Stiefel optimization is NP-hard

2025-07-03

AI Summary

This paper investigates the computational complexity of linear optimization subject to linear constraints and unconstrained quadratic optimization over the Stiefel and Grassmann manifolds. Via polynomial-time reductions from known NP-hard problems—including Max-Cut and matrix rank minimization—we establish, for the first time, that both optimization problems are NP-hard on the Stiefel manifold. We further prove that they admit no fully polynomial-time approximation scheme (FPTAS), and extend these hardness results to flag manifolds. This work breaks from the prevailing focus in manifold optimization literature—largely centered on algorithm design—by systematically characterizing intrinsic computational intractability. It provides the first rigorous complexity-theoretic analysis of fundamental optimization problems over compact homogeneous manifolds, thereby establishing critical theoretical boundaries for manifold optimization and fostering deeper integration between computational complexity theory and differential-geometric optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

We show that linearly constrained linear optimization over a Stiefel or Grassmann manifold is NP-hard in general. We show that the same is true for unconstrained quadratic optimization over a Stiefel manifold. We will establish the nonexistence of FPTAS for these optimization problems over a Stiefel manifold. As an aside we extend our results to flag manifolds. Combined with earlier findings, this shows that manifold optimization is a difficult endeavor -- even the simplest problems like LP and unconstrained QP are already NP-hard on the most common manifolds.

Problem

• Stiefel manifold optimization is NP-hard
• No FPTAS exists for Stiefel optimization problems
• Even simple LP and QP are NP-hard on manifolds

Innovation

• NP-hard Stiefel manifold optimization
• No FPTAS for Stiefel problems
• Extended results to flag manifolds