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.

翻译：我们证明，在一般情况下，Stiefel流形或Grassmann流形上的线性约束线性优化是NP难的。我们还证明，Stiefel流形上的无约束二次优化同样具有NP难度。我们将证明，除非$\mathrm{P}=\mathrm{NP}$，否则这些Stiefel流形上的优化问题不存在$\mathrm{FPTAS}$。作为补充，我们将结果推广到旗流形。结合先前的研究成果，这表明流形优化是一项困难的任务——即使在最常用的流形上，像线性规划（LP）和无约束二次规划（QP）这样最简单的问题也已经是NP难的。