Chapter 9
Multilevel Optimisation
Margarita Antoniou and Peter Korošec
Abstract This chapter is a short introduction to multilevel optimisation problems.
The simplest multilevel problem is the one that has two levels, where one optimisation problem has as part of its constraints a second optimisation problem, known as
bilevel problem. Even this simple version of the problem is from a mathematical
point of view, complicated and difficult to solve. Therefore, approaches and
examples of bilevel problems, as well as special cases and extensions of this problem
that are used widely in literature, are presented. The most common methodologies
used to solve multilevel problems are then described, with more extended reference
to metaheuristic methods.
Keywords Multilevel optimisation · Bilevel · Hierarchical optimisation ·
Minimax problem · Metaheuristic methods
9.1 Introduction
The standard optimisation problem is the one that has a single-objective function
that needs to be optimised while satisfying some constraints and can be considered
as a single-level optimisation problem. Unfortunately, some real-world applications
cannot be represented in this way; therefore it has been extended in its form and
complexity, so it can have more objectives, different constraints, different types of
variable vectors, etc. [1]. One extension is to have multiple levels of optimisation
tasks instead of just one. A large number of application problems require more than
one level of optimisation, where one optimisation task is nested inside the other [2].
In its simplest form, the nested optimisation problem constitutes two levels. These
problems are known as bilevel optimisation problems.
M. Antoniou () · P. Korošec
Jožef Stefan Institute, Ljubljana, Slovenia
Jožef Stefan International Postgraduate School, Ljubljana, Slovenia
e-mail: margarita.antoniou@ijs.si; peter.korosec@ijs.si
© Springer Nature Switzerland AG 2021
M. Vasile (ed.), Optimization Under Uncertainty with Applications to Aerospace
Engineering, https://doi.org/10.1007/978-3-030-60166-9_9
307
Multilevel Optimisation
Margarita Antoniou and Peter Korošec
Abstract This chapter is a short introduction to multilevel optimisation problems.
The simplest multilevel problem is the one that has two levels, where one optimisation problem has as part of its constraints a second optimisation problem, known as
bilevel problem. Even this simple version of the problem is from a mathematical
point of view, complicated and difficult to solve. Therefore, approaches and
examples of bilevel problems, as well as special cases and extensions of this problem
that are used widely in literature, are presented. The most common methodologies
used to solve multilevel problems are then described, with more extended reference
to metaheuristic methods.
Keywords Multilevel optimisation · Bilevel · Hierarchical optimisation ·
Minimax problem · Metaheuristic methods
9.1 Introduction
The standard optimisation problem is the one that has a single-objective function
that needs to be optimised while satisfying some constraints and can be considered
as a single-level optimisation problem. Unfortunately, some real-world applications
cannot be represented in this way; therefore it has been extended in its form and
complexity, so it can have more objectives, different constraints, different types of
variable vectors, etc. [1]. One extension is to have multiple levels of optimisation
tasks instead of just one. A large number of application problems require more than
one level of optimisation, where one optimisation task is nested inside the other [2].
In its simplest form, the nested optimisation problem constitutes two levels. These
problems are known as bilevel optimisation problems.
M. Antoniou () · P. Korošec
Jožef Stefan Institute, Ljubljana, Slovenia
Jožef Stefan International Postgraduate School, Ljubljana, Slovenia
e-mail: margarita.antoniou@ijs.si; peter.korosec@ijs.si
© Springer Nature Switzerland AG 2021
M. Vasile (ed.), Optimization Under Uncertainty with Applications to Aerospace
Engineering, https://doi.org/10.1007/978-3-030-60166-9_9
307