The Maximum Singularity Degree for Linear and Semidefinite Programming
Hao Hu
TL;DR
This work investigates the maximum singularity degree (MSD) of facial reduction sequences for linear and semidefinite programs. It develops LP-specific tools—Swapping and Removal—and proves that a longest FR sequence for LP is necessarily minimal, enabling a complete characterization of the longest sequences. For SDP, it shows that the analogous problem is computationally hard (NP-hard) and demonstrates that minimal FR sequences do not guarantee maximal length, highlighting a fundamental difference from LP. The results illuminate the theoretical and algorithmic implications of using FRA in SDP and LP, including simplification and rank-based perspectives like the Burer–Monteiro method. Overall, the paper advances understanding of FR sequence structure and complexity, with potential impact on solver design and complexity theory in convex optimization.
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.