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G. Filippi and M. Vasile
min d∈D [min u∈U f 1 , min u∈U f 2 , . . . , min u∈U f m ] T
s.t.
max u∈U C(d, u) ≤ 0
(17.39)
In both problem (17.37) and (17.39) the assumption is that the realisations of u that
maximise or minimise the components of f are independent.
17.3.1 Optimising the Worst Case Scenario
In the literature, a number of optimisation techniques have been proposed to solve
problem (17.32). Some are based on mathematical programming [49–60] while
others are using heuristic methods [61–64]. In particular, interesting results have
been obtained with evolutionary algorithms (EAs). As in [65], three main categories
can be identified:
• algorithms for discrete min-max problems evaluating all [66] or a subset [67, 68]
of the uncertain scenarios;
• algorithms directly solving the nested problem max u∈U f (d, u) within
Eq. (17.33) [69, 70]. This approach is highly expensive for costly objective
functions;
• the last approach defined in [65] is the co-evolution strategy: it emulates the
dynamic evolution driven by natural selection where each organism has to
continuously adapt to the other: two different populations are developed in
parallel while information is shared between them. Some ideas are developed
in [71–73] and [74] extends the previous three papers for problems that are
not constrained to satisfy the symmetrical condition min d∈D max u∈U f (d, u) =
max u∈U min d∈D f (d, u).
Furthermore some hybrid methods exist that mix evolutionary strategies and
mathematical programming. Given the considerable computational cost required
to solve problem (17.32) some authors proposed methods to reduce the number of
function evaluations through the use of surrogate models: [65, 75, 76] are SurrogateAssisted Evolutionary Algorithms (SAEA) and [77] proposes a kriging surrogate.
The algorithm proposed in this section solves the more general constrained
problem (17.35) using a combination of optimisation and restoration loops similar
to what can be found in [77]. An extensive description of the approach and its
performance can be found in [78].
The optimisation loop minimises max u e ∈A f (d, u e ) over the decision vector d
and the restoration loop maximises f (d ∗ , u) over the uncertainty vector u for d ∗ =
arg min d max u e ∈A f (d, u e ). The vectors u e are taken from the space of the maxima
over U . The constraint vector C defines the admissible set for both d and u. In
particular, in order for the solution to be robust against all realisations of u one has
G. Filippi and M. Vasile
min d∈D [min u∈U f 1 , min u∈U f 2 , . . . , min u∈U f m ] T
s.t.
max u∈U C(d, u) ≤ 0
(17.39)
In both problem (17.37) and (17.39) the assumption is that the realisations of u that
maximise or minimise the components of f are independent.
17.3.1 Optimising the Worst Case Scenario
In the literature, a number of optimisation techniques have been proposed to solve
problem (17.32). Some are based on mathematical programming [49–60] while
others are using heuristic methods [61–64]. In particular, interesting results have
been obtained with evolutionary algorithms (EAs). As in [65], three main categories
can be identified:
• algorithms for discrete min-max problems evaluating all [66] or a subset [67, 68]
of the uncertain scenarios;
• algorithms directly solving the nested problem max u∈U f (d, u) within
Eq. (17.33) [69, 70]. This approach is highly expensive for costly objective
functions;
• the last approach defined in [65] is the co-evolution strategy: it emulates the
dynamic evolution driven by natural selection where each organism has to
continuously adapt to the other: two different populations are developed in
parallel while information is shared between them. Some ideas are developed
in [71–73] and [74] extends the previous three papers for problems that are
not constrained to satisfy the symmetrical condition min d∈D max u∈U f (d, u) =
max u∈U min d∈D f (d, u).
Furthermore some hybrid methods exist that mix evolutionary strategies and
mathematical programming. Given the considerable computational cost required
to solve problem (17.32) some authors proposed methods to reduce the number of
function evaluations through the use of surrogate models: [65, 75, 76] are SurrogateAssisted Evolutionary Algorithms (SAEA) and [77] proposes a kriging surrogate.
The algorithm proposed in this section solves the more general constrained
problem (17.35) using a combination of optimisation and restoration loops similar
to what can be found in [77]. An extensive description of the approach and its
performance can be found in [78].
The optimisation loop minimises max u e ∈A f (d, u e ) over the decision vector d
and the restoration loop maximises f (d ∗ , u) over the uncertainty vector u for d ∗ =
arg min d max u e ∈A f (d, u e ). The vectors u e are taken from the space of the maxima
over U . The constraint vector C defines the admissible set for both d and u. In
particular, in order for the solution to be robust against all realisations of u one has