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

##### Abstract

In this paper we discuss Pantoja's construction of the Newton direction for discrete time optimal control problems. We show that automatic differentiation (AD) techniques can be used to calculate the Newton direction accurately, without requiring extensive re-writing of user code, and at a surprisingly low computational cost: for an N-step problem with p control variables and q state variables at each step, the worst case cost is 6(p + q + 1) times the computational cost of a single target function evaluation, independent of N, together with at most p3/3 + p2(q + 1) + 2p(q + 1)2 + (q + l)3, i.e. less than (p + q + l)3, floating point multiply-and-add operations per time step. These costs may be considerably reduced if there is significant structural sparsity in the problem dynamics. The systematic use of checkpointing roughly doubles the operation counts, but reduces the total space cost to the order of 4pN floating point stores. A naive approach to finding the Newton step would require the solution of an Np Np system of equations together with a number of function evaluations proportional to Np, so this approach to Pantoja's construction is extremely attractive, especially if q is very small relative to N. Straightforward modifications of the AD algorithms proposed here can be used to implement other discrete time optimal control solution techniques, such as differential dynamic programming (DDP), which use state-control feedback. The same techniques also can be used to determine with certainty, at the cost of a single Newton direction calculation, whether or not the Hessian of the target function is sufficiently positive definite at a point of interest. This allows computationally cheap post-hoc verification that a second-order minimum has been reached to a given accuracy, regardless of what method has been used to obtain it.