Grassmannian optimization is NP-hard

TL;DR

This paper proves that unconstrained quadratic and cubic optimization problems on various manifolds, including Grassmannian, Stiefel, orthogonal group, and positive definite cone, are NP-hard, indicating their computational intractability.

Contribution

It establishes the NP-hardness of several fundamental manifold optimization problems, covering all growth scenarios and fixed parameters, and shows no efficient approximation schemes exist.

Findings

NP-hardness of quadratic optimization over Grassmannian for all scenarios

NP-hardness of cubic optimization over Stiefel and orthogonal groups

No FPTAS exists for these manifold optimization problems

Abstract

We show that unconstrained quadratic optimization over a Grassmannian $\operatorname{Gr}(k,n)$ is NP-hard. Our results cover all scenarios: (i) when $k$ and $n$ are both allowed to grow; (ii) when $k$ is arbitrary but fixed; (iii) when $k$ is fixed at its lowest possible value $1$. We then deduce the NP-hardness of unconstrained cubic optimization over the Stiefel manifold $\operatorname{V}(k,n)$ and the orthogonal group $\operatorname{O}(n)$. As an addendum we demonstrate the NP-hardness of unconstrained quadratic optimization over the Cartan manifold, i.e., the positive definite cone $\mathbb{S}^n_{\scriptscriptstyle++}$ regarded as a Riemannian manifold, another popular example in manifold optimization. We will also establish the nonexistence of $\mathrm{FPTAS}$ in all cases.