13 Risk Measures in the Context of Robust and Reliability Based Optimization
415
13.3 Risk Measures
When random events are modeled by random variables, as in the context of
optimization under uncertainty, a way to measure risk should be figured out. With
this purpose, a functional ρ(X) is going to be defined for risk level quantification.
Subsequently, an acceptable level of risk C must be decided, considering that there
will, inevitably, be adverse events. Thus, the next inequality equation can be defined:
ρ(X) ≤ C
(13.7)
Then, if the random variables representative of the cost depend on a deterministic
decision vector x of size m, the following minimization problem can be stated:
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
min
ρ 0 (X 0 (x))
x ∈ S ⊆ R n
s.to :
ρ i (X i (x)) ≤ c i i = 1, . . . , m
(13.8)
Within this framework, different definitions for risk functional can be established.
This will lead to different approaches of facing optimization problems under
uncertainty. The most immediate and familiar alternative of risk measure is the
expected value. This means that, on average, it should be X ≤ C:
μ(X) ≤ C → ρ(X) = μ(X) = EX
(13.9)
Being more stringent, a condition on the standard deviation or on variance could be
imposed if there is a need to reduce the variation range of the quantity of interest:
μ(X) + λσ (X) ≤ C → ρ(X) = μ(X) + λσ (X)
(13.10)
μ(X) + λσ
2 (X) ≤ C → ρ(X) = μ(X) + λσ
2 (X)
(13.11)
Indeed, classical robust design problem formulation is based on mean, μ, and
variance, σ 2 [11], which can be treated as separated objectives in a multi-objective
framework [9], as a weighted combination, or even cast into a constrained optimization format. However, the use of this classical approach may often generate some
problems, since mean and variance are not independent measures, and it might be
difficult to decide how much the mean must be penalized to get the desired reduction
of variance.
Alternative risk measures are available that offer a better control on the desired
features of the cumulative distribution function of interest. Here, in particular, the
Value-at-Risk or the Conditional Value-at-Risk are introduced and used.
Let X be a random variable and F X (x) = P (X ≤ x) the Cumulative Distribution
Function of X. Thus, the inverse CDF of X can be defined as F
−1
X (γ ) = inf {x :
415
13.3 Risk Measures
When random events are modeled by random variables, as in the context of
optimization under uncertainty, a way to measure risk should be figured out. With
this purpose, a functional ρ(X) is going to be defined for risk level quantification.
Subsequently, an acceptable level of risk C must be decided, considering that there
will, inevitably, be adverse events. Thus, the next inequality equation can be defined:
ρ(X) ≤ C
(13.7)
Then, if the random variables representative of the cost depend on a deterministic
decision vector x of size m, the following minimization problem can be stated:
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
min
ρ 0 (X 0 (x))
x ∈ S ⊆ R n
s.to :
ρ i (X i (x)) ≤ c i i = 1, . . . , m
(13.8)
Within this framework, different definitions for risk functional can be established.
This will lead to different approaches of facing optimization problems under
uncertainty. The most immediate and familiar alternative of risk measure is the
expected value. This means that, on average, it should be X ≤ C:
μ(X) ≤ C → ρ(X) = μ(X) = EX
(13.9)
Being more stringent, a condition on the standard deviation or on variance could be
imposed if there is a need to reduce the variation range of the quantity of interest:
μ(X) + λσ (X) ≤ C → ρ(X) = μ(X) + λσ (X)
(13.10)
μ(X) + λσ
2 (X) ≤ C → ρ(X) = μ(X) + λσ
2 (X)
(13.11)
Indeed, classical robust design problem formulation is based on mean, μ, and
variance, σ 2 [11], which can be treated as separated objectives in a multi-objective
framework [9], as a weighted combination, or even cast into a constrained optimization format. However, the use of this classical approach may often generate some
problems, since mean and variance are not independent measures, and it might be
difficult to decide how much the mean must be penalized to get the desired reduction
of variance.
Alternative risk measures are available that offer a better control on the desired
features of the cumulative distribution function of interest. Here, in particular, the
Value-at-Risk or the Conditional Value-at-Risk are introduced and used.
Let X be a random variable and F X (x) = P (X ≤ x) the Cumulative Distribution
Function of X. Thus, the inverse CDF of X can be defined as F
−1
X (γ ) = inf {x :