Linear programming on the Stiefel manifold

TL;DR

This paper introduces the first study of linear programming on the Stiefel manifold, revealing new exactness conditions for semidefinite relaxations and conditions under which local optimality implies global optimality.

Contribution

It extends classical results by strengthening the conditions for exact SDP relaxation and characterizes when local optima are globally optimal on the Stiefel manifold.

Findings

SDP relaxation is exact when p ≤ n-k, improving previous bounds.

Local optimality conditions are sufficient for global optimality under certain linear independence constraints.

The study covers both classical and new cases of linear programming on the Stiefel manifold.

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 $p$-tuples of orthonormal vectors in ${\mathbb R}^n$ 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 $p(p+1)/2\le n-k$, which is tight when $p=1$. Surprisingly, we can greatly strengthen this sufficient exactness condition to $p\le n-k$, which covers the classical case $p\le n$ 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 $p+1\le n-k$.