17 Introduction to Evidence-Based Robust Optimisation
561
We say that the pair (f, Ω) is M-decomposable if, given an M > 0, for k > M we
have that:
(x k , y k ) = arg max
δΩ pq
f (x, y)
(17.51)
with δΩ pq = δΩ
p
x × δΩ
q
y .
The ENMs has been constructed under the following properties:
1. The contribution of the coupled variable u ij to the value f manifests through the
scalar functions h ij and h ji .
2. The pair (f, U ) is M-decomposable. In particular in this chapter we consider the
case in which M = 0.
17.4.4.2 Decomposition Method
The decomposition algorithm aims at decoupling the sub-systems over the uncertain
variables in order to optimise only over a small subset of the Focal Elements
(Algorithm 2); this procedure requires the following steps:
1. Solution of the optimal worst case scenario problem:
min
d∈D
max
u∈U
F (d, u)
(17.52)
2. Maximisation over the coupled variables and computation of Bel c (A).
3. Maximisation over the uncoupled variables.
4. Reconstruction of the approximation
Bel(A).
Point 1 has been already discussed in Sect. 17.3.1. In the following the solution
of problem (17.52) is represented by the values ˜
d and u and it is assumed that ˜
d is
already available.
For each coupled vector u ij a maximisation is run over each Focal Element
θ k,ij ⊆ Θ ij ⊆ U , given d and keeping fixed all the other components to u k and
u lm ∀k, l, m \ {l, m} = {i, j }. Taking again the example in Fig. 17.1 we have
ˆ
u k,12 = arg max
u 12 ∈θ k,12
F ( d, u 1 , u 2 , u 3 , u 12 , u 13 , u 23 ), ∀θ k,12 ⊂ Θ 12
ˆ
u k,13 = arg max
u 13 ∈θ k,13
F ( d, u 1 , u 2 , u 3 , u 12 , u 13 , u 23 ), ∀θ k,13 ⊂ Θ 13
ˆ
u k,23 = arg max
u 23 ∈θ k,23
F ( d, u 1 , u 2 , u 3 , u 12 , u 13 , u 23 ), ∀θ k,23 ⊂ Θ 23
(17.53)
For easiness in the notation we will indicate with
F (u ij ) := F ( ˜
d, u 1 , . . . , u ij , . . . , u i+1j , . . .).
561
We say that the pair (f, Ω) is M-decomposable if, given an M > 0, for k > M we
have that:
(x k , y k ) = arg max
δΩ pq
f (x, y)
(17.51)
with δΩ pq = δΩ
p
x × δΩ
q
y .
The ENMs has been constructed under the following properties:
1. The contribution of the coupled variable u ij to the value f manifests through the
scalar functions h ij and h ji .
2. The pair (f, U ) is M-decomposable. In particular in this chapter we consider the
case in which M = 0.
17.4.4.2 Decomposition Method
The decomposition algorithm aims at decoupling the sub-systems over the uncertain
variables in order to optimise only over a small subset of the Focal Elements
(Algorithm 2); this procedure requires the following steps:
1. Solution of the optimal worst case scenario problem:
min
d∈D
max
u∈U
F (d, u)
(17.52)
2. Maximisation over the coupled variables and computation of Bel c (A).
3. Maximisation over the uncoupled variables.
4. Reconstruction of the approximation
Bel(A).
Point 1 has been already discussed in Sect. 17.3.1. In the following the solution
of problem (17.52) is represented by the values ˜
d and u and it is assumed that ˜
d is
already available.
For each coupled vector u ij a maximisation is run over each Focal Element
θ k,ij ⊆ Θ ij ⊆ U , given d and keeping fixed all the other components to u k and
u lm ∀k, l, m \ {l, m} = {i, j }. Taking again the example in Fig. 17.1 we have
ˆ
u k,12 = arg max
u 12 ∈θ k,12
F ( d, u 1 , u 2 , u 3 , u 12 , u 13 , u 23 ), ∀θ k,12 ⊂ Θ 12
ˆ
u k,13 = arg max
u 13 ∈θ k,13
F ( d, u 1 , u 2 , u 3 , u 12 , u 13 , u 23 ), ∀θ k,13 ⊂ Θ 13
ˆ
u k,23 = arg max
u 23 ∈θ k,23
F ( d, u 1 , u 2 , u 3 , u 12 , u 13 , u 23 ), ∀θ k,23 ⊂ Θ 23
(17.53)
For easiness in the notation we will indicate with
F (u ij ) := F ( ˜
d, u 1 , . . . , u ij , . . . , u i+1j , . . .).