A Massively Parallel Interior-Point Method for Arrowhead Linear Programs with Local Linking Structure

Abstract. In practice, nonspecialized interior-point algorithms often cannot utilize the massively parallel compute resources offered by modern many- and multicore compute platforms. However, efficient distributed solution techniques are required, especially for large-scale linear programs. This article describes a new decomposition technique for systems of linear equations implemented in the parallel interior-point solver PIPS-IPM++. The algorithm exploits a matrix structure commonly found in optimization problems: a doubly bordered block-diagonal or arrowhead structure with linking constraints and variables often only linking few, consecutive blocks. This structure is preserved in the linear Karush-Kuhn-Tucker (KKT) systems solved during each iteration of the interior-point method. We present a hierarchical Schur complement decomposition that distributes and solves the linear optimization problem. It is designed for high-performance architectures and scales well with the availability of additional computing resources. The decomposition approach uses the border constraints’ locality to decouple the factorization process. Our approach is motivated by large-scale economic dispatch problems but can also be applied to other problem classes. We demonstrate the performance of our method on a set of mid- to large-scale instances, some of which have more than [Formula: see text] nonzeros in their constraint matrices.