17 Introduction to Evidence-Based Robust Optimisation
545
satisfying the constraints under uncertainty. In both cases, from a computational
point of view, uncertainty can be treated in two different ways
1. deterministically, where the quantity of interest is computed for a deterministic
variation of, for example, the uncertain parameters within some deterministically
defined sets;
2. non-deterministically, where some uncertainty measures are computed as a
function of the uncertain parameters;
Consider now a generic deterministic optimisation problem associated to the
design of a system characterised by the performance index f and decision vector d:
min
d∈D
f (d)
(17.1)
subjected to
g j (d) ≤ 0
j = 1, . . . , r
(17.2)
where D is a design space and g j are constraint functions. If the system is affected
by some form of uncertainty, characterised by the uncertainty vector u ∈ U , with U
the uncertainty space, the problem can be re-formulated as:
min
d∈D
φ(d, u)
(17.3)
subjected to
γ j (d, u) ≤ 0
j = 1, . . . , r
(17.4)
where φ and γ j are some measures that account for the effect of u, respectively,
on the quantities of interest f and g j . If uncertainty is treated deterministically and
there are no constraints, then φ in problem (17.3) can be written as:
φ(d, u
∗ ) = sup u∈U f (d, u)
(17.5)
where u ∗ is the uncertain vector for which f attains the maximum value. This
approach is called robust regularisation in [32]. In [32] the function φ is defined as
the worst case scenario in a neighbourhood U(() where is called the regularisation
parameter. This approach can also be found in [33–36] and can be generalised, as
in Eq. (17.5), to include the whole uncertainty space. In this case the value φ is the
global worst case scenario.
If there is enough information to model uncertainty with probability distributions,
a probabilistic approach can be used. Two examples of probabilistic measures of
robustness are:
545
satisfying the constraints under uncertainty. In both cases, from a computational
point of view, uncertainty can be treated in two different ways
1. deterministically, where the quantity of interest is computed for a deterministic
variation of, for example, the uncertain parameters within some deterministically
defined sets;
2. non-deterministically, where some uncertainty measures are computed as a
function of the uncertain parameters;
Consider now a generic deterministic optimisation problem associated to the
design of a system characterised by the performance index f and decision vector d:
min
d∈D
f (d)
(17.1)
subjected to
g j (d) ≤ 0
j = 1, . . . , r
(17.2)
where D is a design space and g j are constraint functions. If the system is affected
by some form of uncertainty, characterised by the uncertainty vector u ∈ U , with U
the uncertainty space, the problem can be re-formulated as:
min
d∈D
φ(d, u)
(17.3)
subjected to
γ j (d, u) ≤ 0
j = 1, . . . , r
(17.4)
where φ and γ j are some measures that account for the effect of u, respectively,
on the quantities of interest f and g j . If uncertainty is treated deterministically and
there are no constraints, then φ in problem (17.3) can be written as:
φ(d, u
∗ ) = sup u∈U f (d, u)
(17.5)
where u ∗ is the uncertain vector for which f attains the maximum value. This
approach is called robust regularisation in [32]. In [32] the function φ is defined as
the worst case scenario in a neighbourhood U(() where is called the regularisation
parameter. This approach can also be found in [33–36] and can be generalised, as
in Eq. (17.5), to include the whole uncertainty space. In this case the value φ is the
global worst case scenario.
If there is enough information to model uncertainty with probability distributions,
a probabilistic approach can be used. Two examples of probabilistic measures of
robustness are: