Grassmannian optimization is NP-hard
Zehua Lai, Lek-Heng Lim, Ke Ye
TL;DR
This work proves that unconstrained quadratic optimization over the Grassmannian $\mathrm{Gr}(k,n)$ is NP-hard in all regimes, including when both dimensions grow, when $k$ is fixed, and in the base case $k=1$. It extends the hardness to unconstrained cubic problems on the Stiefel manifold $\mathrm{V}(k,n)$ and the orthogonal group $\mathrm{O}(n)$, and to the Cartan manifold $\mathbb{S}^n_{++}$, establishing the nonexistence of a $\mathrm{FPTAS}$ in all cases. The authors develop a framework based on matrix-model representations, polynomial definitions on submanifolds and quotients, and polynomial-time diffeomorphisms between models to transfer hardness across representations. A key set of results connects combinatorial problems (e.g., clique) to Grassmannian optimization via Motzkin–Straus-type formulations, and leverages Nesterov’s cubic constructions to prove hardness for fixed dimensions. Overall, the paper delineates fundamental limits for global optimization on common manifold models and highlights the reliance on local optimization methods in practical manifold optimization tasks.
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.