This book examines application and methods to incorporating stochastic
parameter variations into the optimization process to decrease expense
in corrective measures. Basic types of deterministic substitute problems
occurring mostly in practice involve i) minimization of the expected
primary costs subject to expected recourse cost constraints (reliability
constraints) and remaining deterministic constraints, e.g. box
constraints, as well as ii) minimization of the expected total costs
(costs of construction, design, recourse costs, etc.) subject to the
remaining deterministic constraints.
After an introduction into the theory of dynamic control systems with
random parameters, the major control laws are described, as open-loop
control, closed-loop, feedback control and open-loop feedback control,
used for iterative construction of feedback controls. For approximate
solution of optimization and control problems with random parameters and
involving expected cost/loss-type objective, constraint functions,
Taylor expansion procedures, and Homotopy methods are considered,
Examples and applications to stochastic optimization of regulators are
given. Moreover, for reliability-based analysis and optimal design
problems, corresponding optimization-based limit state functions are
constructed. Because of the complexity of concrete optimization/control
problems and their lack of the mathematical regularity as required of
Mathematical Programming (MP) techniques, other optimization techniques,
like random search methods (RSM) became increasingly important.
Basic results on the convergence and convergence rates of random search
methods are presented. Moreover, for the improvement of the - sometimes
very low - convergence rate of RSM, search methods based on optimal
stochastic decision processes are presented. In order to improve the
convergence behavior of RSM, the random search procedure is embedded
into a stochastic decision process for an optimal control of the
probability distributions of the search variates (mutation random
variables).
