The Maximum Singularity Degree for Linear and Semidefinite Programming
arXiv:2402.11795
Abstract
Facial reduction (FR) is an important tool in linear and semidefinite programming, providing both algorithmic and theoretical insights into these problems. The maximum length of an FR sequence for a convex set is referred to as the maximum singularity degree (MSD). The MSD gives a choice-robust worst-case bound on the number of nontrivial FR steps. It also yields a sufficient rank for a low-rank formulation of the SDP exposing-vector search, while upper bounds on the MSD can serve as proof devices for bounding the singularity degree in structured problem classes. These concrete roles motivate our study of its fundamental properties. In this work, we show that if an FR sequence has the longest length, then it satisfies a certain minimal property. For linear programming (LP), we prove that every minimal FR sequence forms a basis of a fixed vector space. This yields a direct characterization of the longest FR sequences. To study the MSD for semidefinite programming (SDP), we provide several useful tools including simplification and upper-bounding techniques. By leveraging these tools and the characterization for LP problems, we prove that finding a longest FR sequence for SDP problems is NP-hard. This complexity result highlights a striking difference between the shortest and the longest FR sequences for SDP problems.