Compact Lifted Relaxations for Low-Rank Optimization

2026-03-05

AI Summary

This work addresses the challenge of lacking a general computable convex relaxation for quadratic matrix optimization problems with rank constraints by proposing a lifted semidefinite relaxation framework that does not rely on spectral structure assumptions. By analyzing block redundancies in moment matrices, the authors derive a compact equivalent formulation involving only two small-scale semidefinite constraints. They further design projection-based cutting planes that exploit rank inheritance under linear mappings to strengthen the low-rank constraint. This approach significantly enhances scalability and enables efficient solutions for problems such as matrix completion and reduced-rank regression, handling instances with dimensions up to \(n + m\).

Abstract

We develop tractable convex relaxations for rank-constrained quadratic optimization problems over $n \times m$ matrices, a setting for which tractable relaxations are typically only available when the objective or constraints admit spectral (permutation-invariant) structure. We derive lifted semidefinite relaxations that do not require such spectral terms. Although a direct lifting introduces a large semidefinite constraint in dimension $n^2 + nm + 1$, we prove that many blocks of moment matrix are redundant and derive an equivalent compact relaxation that only involves two semidefinite constraints of dimension $nm + 1$ and $n+m$ respectively. For matrix completion, basis pursuit, and reduced-rank regression problems, we exploit additional structure to obtain even more compact formulations involving semidefinite matrices of dimension at most $2\max(n,m)$. Overall, we obtain scalable semidefinite bounds for a broad class of low-rank quadratic problems.

Problem

• low-rank optimization
• rank-constrained quadratic optimization
• convex relaxation
• semidefinite programming
• matrix completion

Innovation

• lifted semidefinite relaxation
• low-rank optimization
• projection cuts
• compact formulation
• rank-constrained quadratic optimization