9 Multilevel Optimisation
327
can be considered as an upper level objective with the decision variables such
as the shape of the structure, choice of materials, amount of material, etc. being
the lower level [2]. In [57], the minimisation of the final mass of the structure is
defined as the upper level and the location of the nodes and struts as the lower level.
In aerospace engineering, an example of bilevel formulation of a problem can be
found in [58], where a bilevel optimisation strategy for wing design was developed.
Optimisations of the wing planform and wing airfoil shapes were decoupled from
each other, formulating a bilevel optimisation problem. In [59] the formulation of
the truss topology problem with additional constraints on the displacements was
approached as a bilevel problem. There, satisfying the displacement constraint was
defined as the upper level problem and the minimisation of the compliance as
the lower level problem. Herskovits et al. [60] formulated the problem of shape
optimisation of nonlinear elastic solids in contact, as a bilevel problem, optimising
simultaneously the shape and the nonlinear contact analysis, by taking the cost
function minimisation with respect to the design variables as the upper level and,
the minimisation of the elastic energy.
In a completely different research area, a very interesting application was
proposed by Sinha et al. [61], where the parameter tuning of optimisation algorithms
was presented as a bilevel problem. They tested this approach on two commonly
used optimisation algorithms, differential evolution and Nelder-Mead, and they
found that the approach converges towards the efficient parameters. This is very
promising, as a suitable choice of parameters can have an important impact on the
efficiency of the algorithm. More research in this area is needed though.
Other areas where bilevel optimisation is applied can be found in transport,
for solving the toll setting problem, in the chemical industry, in environmental
economics, in seller-buyer strategies, in optimal design, etc. A recent list of areas of
application can be found in [5].
9.8 Summary
This chapter is a short introduction to multilevel and especially to bilevel optimisation problems. For a better understanding, a bilevel linear example is presented
and compared to a biobjective equivalent. Moreover, special cases of bilevel
problems are introduced, namely, multiobjective, multileader/multifollower and
bilevel optimisation, under uncertainty problems along with minmax (worst-case
scenario) as bilevel optimisation problem. The main solution algorithms categories,
the classical and metaheuristic approaches are noted. More focus is given to
metaheuristic approaches, providing a list of the algorithms that can be found in
the literature. Last but not the least, some applications that have been formulated
and solved as bilevel problems are shortly mentioned.
327
can be considered as an upper level objective with the decision variables such
as the shape of the structure, choice of materials, amount of material, etc. being
the lower level [2]. In [57], the minimisation of the final mass of the structure is
defined as the upper level and the location of the nodes and struts as the lower level.
In aerospace engineering, an example of bilevel formulation of a problem can be
found in [58], where a bilevel optimisation strategy for wing design was developed.
Optimisations of the wing planform and wing airfoil shapes were decoupled from
each other, formulating a bilevel optimisation problem. In [59] the formulation of
the truss topology problem with additional constraints on the displacements was
approached as a bilevel problem. There, satisfying the displacement constraint was
defined as the upper level problem and the minimisation of the compliance as
the lower level problem. Herskovits et al. [60] formulated the problem of shape
optimisation of nonlinear elastic solids in contact, as a bilevel problem, optimising
simultaneously the shape and the nonlinear contact analysis, by taking the cost
function minimisation with respect to the design variables as the upper level and,
the minimisation of the elastic energy.
In a completely different research area, a very interesting application was
proposed by Sinha et al. [61], where the parameter tuning of optimisation algorithms
was presented as a bilevel problem. They tested this approach on two commonly
used optimisation algorithms, differential evolution and Nelder-Mead, and they
found that the approach converges towards the efficient parameters. This is very
promising, as a suitable choice of parameters can have an important impact on the
efficiency of the algorithm. More research in this area is needed though.
Other areas where bilevel optimisation is applied can be found in transport,
for solving the toll setting problem, in the chemical industry, in environmental
economics, in seller-buyer strategies, in optimal design, etc. A recent list of areas of
application can be found in [5].
9.8 Summary
This chapter is a short introduction to multilevel and especially to bilevel optimisation problems. For a better understanding, a bilevel linear example is presented
and compared to a biobjective equivalent. Moreover, special cases of bilevel
problems are introduced, namely, multiobjective, multileader/multifollower and
bilevel optimisation, under uncertainty problems along with minmax (worst-case
scenario) as bilevel optimisation problem. The main solution algorithms categories,
the classical and metaheuristic approaches are noted. More focus is given to
metaheuristic approaches, providing a list of the algorithms that can be found in
the literature. Last but not the least, some applications that have been formulated
and solved as bilevel problems are shortly mentioned.