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
p-tuples of orthonormal vectors in Rn satisfying k additional
linear constraints. Despite the classical polynomial-time solvable case k=0,
general (LPS) is NP-hard. According to the Shapiro-Barvinok-Pataki theorem,
(LPS) admits an exact semidefinite programming (SDP) relaxation when
p(p+1)/2β€nβk, which is tight when p=1. Surprisingly, we can greatly
strengthen this sufficient exactness condition to pβ€nβk, which covers the
classical case pβ€n and k=0. 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
{\it local} necessary optimality conditions are sufficient for {\it global}
optimality when p+1β€nβk