9 Multilevel Optimisation
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too. The nested nature of the bilevel optimisation problems makes the search of
robust solutions substantially more challenging compared to single-level optimisation problems [5, 31]. Suggested definitions of bilevel optimisation problems under
uncertainties and ideas for tackling variable uncertainty are presented in [31]. Test
problems can be found in [32].
9.5.4 Minimax (Worst-Case Scenario) as Bilevel Optimisation
Problem
In many real-world applications, uncertainties occur that are impossible to avoid,
and they might be due to several reasons such as reduced accuracy of the simulations, manufacture tolerances, unknown conditions of the problem, etc. A way
to manage these uncertainties is robust optimisation 2 , by taking into consideration
these uncertain parameters. One way to do this is by transforming them into decision
variables. Having that said, these uncertain parameters must be optimised so that
they can respond to the worst-case scenario they are able to describe (where as
scenario we mean a specific realisation of all parameters of the problem) [34].
In the worst-case scenario, the set of possible scenarios is described, and the
objective is to find a solution that performs reasonably well for all scenarios. This
solution is the one that has the best “worst-case” performance and performs well
even in the most hostile scenario [35].
Using minimax [36] as an approach to tackling uncertainty, the optimisation of
the worst-case scenario can be translated into solving a bilevel optimisation problem
where the worst case is defined as a maximum in the uncertain space (uncertain
parameters) and the optimal design corresponds to the minimum of all the maxima
in the design space (candidate solutions) [37, 38].
The minimax problem can be described as:
min
x∈X
f (x, y)
where
f (x, y) = max
y∈Y
f (x, y)
and X ∈ R m represent the set of candidate solutions and Y ∈ R n the set of
all possible scenarios [39]. Note that the upper and the lower levels have the
same objective function f (x, y), where the upper level is minimising according to
the actual variables x and the lower level maximising according to the uncertain
2 Note that this is one approach of robust optimisation. A practical guide for robust optimisation is
given by [33].
321
too. The nested nature of the bilevel optimisation problems makes the search of
robust solutions substantially more challenging compared to single-level optimisation problems [5, 31]. Suggested definitions of bilevel optimisation problems under
uncertainties and ideas for tackling variable uncertainty are presented in [31]. Test
problems can be found in [32].
9.5.4 Minimax (Worst-Case Scenario) as Bilevel Optimisation
Problem
In many real-world applications, uncertainties occur that are impossible to avoid,
and they might be due to several reasons such as reduced accuracy of the simulations, manufacture tolerances, unknown conditions of the problem, etc. A way
to manage these uncertainties is robust optimisation 2 , by taking into consideration
these uncertain parameters. One way to do this is by transforming them into decision
variables. Having that said, these uncertain parameters must be optimised so that
they can respond to the worst-case scenario they are able to describe (where as
scenario we mean a specific realisation of all parameters of the problem) [34].
In the worst-case scenario, the set of possible scenarios is described, and the
objective is to find a solution that performs reasonably well for all scenarios. This
solution is the one that has the best “worst-case” performance and performs well
even in the most hostile scenario [35].
Using minimax [36] as an approach to tackling uncertainty, the optimisation of
the worst-case scenario can be translated into solving a bilevel optimisation problem
where the worst case is defined as a maximum in the uncertain space (uncertain
parameters) and the optimal design corresponds to the minimum of all the maxima
in the design space (candidate solutions) [37, 38].
The minimax problem can be described as:
min
x∈X
f (x, y)
where
f (x, y) = max
y∈Y
f (x, y)
and X ∈ R m represent the set of candidate solutions and Y ∈ R n the set of
all possible scenarios [39]. Note that the upper and the lower levels have the
same objective function f (x, y), where the upper level is minimising according to
the actual variables x and the lower level maximising according to the uncertain
2 Note that this is one approach of robust optimisation. A practical guide for robust optimisation is
given by [33].