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

Original article can be found at: http://www.informaworld.com/smpp/title~content=t713645924~db=all Copyright Taylor and Francis / Informa.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.Peer reviewe

An empirical study of the variability in the composition of British freight trains

As part of the broader sustainability and economic efficiency agenda, European transport policy places considerable emphasis on improving rail’s competitiveness to increase its share of the freight market. Much attention is devoted to infrastructure characteristics which determine the number of freight trains which can operate and influence the operating characteristics of these trains. However, little attention has been devoted to the composition of the freight trains themselves, with scant published data relating to the practicalities of this important component of system utilisation and its impacts on rail freight viability and sustainability. This paper develops a better understanding of the extent to which freight train composition varies, through a large-scale empirical study of the composition of British freight trains. The investigation is based on a survey of almost 3,000 individual freight trains, with analysis at four levels of disaggregation, from the commodity groupings used in official statistics down to individual services. This provides considerable insight into rail freight operations with particular relevance to the efficiency of utilisation of trains using the available network paths. The results demonstrate the limitations of generalising about freight train formations since, within certain commodity groupings, considerable variability was identified even at fairly high levels of disaggregation

The restructuring and privatisation of British Rail: Was it really that bad?

This paper uses a social cost-benefit analysis (SCBA) framework to assess whether rail privatisation in Britain has produced savings in operating costs. The paper shows that major efficiencies have been achieved, consumers have benefited through lower prices, whilst the increased government subsidy has been largely recouped through privatisation proceeds. We also find that output quality is no lower (and is probably better) than under the counterfactual scenario of public ownership (pre-Hatfield). The achievement of further savings is key to delivering improved rail services in the future. This paper finds that a privatised structure, where shareholders demand a return on their investment, has led to significant improvements in operating efficiency - it remains to be seen whether the new regime, with a not-for-profit infrastructure owner, will deliver the same efficiency improvements