Golden-Ratio Resolvent Algorithm for Semiconvex Optimization in CAT(1) Spaces

Abstract

We introduce a Golden-Ratio Resolvent Algorithm (GRRA) for minimizing proper lower semicontinuous semiconvex functions on admissible regions of complete CAT(1) spaces. The method first evaluates the Moreau-Yosida resolvent and then performs a geodesic relaxation with the fixed parameter given by the reciprocal of the golden ratio. To avoid using a curvature estimate outside its valid range, the analysis is formulated on a common geodesically convex region on which a uniform local strong-convexity inequality for the squared distance is explicitly assumed. For semiconvex objectives, we do not identify resolvent fixed points with global minimizers; instead, the minimizer set is treated directly and the quasi-nonexpansiveness and \(\Delta\)-demiclosedness properties needed for global convergence are stated as explicit assumptions. Under these hypotheses, we establish a Fej\({e}\'\)r-type descent inequality, residual summability, asymptotic regularity, and \(\Delta\)-convergence to a minimizer. Under an additional contractivity assumption on the resolvent, strong convergence with an explicit linear rate follows. We further study stability under Mosco perturbations, including convergence of every fixed finite GRRA trajectory and a quantitative perturbation estimate under a local Lipschitz condition for geodesic interpolation. A spherical quadratic example gives an exact contraction factor, illustrates Mosco stability, and compares the golden-ratio relaxation with two fixed-parameter Krasnosel'ski\(\breve{i}\)-Mann-type resolvent relaxations. The comparison is intended as an illustration and does not assert universal optimality of the golden-ratio parameter.