414
E. M. Tirado and D. Quagliarella
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min
supf (z, X(ω))
z ∈ S ω ∈ Ω
s.to :
c i (z, X(ω)) ≤ 0 i = 1, . . . , m
S ⊆ R n
(13.4)
Two main disadvantages of this method are that minmax problems are generally
very computationally expensive and that the obtained solution is too conservative.
In addition, there is a high probability to face a nonfeasible problem.
Expected Values with Safety Margins In this case, expectations, as well as standard
deviations are introduced in a weighted sum. Therefore, the robust design problem
is cast in the following form:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min
E[f (z, X)] + λ 0 σ (f (z, X))
z ∈ S
s.to :
E[c i (z, X)] + λ i σ (c i (z, X)) ≤ 0 i = 1, . . . , m
S ⊆ R n
(13.5)
In this formulation, weighted sums of standard deviations can be interpreted as
safety margins. Furthermore, this approach is widely used, although it can lead to
serious problems such as the convergence to sub-optimal solutions due to the use
of expectations that penalize favourable and unfavourable candidate solutions in the
same way.
Performance Index on Cumulative Distribution Function (CDF) This approach is
based on the definition of an ad hoc performance index (or risk measure) as a
function of the Cumulative Distribution Function related to the quantity of interest
under investigation. For the sake of completeness, let us give the definition of
the Cumulative Distribution Function (CDF): the CDF gives the area under the
probability density function from minus infinity to x. It describes the probability
that a real-valued random variable X with a given distribution will be found at a
value less than or equal to x. Mathematically, this is expressed by Eq. 13.6.
F X (x) = P (X ≤ x)
(13.6)
A performance index (or risk measurement) allows the comparison of different
CDF shapes according to the risk criterion defined by the user. In this chapter, the
risk measures used are the Value-at-Risk (VaR) and the Conditional Value-at-Risk
(CVaR), also known as quantile and superquantile, respectively (their definitions are
given in Sect. 13.3).
E. M. Tirado and D. Quagliarella
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min
supf (z, X(ω))
z ∈ S ω ∈ Ω
s.to :
c i (z, X(ω)) ≤ 0 i = 1, . . . , m
S ⊆ R n
(13.4)
Two main disadvantages of this method are that minmax problems are generally
very computationally expensive and that the obtained solution is too conservative.
In addition, there is a high probability to face a nonfeasible problem.
Expected Values with Safety Margins In this case, expectations, as well as standard
deviations are introduced in a weighted sum. Therefore, the robust design problem
is cast in the following form:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
min
E[f (z, X)] + λ 0 σ (f (z, X))
z ∈ S
s.to :
E[c i (z, X)] + λ i σ (c i (z, X)) ≤ 0 i = 1, . . . , m
S ⊆ R n
(13.5)
In this formulation, weighted sums of standard deviations can be interpreted as
safety margins. Furthermore, this approach is widely used, although it can lead to
serious problems such as the convergence to sub-optimal solutions due to the use
of expectations that penalize favourable and unfavourable candidate solutions in the
same way.
Performance Index on Cumulative Distribution Function (CDF) This approach is
based on the definition of an ad hoc performance index (or risk measure) as a
function of the Cumulative Distribution Function related to the quantity of interest
under investigation. For the sake of completeness, let us give the definition of
the Cumulative Distribution Function (CDF): the CDF gives the area under the
probability density function from minus infinity to x. It describes the probability
that a real-valued random variable X with a given distribution will be found at a
value less than or equal to x. Mathematically, this is expressed by Eq. 13.6.
F X (x) = P (X ≤ x)
(13.6)
A performance index (or risk measurement) allows the comparison of different
CDF shapes according to the risk criterion defined by the user. In this chapter, the
risk measures used are the Value-at-Risk (VaR) and the Conditional Value-at-Risk
(CVaR), also known as quantile and superquantile, respectively (their definitions are
given in Sect. 13.3).