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.

翻译：我们证明了在格拉斯曼流形 $\operatorname{Gr}(k,n)$ 上的无约束二次优化是NP难的。我们的结果涵盖了所有情形：(i) 当 $k$ 和 $n$ 均允许增长时；(ii) 当 $k$ 为任意固定值时；(iii) 当 $k$ 固定为其最小值 $1$ 时。由此我们进一步推导出在斯蒂弗尔流形 $\operatorname{V}(k,n)$ 和正交群 $\operatorname{O}(n)$ 上的无约束三次优化也是NP难的。作为补充，我们证明了在嘉当流形（即被视为黎曼流形的正定锥 $\mathbb{S}^n_{\scriptscriptstyle++}$，这是流形优化中另一个常见示例）上的无约束二次优化同样是NP难的。我们还将确立所有情况下均不存在 $\mathrm{FPTAS}$。