9 Multilevel Optimisation
317
Fig. 9.5 Feasible objective space
Bilevel multiobjective, bilevel multileader/multifollower, bilevel under uncertainty,
and minimax as a bilevel optimisation problem are introduced as special cases of
the BOP.
9.5.1 Bilevel Multiobjective Optimisation Problems
It is possible that upper and/or lower levels have multiple objectives. The mathematical definition of such problem is:
min
x u ∈X U ,x l ∈X L
F (x u , x l ) = (F 1 (x u , x l ), . . . , F p (x u , x l ))
subject to
x l ∈ arg min
x l ∈X L
l ) = (f 1 (x u , x l ), . . . , f q (x u , x l )) :
317
Fig. 9.5 Feasible objective space
Bilevel multiobjective, bilevel multileader/multifollower, bilevel under uncertainty,
and minimax as a bilevel optimisation problem are introduced as special cases of
the BOP.
9.5.1 Bilevel Multiobjective Optimisation Problems
It is possible that upper and/or lower levels have multiple objectives. The mathematical definition of such problem is:
min
x u ∈X U ,x l ∈X L
F (x u , x l ) = (F 1 (x u , x l ), . . . , F p (x u , x l ))
subject to
x l ∈ arg min
x l ∈X L
l ) = (f 1 (x u , x l ), . . . , f q (x u , x l )) :