Tight Error Bounds for the Sign-Constrained Stiefel Manifold

The sign-constrained Stiefel manifold in ℝn× 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 ℝn× 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 ℓ1 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.