Stiefel optimization is NP-hard

TL;DR

This paper proves that optimization problems over Stiefel and Grassmann manifolds are NP-hard, indicating that even simple manifold optimization tasks are computationally intractable in the worst case.

Contribution

It establishes the NP-hardness of linearly constrained and unconstrained quadratic optimization over Stiefel and Grassmann manifolds, extending to flag manifolds, highlighting the computational difficulty of manifold optimization.

Findings

Linearly constrained optimization over Stiefel/Grassmann manifolds is NP-hard.

Unconstrained quadratic optimization over Stiefel manifolds is NP-hard.

Manifold optimization problems are computationally intractable in general.

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 show that unless $\mathrm{P}=\mathrm{NP}$, these optimization problems over a Stiefel manifold do not have $\mathrm{FPTAS}$. 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.