Progressively Interactive Evolutionary Multi-Objective Optimization ...

F2
Step 3: Map (U1,U2)
to (f1*,f2*)
V2
B’ A’
t
C’
u
f2
Derived Upper level
Pareto−optimal front
A
s
C
Lower
level
B
Step 2: Form envelope (U1,U2) at every (v1,v2)
f1
U1 Step 1: Choose ΦU
(v1,v2)
B’A’
Step 4: Form lower
level problem from (f1*,f2*)
f2
Step 5: Form upper
level problem from
(v1,v2) and (U1,U2)
A
D
Lower
level
Figure 3: A multi-objective bilevel test problem construction procedure is illustrated
throughtwo objectives inbothupper andlower levels.
D’
ej(xl\s) with ej ≥ 0. The task of the lower level optimization task would be to
makethe ej termzeroforeachobjective. Theterm ej canbemadecomplex(multimodal,
non-linear, or large-dimensional) to make the convergence to the lower
level Pareto-optimalfront difficultby anoptimization algorithm.
Step5: Finally,theupperlevelobjectivescanbeformedfrom uj functionsbyincluding
additional terms fromother upperlevel decisionvariables. Anadditiveformis as
follows: Fj(x) = uj(u) + Ej(xu\u) with Ej ≥ 0. Like the ej term, the term Ej can
also be made complex for an algorithm to properly converge to the upper level
Pareto-optimalfront.
Step6: Additionally,anumberoflinkedterms lj(xu\u, xl\s)and Lj(xu\u, xl\s)(nonnegative
terms) involving remaining xu (without u) and xl (without s) variables
canbeaddedtobothlowerandupperlevelproblems,respectively,tomakesurea
proper coordination between lower and upper level optimization tasks is needed
toconvergeto therespectivePareto-optimalfronts.
Another interesting yet a difficult scenario can be created with the linked terms. An
identical link term can be added to the lower level problem, but subtracted from the
the upper level problem. Thus, an effort to reduce the value of the linked term will
make an improvement in the lower level, whereas it will cause a deterioration in the
upper level. This will create a conflict in the working of both levels of optimization.
The following two-objective test problems areconstructed using the aboveprocedure.
84
B
F1
f1