Linear Programming on the Stiefel Manifold

SIAM Journal on Optimization, Volume 34, Issue 1, Page 718-741, March 2024.
Abstract. Linear programming on the Stiefel manifold (LPS) is studied for the first time. It aims at minimizing a linear objective function over the set of all [math]-tuples of orthonormal vectors in [math] satisfying [math] additional linear constraints. Despite the classical polynomial-time solvable case [math], general (LPS) is NP-hard. According to the Shapiro–Barvinok–Pataki theorem, (LPS) admits an exact semidefinite programming relaxation when [math], which is tight when [math]. Surprisingly, we can greatly strengthen this sufficient exactness condition to [math], which covers the classical case [math] and [math]. Regarding (LPS) as a smooth nonlinear programming problem, we reveal a nice property that under the linear independence constraint qualification, the standard first- and second-order local necessary optimality conditions are sufficient for global optimality when [math].