The Maximum Singularity Degree for Linear and Semidefinite Programming

The singularity degree plays a crucial role in understanding linear and semidefinite programming, providing a theoretical framework for analyzing these problems. It is defined as the minimum number of facial reduction (FR) steps needed to reach strict feasibility for a convex set. On the other hand, the maximum singularity degree (MSD) is the maximum number of steps required. Recent progress in the applications of MSD has motivated us to explore its fundamental properties in this paper. For semidefinite programming, we establish a necessary condition for an FR sequence to be the longest. Additionally, we propose an upper bound for MSD, which can be computed more easily. By leveraging these findings, we prove that computing MSD is NP-hard. This complexity result complements the existing algorithms for computing the singularity degree found in the literature. For linear programming, we provide a characterization for the longest FR sequences, which also serves as a polynomial-time algorithm for constructing such a sequence. In addition, we introduce two operations that ensure the longest FR sequences remain the longest. Lastly, we prove that MSD is equivalent to a novel parameter called the implicit problem singularity.