17 Introduction to Evidence-Based Robust Optimisation
553
min d∈D max u∈U f (d, u)
s.t.
C(d, u) ≤ 0
(17.34)
where this time Ω = {u|C(d, u) ≤ 0} and C : U × D → R n c is a vector function.
In this case, the value of Bel is not 1 and the Bel associated to the solution of
problem (17.34) needs to be evaluated a posteriori. If one requires the problem to be
globally reliable and not just globally robust, then the constrained min-max problem
becomes
min d∈D max u∈U f (d, u)
s.t.
max u∈U C(d, u) ≤ 0
(17.35)
with Ω = {u| max u∈U C(d, u) ≤ 0}. Note that problem (17.34) is not equivalent to
problem (17.26). In fact, consider the constrained problem:
min d∈D max u∈U f (d, u)
s.t.
f (d, u) − ν ≤ 0
(17.36)
In this case the optimal solution does not necessarily correspond to a maximum Bel
value because the Bel value is the sum of all focal elements that satisfy f ≤ ν while
the minimisation with respect to d would only consider a single realisation of u.
Problems (17.34) can be readily extended to the case in which f =
[f 1 , f 2 , . . . , f m ] T is a vector function, as follows:
min d∈D [max u∈U f 1 , max u∈U f 2 , . . . , max u∈U f m ] T
s.t.
max u∈U C(d, u) ≤ 0
(17.37)
Likewise one can extend problem (17.26) to the multi-Belief case:
max d∈D Bel(f 1 ≤ ν 1 )
max d∈D Bel(f 2 ≤ ν 2 )
. . .
max d∈D Bel(f m ≤ ν m )
min ν 1 ∈R ν 1
min ν 2 ∈R ν 2
. . .
min ν m ∈R ν m
(17.38)
Note that the dual to problem (17.37) is the minimisation problem: