  This paper is concerned with a class of optimization problems with the
nonnegative orthogonal constraint, in which the objective function is
$L$-smooth on an open set containing the Stiefel manifold ${\rm St}(n,r)$. We
derive a locally Lipschitzian error bound for the feasible points without zero
rows when $n>r>1$, and when $n>r=1$ or $n=r$ achieve a global Lipschitzian
error bound. Then, we show that the penalty problem induced by the elementwise
$\ell_1$-norm distance to the nonnegative cone is a global exact penalty, and
so is the one induced by its Moreau envelope under a lower second-order
calmness of the objective function. A practical penalty algorithm is developed
by solving approximately a series of smooth penalty problems with a
retraction-based nonmonotone line-search proximal gradient method, and any
cluster point of the generated sequence is shown to be a stationary point of
the original problem. Numerical comparisons with the ALM \citep{Wen13} and the
exact penalty method \citep{JiangM22} indicate that our penalty method has an
advantage in terms of the quality of solutions despite taking a little more
time.
