Progressively Interactive Evolutionary Multi-Objective Optimization ...

more studies are performed, the algorithms must have to be tested and compared
against each other. This process needs an adequate number of test problems with tunabledifficulties.
Inthenextsection,wesuggestagenericprocedurefordevelopingtest
problems andsuggest a testsuite.
4 Test ProblemsforMulti-Objective Bilevel Programming
There does not exist any systematic past study in developing test problems for multiobjectivebilevelprogramming.However,Eichfelder(2007)usedanumberoftestproblems
in her study, in which some problems were not analyzed for their exact Paretooptimal
fronts. Here, we first describe some of these existing test problems (we refer
tothemas ‘TP’ here)andthen discuss anextension ofour recentlyproposedtest problem
development procedure (Deb and Sinha, 2009a) for any number of objectives and
scenarios. Theprincipleisusedtogeneratefivetestproblems(werefertothemas‘DS’
here).
4.1 ProblemTP1
ProblemTP1hasatotalofthreevariableswith xl = (x1, x2) T and xu = (y)(Eichfelder,
2007):
x1 − y
Minimize F(x) =
,
x2
g1(x) = y 2 − x 2 1 − x 2
2 ≥ 0 , (2)
subject to (x1, x2) ∈ argmin (x1,x2)
G1(x) = 1 + x1 + x2 ≥ 0,
−1 ≤ x1, x2 ≤ 1, 0 ≤ y ≤ 1.
x1
f(x) =
x2
ThePareto-optimalsetforthelower-leveloptimizationtaskforafixed y is thebottomleft
quarter of the circle: {(x1, x2) ∈ R 2
| x2 1 + x22 = y2 , x1 ≤ 0, x2 ≤ 0}. The linear
constraint in the upper level optimization task does not allow the entire quarter circle
to be feasible for some y. Thus, at most a couple of points from the quarter circle
belongstothePareto-optimalsetoftheoverallproblem. Eichfelder(2007)reportedthe
following Pareto-optimalsolutions forthis problem:
x ∗
= (x1, x2, y) ∈ R 3
x1 = −1 − x2, x2 = − 1 1
1
± 8y2 − 4, y ∈ √2 , 1 . (3)
2 4
Figure 1 shows the Pareto-optimal front for the problem TP1. Lower level Paretooptimal
fronts of some representative y values are also shown in the figure using
dashedlines, indicating thatatmost twosuchPareto-optimalsolutions (suchaspoints
B and C for y = 0.9) of a lower level optimization problem become the candidate
Pareto-optimalsolutions ofthe upperlevel problem.
4.2 ProblemTP2
Wecreatedasimplisticbileveltwo-objectiveoptimizationproblemelsewhere(Deband
Sinha,2009b),having xl = (x1, . . . , xK) and xu = (y):
Minimize F(x) =
subjectto
(x1, x2, . . . , xK) ∈ argmin (x1,x2,...,x K )
−1 ≤ (x1, x2, . . . , xK, y) ≤ 2.
(x1 − 1) 2 + K
i=2 x2 i + y 2
(x1 − 1) 2 + K
i=2 x2 i + (y − 1) 2
80
f(x) =
,
x 2 1 + K
i=2 x2 i
(x1 − y) 2 + K
i=2 x2 i
,
(4)