Convex analysis and optimisation on hyperbolic space forms

Zhu, Jinzhen (2025). Convex analysis and optimisation on hyperbolic space forms. University of Birmingham. Ph.D.

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Abstract

This thesis is based on two published paper and one submitted paper in collaboration with co-authors Ferreira, O.P. and Németh, S.Z.. Some concepts of convex analysis on hyperbolic spaces are firstly studied. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first and second order characterizations of these functions. An extensive study of the hyperbolically convex quadratic functions is also presented. Unlike in the Euclidean space, the study of intrinsic convexity of non-homogeneous quadratic functions in the hyperbolic space is more elaborate than that of homogeneous quadratic functions. Then, we examine the gradient projection method as a solution approach for constrained optimization problems in \(\kappa\)-hyperbolic space forms. 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. Finally, we explore the properties of the constrained Fermat-Weber problem, demonstrating that the sequence generated by the gradient projection method converges to its unique solution.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):