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Published in:
2020 | OriginalPaper | Chapter
18. Solving Optimization Problems Under Stochastic Uncertainty by Random Search Methods (RSM)
Author : Kurt Marti
Published in: Optimization Under Stochastic Uncertainty
Publisher: Springer International Publishing
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Abstract
In many optimization problems arising in practice only statistical information is available about the objective function F
0. Hence, we have then only access to a realization of a random function f = f(ω, x) on a probability space \(\left (\Omega ,\mathcal {A},\mathcal {P}\right )\) such that F
0(x) is the expectation of f(⋅, x): see, e.g., A special case is given by where n(ω, x) is an additional zero-mean random noise term. Having a certain number m of independent sample functions of the random function in (18.2a), we may use the estimated objective function
$$\displaystyle \begin{aligned} \min \: F_0(x) \quad \mbox{s.t. } x \in D \end{aligned} $$
$$\displaystyle \begin{aligned} F_0(x) = Ef(\omega ,x), \: x \in D, \end{aligned} $$
$$\displaystyle \begin{aligned} f(\omega ,x) = F_0(x) + n(\omega ,x), \end{aligned} $$
$$\displaystyle \begin{aligned} f_k(x) = f(\omega _k,x), \: k \in M, \end{aligned} $$
$$\displaystyle \begin{aligned} \hat {F}(x):= \frac {1}{m}\sum \limits _{k \in M}f_k(x). \end{aligned} $$