Progressively Interactive Evolutionary Multi-Objective Optimization ...

3. Thepreviousapproacheshadafixedpopulationsizeforthelowerlevel. Thismade
the number of upper level variable vectors fixed from one generation to another
for these approaches, whereas in H-BLEMO, the use of a variable population size
for different lower level NSGA-IIs establishes that the upper level population can
hold varying number of upper level variable vectors. This allows the H-BLEMO
proceduretoprovideavaryingimportancebetweentheextentofupperandlower
level optimizations, as maybe demandedby aproblem.
4. The earlierproceduresdid not have aalgorithmically sensible terminationcriteria
fortheupperorthelowerlevelNSGA-IIandhadtorunforafixednumberofgenerations.
In H-BLEMO a hypervolume based termination criteria has been used
where both lower and upper level terminate, based on the dynamic performance
of the algorithms. The criteriaensures the termination of eachNSGA-IIwhenever
the corresponding non-dominated front has stabilized,therebyavoiding unnecessaryfunction
evaluations.
6 Results onTest Problems
We use the following standard NSGA-II parameter values in both lower and upper
levelsonallproblemsof this study: Crossoverprobabilityof0.9,distributionindex for
SBX of 15, mutation probability of 0.1, distribution index for the polynomial mutation
of 20. The upper level population size is set proportional to the total number of variables
(n): Nu = 20n. As described in the algorithm, all other parameters including the
lower level population size, termination criteria are all set in a self-adaptive manner
during the optimization run. In all cases, we have used 21 different simulations startingfromdifferentinitialpopulationsandshowthe0,50,and100%attainmentsurfaces
(Fonseca andFleming, 1996)todescribethe robustness of the proposedprocedure.
6.1 ProblemTP1
This problem has three variables (one for upper level and two for lower level). Thus,
we use Nu = 60 population members. Figure 11 shows the final archive members of
a single run on the upper level objective space. It can be observed that our proposed
procedure is able to find solutions on the true Pareto-optimal front. Conditions for
the exact Pareto-optimal solutions are given in equation (3). For each of the obtained
solutions, we use the upper level variable (y) value to compute lower level optimal
variable (x ∗ 1 and x∗ 2
) values using the exact conditions for Pareto-optimality and com-
pare the values with H-BLEMO solutions. The average error 2
i=1 (xi − x ∗ i )2 /2 for
eachobtained solution is computed and then averagedover all archivesolutions. This
error value for our H-BLEMO is found to be 7.0318 × 10 −5 , whereas the same error
value computed for the solutions reported with our earlier algorithm (Deb and Sinha,
2009a) is found to be 2.2180 × 10 −3 . The closer adherence to optimal variable values
withH-BLEMOindicatesabetterperformanceofourhybridalgorithm. Theminimum,
medianand worst number of function evaluations neededover 21 differentruns of H-
BLEMO are 567,848, 643,753, and 716,780, respectively. Although for a three-variable
problem,thesenumbers mayseemtoolarge,one needstorealizethatthebilevelproblems
have one optimization algorithm nested into another and Pareto-optimality for
the lower level problem is essential for an upper level solution. In comparison, our
previous algorithm (Sinha and Deb, 2009) required 1,030,316, 1,047,769 and 1,065,935
functionevaluations forbest,medianandworst performingruns. Despite using about
40%less functionevaluations, our H-BLEMOalsofinds more accuratesolutions.
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