Optimization and Control (math.OC)

Abstract: Bilevel optimization is a comprehensive framework that bridges single- and multi-objective optimization. It encompasses many general formulations, including, but not limited to, standard nonlinear programs. This work demonstrates how elementary proximal gradient iterations can be used to solve a wide class of convex bilevel optimization problems without involving subroutines. Compared to and improving upon existing methods, ours (1) can handle a wider class of problems, including nonsmooth terms in the upper and lower level problems, (2) does not require strong convexity or global Lipschitz gradient continuity assumptions, and (3) provides a systematic adaptive stepsize selection strategy, allowing for the use of large stepsizes while being insensitive to the choice of parameters.

Abstract: With the aim to stimulate future research, we describe an exploratory study of a railway rescheduling problem. A widely used approach in practice and state of the art is to decompose these complex problems by geographical scope. Instead, we propose defining a core problem that restricts a rescheduling problem in response to a disturbance to only trains that need to be rescheduled, hence restricting the scope in both time and space. In this context, the difficulty resides in defining a scoper that can predict a subset of train services that will be affected by a given disturbance. We report preliminary results using the Flatland simulation environment that highlights the potential and challenges of this idea. We provide an extensible playground open-source implementation based on the Flatland railway environment and Answer-Set Programming.

Abstract: The proximal point algorithm (PPA) is the most widely recognized method for solving inclusion problems and serves as the foundation for many numerical algorithms. Despite this popularity, its convergence results have been largely limited to the monotone setting. In this work, we study the convergence of (relaxed) preconditioned PPA for a class of nonmonotone problems that satisfy an oblique weak Minty condition. Additionally, we study the (relaxed) Douglas-Rachford splitting (DRS) method in the nonmonotone setting by establishing a connection between DRS and the preconditioned PPA with a positive semidefinite preconditioner. To better characterize the class of problems covered by our analysis, we introduce the class of semimonotone operators, offering a natural extension to (hypo)monotone and co(hypo)monotone operators, and describe some of their properties. Sufficient conditions for global convergence of DRS involving the sum of two semimonotone operators are provided. Notably, it is shown that DRS converges even when the sum of the involved operators (or of their inverses) is nonmonotone. Various example problems are provided, demonstrating the tightness of our convergence results and highlighting the wide range of applications our theory is able to cover.