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More than 40 years ago, Lions and Mercier introduced in a seminal paper the Douglas–Rachford algorithm. Today, this method is well-recognized as a classic and highly successful splitting method to find minimizers of the sum of two (not necessarily smooth) convex functions. Whereas the underlying theory has matured, one case remains a mystery: the behavior of the shadow sequence when the given functions have disjoint domains. Building on previous work, we establish for the first time weak and value convergence of the shadow sequence generated by the Douglas–Rachford algorithm in a setting of unprecedented generality. The weak limit point is shown to solve the associated normal problem, which is a minimal perturbation of the original optimization problem. We also present new results on the geometry of the minimal displacement vector. Funding: The research of H. H. Bauschke and W. M. Moursi was partially supported by Discovery Grants of the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-03703 and RGPIN-2019-04803], respectively. [ABSTRACT FROM AUTHOR]

This paper examines the problems of routing freight traffic, scheduling train services and allocating classification work between yards on a rail network. It describes a general optimization model which takes into account the interactions between these activities and develops global strategies for their management over a medium term planning horizon. A heuristic algorithm developed to solve this nonlinear, mixed-integer multicommodity flow problem is also presented with some computational results based on an example at the Canadian National Railroads. [ABSTRACT FROM AUTHOR]