An ordinal optimization approach to optimal control problems
The paper deals with a discrete stochastic optimization problem of the type: Find \[ \min_{\theta \in \Theta} J(\theta), \] where \( J(\theta) ={\mathbb E} [L (x(t;\theta, \xi))]\); \(L (x(t;\theta, \xi))\) represents some sample performance function evaluated through the realization of a system trajectory \(x((t;\theta, \xi)) \) under the design parameter \(\theta; \) \(\xi\) represents the random effects of the system. (The symbol \({\mathbb E} \) is reserved for mathematical expectation.) In particular, it is supposed that \(\Theta\) is a finite set with a huge number of elements. Furthermore, it is supposed that \(J(\theta)\) has little analytical structure but large uncertainty and must be estimated through repeated simulation of sample performances. Evidently, to solve (under these assumptions) the above mentioned type of problems is rather complicated and most of the time impossible. Consequently, the solving procedure is very often constructed to obtain (with high probability) an ``acceptable solution'' instead of the exact optimal one. The aim of the authors is to deal with ``ordinal optimization'' methods that just can guarantee the ``acceptable'' value of the objective function. They describe this approach in the case of a well-known (two-stage) Witsenhausen problem.
scientific article; zbMATH DE number 1325699
scientific article