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 et al. in J Differ Geom 81:297–354, 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.

在本文中，我们解决了 Hadamard 空间上测地凸优化的有界/无界确定。在欧几里德凸优化中，衰退函数是研究无界性的基本工具，并提供目标函数的勒让德-芬切尔共轭域。在 Hadamard 空间中，渐近斜率函数（Kapovich et al. in J Differ Geom 81:297–354, 2009）是无穷远边界上的函数，起着衰退函数的作用。我们通过凸分析和优化扩展了这一概念，并为哈达玛空间上的测地凸优化的无界确定建立了凸分析基础，特别是在非正曲率的对称空间上。我们解释了我们开发的理论如何应用于群轨道上的算子缩放和相关优化，这是我们的动机。