Progressively Interactive Evolutionary Multi-Objective Optimization ...

the lower level, solution y can be better than solution x in the upper level. Due
to these discrepancies, these problems will cause a conflict in converging to the
appropriatePareto-optimalfrontinbothlowerandupperleveloptimizationtasks.
6. Extension to higher level optimizationproblems is possible. Although our focus here is
for bilevel problems only, test problems scalable to three or higher levels would
be interesting, as there may exist some practical problems formulated in three or
higher levels. On the other hand, it will alsobe ideal tohave bilevel test problems
whichwilldegeneratetochallengingsingleleveltestproblems,ifasingleobjective
function is chosenfor eachlevel.
7. Differentlowerlevelproblemsmaycontributedifferentlytotheupperlevelfrontintermsof
theirextentofrepresentativesolutionson theupperlevel Pareto-optimalfront. Thesetest
problemswilltestanalgorithm’sabilitytoemphasisdifferentlowerlevelproblems
differentlyinordertofind awell-distributedsetof Pareto-optimalsolutions atthe
upperlevel.
8. Test problems must include constraints at both levels. This will allow algorithms to be
tested for their ability to handle constraints in both lower and upper level optimizationproblems.
Different principles are possible to construct test problems following the above
guidelines. Here, we present a generalized version of a recently proposed procedure
(Deband Sinha,2009a).
4.5.1 AMulti-ObjectiveBilevelTestProblemConstructionProcedure
We suggest atest problemconstruction procedureforabilevel problemhaving M and
m objectives in the upper and lower level, respectively. The procedure needs at most
threefunctional forms andis describedbelow:
Step1: First, a parametric trade-off function ΦU : R M−1 → R M which determines a
trade-off frontier (v1(u), . . . , vM (u)) on the F-space as a function of (M − 1) parameters
u (can be considered as a subset of xu) is chosen. Figure 3 shows such a
v1-v2 relationship on atwo-objective bilevel problem.
Step2: Next, for every point v on the ΦU-frontier, a (M − 1)-dimensional envelope
(U1(t), . . . , UM(t)) v ) on the F-space as a function of t (having (M − 1)
parameters) is chosen. The non-dominated part of the agglomerate envelope
∪v ∪ t
(v1(u) + U1(t) v ), . . . , (vM (u) + UM (t) v ) constitutes the overall upper
levelPareto-optimalfront. Figure3indicatesthis upperlevelPareto-optimalfront
andsomespecificPareto-optimalpoints(markedwithbiggercircles)derivedfrom
specific v-vectors.
Step3: Next, for every point v on the ΦU-frontier, a mapping function ΦL : R M−1 →
R m−1 which maps every v-point from the U-frontier to the lower level Paretooptimalfront
(f ∗ 1 (s), . . . , f ∗ m (s))v ischosen. Here, sisa(m−1)-dimensionalvector
andcanbeconsideredasasubsetof xl. Figure3showsthismappingTheenvelope
A ′ C ′ B ′ (a circle in the figure) is mapped to the lower level Pareto-optimal frontier
ACB(inlet figureon top).
Step4: After these three functions are defined, the lower level problem can be constructed
by using a bottom-up procedureadopted in Deb et al. (2005)through additional
terms arising from other lower level decision variables: fj(xl) = f ∗ j (s) +
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