume valueover 21runs is only about 3%.
In order to investigate the effect of Nu on the performance of the algorithm, next,
we use different Nu values but maintain an identical termination criteria. Figure 13
showsthefunctionevaluationsneededforlower(includingthelocalsearch)andupper
level optimization tasks for different Nu values ranging from40 to 200. It is clear from
the figure that a population size of Nu = 60 (which we used in Figures 11 and 12)
performsthe best, onanaverage,inbothlower and upperleveloptimization tasks.
6.2 ProblemTP2
The second test problem has n = 15 variables. Thus, we use Nu = 300. Figure 14
shows the final archivepopulation of atypical run. The attainment surfaceplot inFigure15showsthattheproposedalgorithmisfairlyrobustinall21differentsimulations.
The algorithm finds an almost an identical front close to the true Pareto-optimal front
in multiple runs. The hypervolumes for the obtained attainment surfaces are 0.8561,
0.8582 and 0.8589, making a maximum difference of 0.3% only. A comparison of our
currentlocal searchbasedalgorithm with our previously proposedBLEMOprocedure
(SinhaandDeb,2009)intermsofanerrormeasure(asdiscussedforproblemTP1)from
the exact Pareto-optimal solutions indicates a smaller error for our current approach.
Despite the use of 15 variables here, as opposed to 7 variables used in the previous
study, the error in the current approach is 7.920 × 10 −6 , compared to 11.158 × 10 −6 in
the previous study. In terms of the median performance, H-BLEMO requires 338,232
function evaluations for the 15-variableproblem, as opposed to 771,404function evaluations
needed by our previous approach (Sinha and Deb, 2009) on a seven-variable
versionof the problem.
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