  In this paper, we address the bounded/unbounded determination of geodesically
convex optimization on Hadamard spaces. In Euclidean convex optimization, the
recession function is a basic tool to study the unboundedness, and provides the
domain of the Legendre-Fenchel conjugate of the objective function. In a
Hadamard space, the asymptotic slope function (Kapovich, Leeb, and Millson
2009), which is a function on the boundary at infinity, plays a role of the
recession function. We extend this notion by means of convex analysis and
optimization, and develop a convex analysis foundation for the unbounded
determination of geodesically convex optimization on Hadamard spaces,
particularly on symmetric spaces of nonpositive curvature. We explain how our
developed theory is applied to operator scaling and related optimization on
group orbits, which are our motivation.
