In this paper we propose a novel two-step linear optimization model to calculate energy-efficient timetables in metro railway networks. The resultant timetable minimizes the total energy consumed by all trains and maximizes the utilization of regenerative energy produced by braking trains, subject to the constraints in the railway network. In contrast to other existing models, which are N P-hard, our model is computationally the most tractable one being a linear program. We apply our optimization model to different instances of service PES2-SFM2 of line 8 of Shanghai Metro network spanning a full service period of one day (18 hours) with thousands of active trains. For every instance, our model finds an optimal timetable very quickly (largest runtime being less than 13s) with significant reduction in effective energy consumption (the worst case being 19.27%). Code based on the model has been integrated with Thales Timetable Compiler -the industrial timetable compiler of Thales Inc that has the largest installed base of communication-based train control systems worldwide.

In this paper, we study the railway timetabling problem to utilize regenerative braking energy produced by trains in a railway network. An electric train produces regenerative energy while braking, which is often lost in present technology. A positive overlapping time between braking and accelerating phases of a suitable train pair makes it possible to save electrical energy by transferring the regenerative energy of the braking train to the accelerating one. We propose a novel optimization model to determine a timetable that saves energy by maximizing the total overlapping time of all suitable train pairs. We apply our optimization model to different instances of a railway network for a time horizon spanning six hours. For each instance, our model finds an optimal or near-optimal timetable within an acceptable running time. We observe significant increase in the final overlapping time compared to the existing timetable for every instance, thus making it possible to save the associated electrical energy.

We propose a computer-assisted approach to the analysis of the worst-case convergence of nonlinear conjugate gradient methods (NCGMs). Those methods are known for their generally good empirical performances for large-scale optimization, while having relatively incomplete analyses. Using our computer-assisted approach, we establish novel complexity bounds for the Polak-Ribière-Polyak (PRP) and the Fletcher-Reeves (FR) NCGMs for smooth strongly convex minimization. Conversely, we provide examples showing that those methods might behave worse than the regular steepest descent on the same class of problems.

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