Stiefel optimization is NP-hard
Zehua Lai, Lek-Heng Lim, Tianyun Tang
TL;DR
The paper proves that linearly constrained LP and unconstrained QP over the Stiefel, Grassmann, and flag manifolds are NP-hard, and that no $\mathrm{FPTAS}$ exists unless $\mathrm{P}=\mathrm{NP}$, with extensions to Cartan (ellipsoids). The authors leverage reductions from graph problems—stability number, max-cut, and clique number—across multiple manifold models, and establish precise objective correspondences (e.g., $\max_{X\in V(k,n)} \mathrm{diag}(X)^{\top}(I_k-A)\mathrm{diag}(X) = 4\kappa(G_k)-2|E|+k$). These results demonstrate that even the simplest optimization tasks on common manifolds are intractable in general, underscoring the need for convex relaxations or approximate frameworks in manifold optimization. The work also clarifies that unconstrained LP over these manifolds can be solved in closed form, highlighting a sharp boundary between tractable and intractable problems on manifold domains.
Abstract
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.