  In this paper, we study asymptotic behaviors of continuous-time and
discrete-time gradient flows of a ``lower-unbounded" convex function $f$ on a
Hadamard manifold $M$, particularly, their convergence properties to the
boundary $M^{\infty}$ at infinity of $M$. We establish a duality theorem that
the infimum of the gradient-norm $\|\nabla f(x)\|$ of $f$ over $M$ is equal to
the supremum of the negative of the recession function $f^{\infty}$ of $f$ over
the boundary $M^{\infty}$, provided the infimum is positive. Further, the
infimum and the supremum are obtained by the limits of the gradient flows of
$f$, Our results feature convex-optimization ingredients of the moment-weight
inequality for reductive group actions by Georgoulas, Robbin, and Salamon,and
are applied to noncommutative optimization by B\"urgisser et al. FOCS 2019. We
show that the gradient descent of the Kempf-Ness function for an unstable orbit
converges to a 1-parameter subgroup in the Hilbert-Mumford criterion, and the
associated moment-map sequence converges to the mimimum-norm point of the
moment polytope. We show further refinements for operator scaling -- the
left-right action on a matrix tuple $A= (A_1,A_2,\ldots,A_N)$. We characterize
the gradient-flow limit of operator scaling via a vector-space generalization
of the classical Dulmage-Mendelsohn decomposition of a bipartite graph. Also,
for a special case of $N = 2$, we reveal that this limit determines the
Kronecker canonical form of matrix pencils $s A_1+A_2$.
