Computational Intelligence - November 2017 - 17
acceptable level of risk. So o 0.95 # C means
The optimization framework here adopted is based
that, regardless of probability density function
type, the realizations of the quantity of interon the Value-at-risk (VaR) and conditional Value-atest should not exceed the threshold value
risk (CVaR) risk measures, also called quantile and
C in more than the 5% of the possible occursuperquantile respectively, which were originally
rences. If mean n and standard deviation v are
used to set the same constraint, the probability
conceived in financial engineering.
density function type of the quantity of interest must be known. Hence, if the distribution
is normal, o a # C can be expressed in closed form as
❏ use advanced sampling techniques such as multilevel Monte
-1
n + kv # C, with k = 2 erf (2a - 1). For other distribution
Carlo, or other importance sampling methods;
functions, it may not be possible to obtain expressions in closed
❏ use optimization algorithms not quite sensitive to noise
form or higher order moments may be required. The use of
since, from the point of view of the optimization algorithm,
CVaR as constraint or objective has similar implications, and a
the higher or lower quality of an estimate can be seen, in
simple closed form is also available for the normal distribution.
the broad sense, as noise level that influences the value of
objective functions and constraints;
B. VaR and CVaR Estimation
❏ use computational statistics methods to evaluate estimate
In the case where we have a finite number of samples, that is an
accuracy and confidence intervals; here in particular, the
Empirical Cumulative Distribution Function (ECDF) [20], we
bootstrap method is used, which has the important characproceed as described below. If x 1, x 2, f, x n are n independent
teristic of being a non-parametric method [21].
and identically distributed (i.i.d.) observations of the random
These issues, with varying degrees of importance, have been
variable X, then the a -VaR of X can be estimated by
present in all the literature related to the optimization under
uncertainty. The stochastic nature of risk function estimates has
-1
a; n
(8)
led the research on multi-objective methods to several extenot = X^nah: n = F n (a)
sions of the classical Pareto front concept. In [22], for example,
where X i: n is the i -th order statistic from the n observathe Pareto front exploration in presence of uncertainties is
tions, and
faced introducing the concept of probabilistic dominance, which is
an extension of the classical Pareto dominance. While in [23], a
n
Fn (y) = / 1 {X i # y}
(9)
probabilistic ranking mechanism is proposed that introduces
i=1
the probability of wrong decision directly in the formula for rank
is the empirical CDF constructed from the sequence Xu of
computation. An interesting approach, similar in some aspects
x 1, x 2, f, x n, and 1 {·} is the indicator function. The estimation
to the one here described, is found in [24] where a quantile
of a -CVaR of X, according to reference [14], can be directly
based approach is coupled with the probability of Pareto nonobtained using Equation (6):
dominance (already seen in [23]). Here, contrary to the cited
work, the optimization technique relies on the direct estiman
1
a; n
a; n
a; n +
tion of the risk functions obtained through the ECDF and of
ct = ot +
(10)
/ [X - ot ]
n (1 - a ) i = 1 i
its confidence intervals computed using the non-parametric
bootstrap method [21], [25].
III. Optimization Under Uncertainty
The stochastic nature of the sampling for risk function estiThe robust optimization problem (2) has to be redefined in
mation requires the use of an optimization algorithm not quite
terms of estimates of the risk functions in order to be numerisensitive to noise. In any case, if the seed of the Monte Carlo
cally solved:
sampling for the estimation of ECDF is made to change for
each new element in the population to be evaluated, it is quite
min n
tt i; n (z)
i = 1, f, p
likely that, sooner or later, some samples that give an underestiz!Z3R
mation of the risk function in minimization problems will
subject to:
emerge and prevail. The simplest remedy against these "pathott i; n (z) # c i i = p + 1, f, p + q
(11)
logical samples" is to increase the sample size until the estimawhere tt i; n is an estimator of the generic risk measure t i obtained
tion error is lower than a required threshold. If this is not
using a sample of size n.
possible, an alternative possibility is to make sure that a pathoThe quality of the risk function estimate directly influences
logical sampling does not extend its effect beyond a single evothe optimization results. Therefore, the following guidelines
lutionary algorithm generation. Consequently, if it is possible,
should be used in building a robust optimization procedure:
elitism should not be used so that the effect of the bias error of
❏ try to use the largest number of samples compatible
a single estimate is limited to the current generation. If elitism
with the computational budget in the estimation of the
has to be used, it is necessary to change the activation mecharisk function;
nism considering confidence intervals. In any case, even an
NOVEMBER 2017 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE
17
Table of Contents for the Digital Edition of Computational Intelligence - November 2017
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