Progressively Interactive Evolutionary Multi-Objective Optimization ...

(derived from tmax l , as calculated below) or the above Hl ≤ ǫl is satisfied. We bound
the limiting generation (tl) to be proportional to the distance of current xu from the
archive,as follows:
tl = (int) δu
δU
t max
l . (18)
For terminating the upper level NSGA-II,the normalized change in hypervolume
measure Hu oftheupperlevelpopulation(asinequation(17)exceptthatthehypervolume
measure is computed in the upper level objective space) is computed in every τ
consecutive generations. When Hu ≤ ǫu (a threshold parameter)is obtained, the overallalgorithmisterminated.
Wehaveused τ = 10, ǫl = 0.1(foraquicktermination)and
ǫu = 0.0001(for a reliable convergence of the upper level problem) for all problems in
this study.
Now, we are ready to describe the overall algorithm for a typical generation in a
step-by-stepformat.
5.3 Step-by-StepProcedure
At the start of the upper level NSGA-II generation T, we have a population PT of size
Nu. Every population member has the following quantities computed from the previousiteration:
(i)anon-dominatedrank NDu correspondingto Fand G,(ii)acrowding
distance value CDu corresponding to F, (iii) a non-dominated rank NDl corresponding
to f and g, and (iv) a crowding distance value CDl using f. In addition to these
quantities, for the members stored in the archive AT, we have also computed (v) a
crowding distance value CDa corresponding to F and (vi) a non-dominated rank NDa
corresponding to F and G.
Step1a: Creationofnew xu: Weapplytwobinarytournamentselectionoperationson
members (x = (xu, xl)) of PT using NDu and CDu lexicographically. Also, we apply
two binary tournament selections on the archive population AT using NDa
and CDa lexicographically. Of the four selected members, two participate in the
recombination operator based on stochastic events. The members from AT par-
|AT |
ticipate as parents with a probability of |AT |+|PT | , otherwise the members from
PT become the parents for recombination. The upper level variable vectors xu of
the two selected parents are then recombined using the SBX operator (Deb and
Agrawal, 1995) to obtain two new vectors of which one is chosen for further processingatrandom.
Thechosenvectoristhenmutatedbythepolynomial mutation
operator(Deb,2001)toobtainachild vector (say, x (1)
u ).
Step1b: Creationofnew xl: First,the populationsize(Nl(x (1)
u ))forthe child solution
x (1)
u is determined by equation (16). The creation of xl depends on how close the
newvariableset x (1)
u iscomparedtothecurrentarchive, AT. If Nl = N (0)
l (indicating
that the xu is away from the archive members), new lower level variable vectors
x (i)
l (for i = 1, . . . , Nl(x (1)
u )) are created by applying selection-recombinationmutation
operations on members of PT and AT. Here, a parent member is cho-
sen from AT with a probability
|AT |
|AT |+|PT | , otherwise a member from PT is cho-
sen at random. A total of Nl(x (1)
u ) child solutions are created by concatenating
upper and lower level variable vectors together, as follows: ci = (x (1)
u , x (i)
l ) for
i = 1, . . . , Nl(x (1)
u ). Thus, for the new upper level variable vector x (1)
u , a subpop-
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