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2025. On projection mappings and the gradient projection method on hyperbolic space forms

Invited abstract in session TB-41: Optimization on Manifolds, stream Optimization on Geodesic Metric Spaces: Smooth and Nonsmooth.

Tuesday, 10:30-12:00
Room: 97 (building: 306)

Authors (first author is the speaker)

1. Sandor Zoltan Nemeth
School of Mathematics, University of Birmingham
2. Orizon P Ferreira
IME-Instituto de Matemática e Estatística, Universidade Federal de Goiás
3. Jinzhen Zhu
School of Mathematics, University of Birmingham

Abstract

This talk examines the gradient projection method as a solution approach for constrained optimization problems in kappa-hyperbolic space forms, particularly for potentially non-convex objective functions. We consider both constant and backtracking step sizes in our analysis. Our studies are based on the hyperboloid model, commonly referred to as the Lorentz model. In our investigation, we present several innovative properties of the intrinsic kappa-projection into convex sets of kappa-hyperbolic space forms. These properties are crucial for analyzing the method and also hold independent significance. We discuss the relationship between the intrinsic kappa-projection and the Euclidean orthogonal projection as well as the Lorentz projection. Moreover, we provide formulas for the intrinsic kappa-projection into specific convex sets using the Euclidean orthogonal projection and the Lorentz projection. Regarding the convergence results of the gradient projection method, we establish two main findings. Firstly, we demonstrate that every accumulation point of the sequence generated by the method with backtracking step sizes is a stationary point for the given problem. Secondly, assuming the Lipschitz continuity of the gradient of the objective function, we show that each accumulation point of the sequence generated by the gradient projection method with a constant step size is also a stationary point. Additionally, we provide an iteration complexity bound.

Keywords

Status: accepted


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