Quasi Newton methods and unconstrained optimal control problems

Abstract

Quasi Newton methods play an important role in the numerical solution of problems in unconstrained optimization. Optimal control problems in their discretized form can be viewed as optimization problems and therefore be solved by quasi Newton methods. Since the discretized problems do not solve the original infinite-dimensional control problem but rather approximate it up to a certain accuracy, various approximations of the control problem need to be considered. It is known that an increase in the dimension of optimization problems can have a negative effect on the convergence rate of the quasi Newton method which is used to solve the problem. We want to investigate this behavior and to explain how this drawback can be avoided for a class of optimal control problems. We show how to use the infinite dimensional original problem to predict the speed of convergence of the BFGS-method [1, 7, 10, 22] for the finite-dimensional approximations. In several papers [6, 14, 24, 27] the DFP-method [4, 8] and its application to optimal control problems were considered but rates of convergence were given at best for quadratic problems. In [25, 26] a linear rate of convergence was proved in Hilbert spaces and applied to optimal control. All the applications to optimal control problems were carried out for finite dimensional approximations. This fact is important, because in [23] it was shown, that contrary to the finite dimensional case [2], the BFGS-method can converge very slowly when applied to an infinite dimensional problem. Hence it is desirable to know whether this convergence behavior can occur also for fine discretizations of control problems. Sufficient ([19]) and characteristic ([12]) conditions for the superlinear rate were given in other analyses. Like in the linear case for Broyden's method [28] and the conjugate gradient method [3], [9] an additional assumption on the initial approximation of the Hessian, i.e. it approximates the true Hessian up to a compact operator, is needed to guarantee superlinear convergence, see [11]. In [9] a connection to quadratic control problems is shown. Here we want to consider nonlinear control problems and their discretization.