42
D. Krpelík and T. Basu
Fig. 2.2 The evolution of quantiles of U(t) (left) and CDFs of U(t) at various times (right)
F U(t) (x) = P r(U (t) < x) = P A ({a : u(t; a) < x})
= P r(A > u
−1
t (x)) = 1 − F A (u
−1
t (x))
= exp
−λu
−1
t (x)
= x
λ
t ,
where u
−1
t (x) =
log x
t .
Examples of derived CDFs and evolutions of α-quantiles of U(t), the values x
such that F U(t) (x) = α, are depicted in Fig. 2.2.
Since we have drifted away from the crisp true–false values of boolean logic
towards a multi-valued logic in which each statement is assigned a probability,
the degree of faith, the meaning of comparative statements becomes unclear (apart
from some special cases). Several possible orderings are available for pairs of RVs.
We are able to evaluate the probability that they will be ordered once realised,
P r(X < Y ). Another widely used one is the stochastic dominance for which
X ≥ st Y iff ∀x : F X (x) ≤ F Y (x). Nevertheless, given two RVs X, Y , the order
among their realisations x = X(ω), y = Y (ω) may differ depending on ω ∈ Ω,
where Ω represents the sample space (Sect. 2.3.1). Generally, there is no unique
way of defining which RV is greater in a pair. For us to be able to compare them in
an optimisation algorithm, we need to redefine the problem, so that we obtain a total
ordering in the range of the objective function (i.e. so that we can compare any pair
of proposals by their fitnesses), and so that we uniquely determine whether possible
constraints of the problem were violated.
Both may be achieved by replacing the RVs with some meaningful functionals
derived from the distribution of the original RV, such as the expected values
(Definition 2.4). This approach is justified in mass production scenarios where
we try to optimise our long-run (financial) gains according to the law of large
numbers (Theorem 2.2). But a different approach should be taken in the case
of robust design optimisation, which is often solved by deriving the worst-case
scenario as the objective function. A crisp worst case may not be available if
the original objective function is a RV, but we can study what is the likely worst
D. Krpelík and T. Basu
Fig. 2.2 The evolution of quantiles of U(t) (left) and CDFs of U(t) at various times (right)
F U(t) (x) = P r(U (t) < x) = P A ({a : u(t; a) < x})
= P r(A > u
−1
t (x)) = 1 − F A (u
−1
t (x))
= exp
−λu
−1
t (x)
= x
λ
t ,
where u
−1
t (x) =
log x
t .
Examples of derived CDFs and evolutions of α-quantiles of U(t), the values x
such that F U(t) (x) = α, are depicted in Fig. 2.2.
Since we have drifted away from the crisp true–false values of boolean logic
towards a multi-valued logic in which each statement is assigned a probability,
the degree of faith, the meaning of comparative statements becomes unclear (apart
from some special cases). Several possible orderings are available for pairs of RVs.
We are able to evaluate the probability that they will be ordered once realised,
P r(X < Y ). Another widely used one is the stochastic dominance for which
X ≥ st Y iff ∀x : F X (x) ≤ F Y (x). Nevertheless, given two RVs X, Y , the order
among their realisations x = X(ω), y = Y (ω) may differ depending on ω ∈ Ω,
where Ω represents the sample space (Sect. 2.3.1). Generally, there is no unique
way of defining which RV is greater in a pair. For us to be able to compare them in
an optimisation algorithm, we need to redefine the problem, so that we obtain a total
ordering in the range of the objective function (i.e. so that we can compare any pair
of proposals by their fitnesses), and so that we uniquely determine whether possible
constraints of the problem were violated.
Both may be achieved by replacing the RVs with some meaningful functionals
derived from the distribution of the original RV, such as the expected values
(Definition 2.4). This approach is justified in mass production scenarios where
we try to optimise our long-run (financial) gains according to the law of large
numbers (Theorem 2.2). But a different approach should be taken in the case
of robust design optimisation, which is often solved by deriving the worst-case
scenario as the objective function. A crisp worst case may not be available if
the original objective function is a RV, but we can study what is the likely worst