# Cheap Newton steps for discrete time optimal control problems: automatic differentiation and Pantoja's algorithm

20 January 2000
14:00
Prof Bruce Christianson
Abstract
In 1983 Pantoja described a stagewise construction of the exact Newton direction for a discrete time optimal control problem. His algorithm requires the solution of linear equations with coefficients given by recurrences involving second derivatives, for which accurate values are therefore required. \\ \\ Automatic differentiation is a set of techniques for obtaining derivatives of functions which are calculated by a program, including loops and subroutine calls, by transforming the text of the program. \\ \\ In this talk we show how automatic differentiation can be used to evaluate exactly the quantities required by Pantoja's algorithm, thus avoiding the labour of forming and differentiating adjoint equations by hand. \\ \\ The cost of calculating the newton direction amounts to the cost of solving one set of linear equations, of the order of the number of control variables, for each time step. The working storage cost can be made smaller than that required to hold the solution.
• Computational Mathematics and Applications Seminar