The Maximum Singularity Degree for Linear and Semidefinite Programming
arxiv(2024)
摘要
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.
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