Grassmannian optimization is NP-hard
<|reference_start|>Grassmannian optimization is NP-hard: 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.<|reference_end|>
@article{lai2024grassmannian,
    title={Grassmannian optimization is NP-hard},
    author={Zehua Lai, Lek-Heng Lim, Ke Ye},
    journal={arXiv preprint arXiv:2406.19377},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.19377},
    primaryClass={math.OC cs.NA math.NA}
}