The Maximum Singularity Degree for Linear and Semidefinite Programming

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). We observe that the behavior of certain FR algorithms can be explained through the MSD. Combined with recent applications of the MSD in the literature, this motivates our study of its fundamental properties in this paper. In this work, we show that an FR sequence has the longest length implies that it satisfies a certain minimal property. For linear programming (LP), we introduce two operations for manipulating the longest FR sequences. These operations enable us to characterize the longest FR sequences for LP problems. 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.