Archive of Issues
Russia Nizhni Novgorod
Section  Mathematics 
Title  On applicability of control parametrization technique to solving distributed optimization problems 
Author(s)  Chernov A.V.^{ab} 
Affiliations  Nizhni Novgorod State Technical University^{a}, Nizhni Novgorod State University^{b} 
Abstract  We study approximating finitedimensional 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 CauchyDarboux 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 fmincon and also an authordeveloped 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 fminunc and also two authordeveloped programs based on the steepest descent and BFGS methods, respectively. The results of all numerical experiments are analyzed in detail. 
Keywords  distributed parameter systems optimization, functional differentiation, piecewise constant approximation of control, control parametrization technique 
UDC  517.957, 517.988, 517.977.56 
MSC  47J05, 47J35, 47N10, 49M25, 49M37 
DOI  10.20537/vm140109 
Received  19 December 2013 
Language  Russian 
Citation  Chernov A.V. On applicability of control parametrization technique to solving distributed optimization problems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 102117. 
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