Tight Error Bounds for the Sign-Constrained Stiefel Manifold

TL;DR

This paper establishes tight, explicit error bounds for matrices on the sign-constrained Stiefel manifold, explaining the necessity of certain terms and norms, and applies these bounds to develop exact penalty methods for constrained optimization.

Contribution

It provides the first comprehensive set of global and local error bounds for the sign-constrained Stiefel manifold with explicit residual functions and coefficients, including their optimality.

Findings

Error bounds include necessary square-root terms for certain dimensions.

The bounds are tight and cannot be improved under mild conditions.

Application to exact penalty methods for constrained optimization.

Abstract

The sign-constrained Stiefel manifold in $\mathbb{R}^{n\times r}$ is a segment of the Stiefel manifold with fixed signs (nonnegative or nonpositive) for some columns of the matrices. It includes the nonnegative Stiefel manifold as a special case. We present global and local error bounds that provide an inequality with easily computable residual functions and explicit coefficients to bound the distance from matrices in $\mathbb{R}^{n\times r}$ to the sign-constrained Stiefel manifold. Moreover, we show that the error bounds cannot be improved except for the multiplicative constants under some mild conditions, which explains why two square-root terms are necessary in the bounds when $1< r <n$ and why the $\ell_1$ norm can be used in the bounds when $r = n$ or $r = 1$ for the sign constraints and orthogonality, respectively. The error bounds are applied to derive exact penalty methods for minimizing a Lipschitz continuous function with orthogonality and sign constraints.