Solving Optimization Problems Under Stochastic Uncertainty by Random Search Methods (RSM)

Kurt Marti
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Kurt Marti: University of Bundeswehr Munich

Chapter Chapter 18 in Optimization Under Stochastic Uncertainty, 2020, pp 341-349 from Springer

Abstract: Abstract In many optimization problems arising in practice min F 0 ( x ) s.t. x ∈ D $$\displaystyle \begin{aligned} \min \: F_0(x) \quad \mbox{s.t. } x \in D \end{aligned} $$ 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 Ω , A , P $$\left (\Omega ,\mathcal {A},\mathcal {P}\right )$$ such that F 0(x) is the expectation of f(⋅, x): F 0 ( x ) = E f ( ω , x ) , x ∈ D , $$\displaystyle \begin{aligned} F_0(x) = Ef(\omega ,x), \: x \in D, \end{aligned} $$ see, e.g., A special case is given by f ( ω , x ) = F 0 ( x ) + n ( ω , x ) , $$\displaystyle \begin{aligned} f(\omega ,x) = F_0(x) + n(\omega ,x), \end{aligned} $$ where n(ω, x) is an additional zero-mean random noise term. Having a certain number m of independent sample functions f k ( x ) = f ( ω k , x ) , k ∈ M , $$\displaystyle \begin{aligned} f_k(x) = f(\omega _k,x), \: k \in M, \end{aligned} $$ of the random function in (18.2a), we may use the estimated objective function F ̂ ( x ) : = 1 m ∑ k ∈ M f k ( x ) . $$\displaystyle \begin{aligned} \hat {F}(x):= \frac {1}{m}\sum \limits _{k \in M}f_k(x). \end{aligned} $$

Date: 2020
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DOI: 10.1007/978-3-030-55662-4_18

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