For the intersection of the Stiefel manifold and the set of nonnegative matrices in $\mathbb{R}^{n\times r}$, we present global and local error bounds with easily computable residual functions and explicit coefficients. Moreover, we show that the error bounds cannot be improved except for the coefficients, which explains why two square-root terms are necessary in the bounds when $1 < r < n$ for the nonnegativity and orthogonality, respectively. The error bounds are applied to penalty methods for minimizing a Lipschitz continuous function with nonnegative orthogonality constraints. Under only the Lipschitz continuity of the objective function, we prove the exactness of penalty problems that penalize the nonnegativity constraint, or the orthogonality constraint, or both constraints. Our results cover both global and local minimizers.

翻译：对于Stiefel 元件和一套非负矩阵的交叉点, $\mathbb{R ⁇ n\times r}$, 我们用容易计算剩余函数和明确系数来显示全球和局部误差界限。 此外, 我们显示, 错误界限除系数外是无法改进的, 这解释了为什么在界限上需要两个正方根术语, 分别用于非惯性和非正向性 $ 1 < r < n$ 。 错误界限适用于以非正向性或异向性限制来尽量减少利普西茨连续函数的处罚方法。 在目标功能的利普西茨连续性下, 我们只证明惩罚非惯性限制、 或正向性制约或两种制约的处罚问题的确切性。 我们的结果涵盖全球和地方最小化者 。