An Interior Point–Inspired Algorithm for Linear Programs Arising in Discrete Optimal Transport

Discrete optimal transport problems give rise to very large linear programs (LPs) with a particular structure of the constraint matrix. In this paper, we present hybrid algorithm that mixes an interior point method (IPM) and column generation, specialized for LP originating from Kantorovich problem. Knowing solutions such display high degree sparsity, propose generation–like technique force all intermediate iterates be as sparse possible. The is implemented nearly matrix-free. Indeed, most computations avoid forming huge matrices involved solve Newton system using only much smaller Schur complement normal equations. We prove theoretical results about sparsity pattern solution, exploiting graph underlying use these mix iterative direct solvers efficiently in way avoids producing preconditioners or factorizations excessive fill-in at same time guaranteeing low number conjugate gradient iterations. compare proposed two state-of-the-art show it can compete best network optimization tools terms computational memory use. perform experiments reaching more than four billion variables demonstrate robustness method. History: Accepted by Antonio Frangioni, Area Editor Design & Analysis Algorithms–Continuous. Funding: F. Zanetti received funding University Edinburgh, form PhD scholarship. Supplemental Material: software supports findings study available within paper its Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0184 ) well IJOC GitHub repository https://github.com/INFORMSJoC/2022.0184 ). complete Software Data Repository https://informsjoc.github.io/ .