17 Introduction to Evidence-Based Robust Optimisation
565
The multidisciplinary nature of a generic complex system is captured by the Evidence Network Model (ENM) described in Sect. 17.4.4.1 where the component’s
QoI contribute to the overall performance as stated in Eq.(17.46).
The analytical function f =
i g i is here used as test case. The i-th node’
performance and the generic coupling function h ij between couples of nodes are,
respectively, formulated as:
g i = d i +
N u
F E,i
k=1
sin u ik −
N u
F E,i
k=1
u ik
+
m c
k=1
h ik
3
(17.62)
h ik =
N u
F E,i
k=1
sin
u ik
+ u ik
f is multi-modal with respect of Θ u and Θ c separately while monotonic with
respect of the coupling functions h.
The scaling behaviour of the method is tested applying f to the different network
topology represented in Fig. 17.4. In particular, each i-node depends on a pair of
uncoupling uncertain variables u u,i = [u 1
u,i , u 2
u,i ] and each link shares between two
nodes a pair of coupling uncertain variables u c,ij = [u 1
c,ij , u 2
c,ij ]. Then, being n the
number of nodes and l the number of links for a selected topology, the total number
of uncertain variables is: n u,tot = 2(n + l).
The robust optimisation approach first evaluates the optimal design configuration d opt solving the constrained min-max optimisation problem described by
Eq. (17.35) by means of Algorithm 1. Figure 17.3 presents, for example, the
convergence of the algorithm for f applied to topology (c) where the constraint
function has been defined as:
C(d, u) = −
i
d i + 5
i
sin(d i u i )
≤ 0
(17.63)
The figure shows the convergence of the algorithm to the optimal solution trading
the conflict between the performance indicator
f max = max
u∈U
f (d, u)
(17.64)
and the constraint function
c max = max
u∈U
c(d, u) ≤ 0
(17.65)
At convergence, d opt gives the minimum worst case value of f max while pushing
c max at the edge of the feasible set ] − ∞, 0]. The curve c, instead, represents
the value of the constraint function corresponding to d opt and the worst uncertain
scenario for f .
565
The multidisciplinary nature of a generic complex system is captured by the Evidence Network Model (ENM) described in Sect. 17.4.4.1 where the component’s
QoI contribute to the overall performance as stated in Eq.(17.46).
The analytical function f =
i g i is here used as test case. The i-th node’
performance and the generic coupling function h ij between couples of nodes are,
respectively, formulated as:
g i = d i +
N u
F E,i
k=1
sin u ik −
N u
F E,i
k=1
u ik
+
m c
k=1
h ik
3
(17.62)
h ik =
N u
F E,i
k=1
sin
u ik
+ u ik
f is multi-modal with respect of Θ u and Θ c separately while monotonic with
respect of the coupling functions h.
The scaling behaviour of the method is tested applying f to the different network
topology represented in Fig. 17.4. In particular, each i-node depends on a pair of
uncoupling uncertain variables u u,i = [u 1
u,i , u 2
u,i ] and each link shares between two
nodes a pair of coupling uncertain variables u c,ij = [u 1
c,ij , u 2
c,ij ]. Then, being n the
number of nodes and l the number of links for a selected topology, the total number
of uncertain variables is: n u,tot = 2(n + l).
The robust optimisation approach first evaluates the optimal design configuration d opt solving the constrained min-max optimisation problem described by
Eq. (17.35) by means of Algorithm 1. Figure 17.3 presents, for example, the
convergence of the algorithm for f applied to topology (c) where the constraint
function has been defined as:
C(d, u) = −
i
d i + 5
i
sin(d i u i )
≤ 0
(17.63)
The figure shows the convergence of the algorithm to the optimal solution trading
the conflict between the performance indicator
f max = max
u∈U
f (d, u)
(17.64)
and the constraint function
c max = max
u∈U
c(d, u) ≤ 0
(17.65)
At convergence, d opt gives the minimum worst case value of f max while pushing
c max at the edge of the feasible set ] − ∞, 0]. The curve c, instead, represents
the value of the constraint function corresponding to d opt and the worst uncertain
scenario for f .