An ${\ell}_{0}$ -Norm Minimization for Energy-Efficient Timetabling in Subway Systems

In this paper, a sparse optimization model with <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula>-norm and the squared <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-norm as the objective function is proposed for energy-efficient timetabling in subway systems by means of improving the regenerative braking energy utilization. Optimality analysis is addressed for the proposed sparse optimization problem. Specifically, the local minimizer is shown to be a KKT point without any additional constraint qualification. Moreover, based on the hard-thresholding operator, we yield an explicit formula for the Lagrangian dual problem for the proposed non-convex discontinuous optimization problem and achieve the strong duality under the stationarity condition. To evaluate the effectiveness of our proposed model for the energy-efficient timetabling, we design a hard-thresholding based alternating direction method of multipliers for solving the proposed model. The case study on Beijing Metro Yizhuang Line is conducted and the comparison to some recent existing approaches illustrates the effectiveness of our model in terms of energy saving rate and the efficiency of our proposed algorithm in terms of computation time.