On applicability of control parametrization technique to solving distributed optimization problems

A. V. Chernov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 1 (2014), pp. 102-117

Résumé

We study approximating finite-dimensional mathematical programming problems arising from piecewise constant discretization of the control (in the framework of control parametrization technique) in the course of optimization of distributed parameter systems of a rather wide class. We establish the Lipschitz continuity for gradients of approximating problems. We present their formulas involving analytical solutions of an original controlled system and their adjoint one, thereby giving the opportunity for algorithmic separation of the optimization problem itself and the problem of solving a controlled system. Application of the approach under study to numerical optimization of distributed systems is illustrated by example of the Cauchy–Darboux system controlled by an integral criterion. We present the results of numerical solving the corresponding approximation problem in MatLab with the help of the program {\tt fmincon} and also an author-developed program based on the conditional gradient method. Moreover, the unconstrained minimization problem is investigated that arises from the constrained approximation problem with applying the sine parametrization method. We present the results of numerical solving this problem in MatLab with the help of the program {\tt fminunc} and also two author-developed programs based on the steepest descent and BFGS methods, respectively. The results of all numerical experiments are analyzed in detail.

Détail

Keywords: distributed parameter systems optimization, functional differentiation, piecewise constant approximation of control, control parametrization technique.