Convex analysis on Hadamard spaces and scaling problems

Understand

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.