A survey of methods of feasible directions for the solution of optimal control problems

Three methods of feasible directions for optimal control are reviewed. These methods are an extension of the Frank-Wolfe method, a dual method devised by Pironneau and Polack, and a Zontendijk method. The categories of continuous optimal control problems are shown as: (1) fixed time problems with fixed initial state, free terminal state, and simple constraints on the control; (2) fixed time problems with inequality constraints on both the initial and the terminal state and no control constraints; (3) free time problems with inequality constraints on the initial and terminal states and simple constraints on the control; and (4) fixed time problems with inequality state space contraints and constraints on the control. The nonlinear programming algorithms are derived for each of the methods in its associated category

On the removal of ill conditioning effects in the computation of optimal controls

Ill conditioning effects eliminated in nonlinear programming algorithms for optimal control

Exploring the potential for cross-nesting structures in airport-choice analysis: A case-study of the Greater London area

The analysis of air-passengers’ choices of departure airport in multi-airport regions is a crucial component of transportation planning in many large metropolitan areas, and has been the topic of an increasing number of studies over recent years. In this paper, we advance the state of the art of modelling in this area of research by making use of a Cross-Nested Logit (CNL) structure that allows for the joint representation of inter-alternative correlation along the three choice dimensions of airport, airline and access-mode. The analysis uses data collected in the Greater London area, which arguably has the highest levels of inter-airport competition of any multi-airport region; the authors of this paper are not aware of any previous effort to jointly analyse the choice of airport, airline and access-mode in this area. The results of the analysis reveal significant influences on passenger behaviour by access-time, access-cost, flight-frequency and flight-time. A structural comparison of the different models shows that the cross-nested structure offers significant improvements over simple Nested Logit (NL) models, which in turn outperform the Multinomial Logit (MNL) model used as the base model

The National Transport Data Framework

Report by Professor Peter Landshoff (Cambridge University) and Professor John Polak (Imperial College London) on a project for the Department for Transport. emails: [email protected] [email protected] NTDF is designed to be a resource for data owners to deposit descriptions into a central catalogue, so that people can search for data and find data and understand their characteristics. The value of this is to individuals, to commercial organizations, and to public bodies. For example, services that provide better information to travellers will help to make their journey less stressful and persuade them to make more use of public transport. Transport operators need very diverse information to help them plan developments to their services: demographic, geographical, economic etc. And policy makers need a similar range of information to help them decide how to divide their budget and afterwards to evaluate how valuable it has been.This work was supported by the Department for Transport (DfT)

Semidefinite programming bounds for Lee codes

For $q,n,d \in \mathbb{N}$, let $A_q^L(n,d)$ denote the maximum cardinality of a code $C \subseteq \mathbb{Z}_q^n$ with minimum Lee distance at least $d$, where $\mathbb{Z}_q$ denotes the cyclic group of order $q$. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on $A_q^L(n,d)$. The technique also yields an upper bound on the independent set number of the $n$-th strong product power of the circular graph $C_{d,q}$, which number is related to the Shannon capacity of $C_{d,q}$. Here $C_{d,q}$ is the graph with vertex set $\mathbb{Z}_q$, in which two vertices are adjacent if and only if their distance (mod $q$) is strictly less than $d$. The new bound does not seem to improve significantly over the bound obtained from Lov\'asz theta-function, except for very small $n$.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1703.0517

An alternative method to the scrambled Halton sequence for removing correlation between standard Halton sequences in high dimensions

Halton sequences were first introduced in the 1960s as an alternative to pseudo-random number sequences, with the aim of providing better coverage of the area of integration and negative correlation in the simulated probabilities between observations. This is needed in order to achieve variance reduction when using simulation to approximate an integral that does not have a closed-form expression. Such integrals arise in many areas of regional science, for example in the evaluation and estimation of certain types of discrete choice models. While the performance of standard Halton sequences is very good in low dimensions, problems with correlation have been observed between sequences generated from higher primes. This can cause serious problems in the estimation of models with high-dimensional integrals (e.g., models of aspects of spatial choice, such as route or location). Various methods have been proposed to deal with this; one of the most prominent solutions is the scrambled Halton sequence, which uses special predetermined permutations of the coefficients used in the construction of the standard sequence. In this paper, we conduct a detailed analysis of the ability of scrambled Halton sequences to remove the problematic correlation that exists between standard Halton sequences for high primes in the two-dimensional space. The analysis shows that although the scrambled sequences exhibit a lower degree of overall correlation than the standard sequences, for some choices of primes, correlation remains at an unacceptably high level. This paper then proposes an alternative method, based on the idea of using randomly shuffled versions of the one-dimensional standard Halton sequences in the construction of multi-dimensional sequences. We show that the new shuffled sequences produce a significantly higher reduction in correlation than the scrambled sequences, without loss of quality of coverage. Another substantial advantage of this new method is that it can, without any modifications, be used for any number of dimensions, while the use of the scrambled sequences requires the a-priori computation of a matrix of permutations, which for high dimensional problems could lead to significant runtime disadvantages. Repeated runs of the shuffling algorithm will also produce different sequences in different runs, which nevertheless maintain the same quality of one-dimensional coverage. This is not at all the case for the scrambled sequences. In view of the clear advantages in its ability to remove correlation, combined with its runtime and generalization advantages, this paper recommends that this new algorithm should be preferred to the scrambled Halton sequences when dealing with high correlation between standard Halton sequences.

Adaptive Horizon Model Predictive Control and Al'brekht's Method

A standard way of finding a feedback law that stabilizes a control system to an operating point is to recast the problem as an infinite horizon optimal control problem. If the optimal cost and the optmal feedback can be found on a large domain around the operating point then a Lyapunov argument can be used to verify the asymptotic stability of the closed loop dynamics. The problem with this approach is that is usually very difficult to find the optimal cost and the optmal feedback on a large domain for nonlinear problems with or without constraints. Hence the increasing interest in Model Predictive Control (MPC). In standard MPC a finite horizon optimal control problem is solved in real time but just at the current state, the first control action is implimented, the system evolves one time step and the process is repeated. A terminal cost and terminal feedback found by Al'brekht's methoddefined in a neighborhood of the operating point is used to shorten the horizon and thereby make the nonlinear programs easier to solve because they have less decision variables. Adaptive Horizon Model Predictive Control (AHMPC) is a scheme for varying the horizon length of Model Predictive Control (MPC) as needed. Its goal is to achieve stabilization with horizons as small as possible so that MPC methods can be used on faster and/or more complicated dynamic processes.Comment: arXiv admin note: text overlap with arXiv:1602.0861

Mean-Dispersion Preferences and Constant Absolute Uncertainty Aversion

We axiomatize, in an Anscombe-Aumann framework, the class of preferences that admit a representation of the form V(f) = mu - rho(d), where mu is the mean utility of the act f with respect to a given probability, d is the vector of state-by-state utility deviations from the mean, and rho(d) is a measure of (aversion to) dispersion that corresponds to an uncertainty premium. The key feature of these mean-dispersion preferences is that they exhibit constant absolute uncertainty aversion. This class includes many well-known models of preferences from the literature on ambiguity. We show what properties of the dispersion function rho(dot) correspond to known models, to probabilistic sophistication, and to some new notions of uncertainty aversion.Ambiguity aversion, Translation invariance, Dispersion, Uncertainty, Probabilistic sophistication