Low-rank matrix recovery is well-known to exhibit benign nonconvexity under the restricted isometry property (RIP): every second-order critical point is globally optimal, so local methods provably recover the ground truth. Motivated by the strong empirical performance of projected gradient methods for nonnegative low-rank recovery problems, we investigate whether this benign geometry persists when the factor matrices are constrained to be elementwise nonnegative. In the simple setting of a rank-1 nonnegative ground truth, we confirm that benign nonconvexity holds in the fully-observed case with RIP constant $$\delta =0$$ . This benign nonconvexity, however, is unstable. It fails to extend to the partially-observed case with any arbitrarily small RIP constant $$\delta >0$$ , and to higher-rank ground truths $$r^{\star }>1$$ , regardless of how much the search rank $$r\ge r^{\star }$$ is overparameterized. Together, these results undermine the standard stability-based explanation for the empirical success of nonconvex methods and suggest that fundamentally different tools are needed to analyze nonnegative low-rank recovery.

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 $$\textrm{P}=\textrm{NP}$$ , these optimization problems over a Stiefel manifold do not have $$\textrm{FPTAS}$$ . 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.

Abstract The quadratic linear ordering problem models a large number of applications in a variety of different domains. To solve it exactly, polyhedral methods have been proposed but their development is still in its beginning. Specifically, while it is evident that only a fraction of the triangle inequalities, which are most commonly known from the boolean quadric polytope, take part in a minimal linear description of the polytope associated with a canonical formulation of the quadratic linear ordering problem, it is broadly unclear to which of these inequalities this applies and how to distinguish them from the others. At the same time, these inequalities are essential to build strong polyhedral and semidefinite programming relaxations. Addressing these open questions and potentials, we reveal the desired combinatorial pattern that enables to identify the triangle inequalities which are facet-inducing and deduce a corresponding exact polynomial-time separation algorithm.

Abstract We consider two single-machine scheduling problems with the late work criterion, where each job’s processing time follows a decreasing convex function of the resource consumption amount, and each job can be interrupted and resumed later. The first objective is to minimize the sum of late work and resource consumption amount, while the second objective is to minimize the total late work with a constraint on the total resource consumption amount. We show that both problems can be solved in strongly polynomial time.