Linear programming on the Stiefel manifold

Mengmeng Song; Yong Xia

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arXiv · 2301.06918

Linear programming on the Stiefel manifold

Mengmeng Song·Yong Xia

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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 $p+1\le n-k$.

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Mengmeng Song, Yong Xia. 2023-01-17. Linear programming on the Stiefel manifold. https://arxiv.org/abs/2301.06918

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