Optimization methods on Riemannian manifolds with lower bound curvature: gradient for scalar and multi-objective functions and subgradient for scalar functions - Louzeiro, Maurício Silva (2019)
Let M a Riemannian manifolds with lower bounded curvature. In this thesis, we consider first-order iterative methods to solve optimization problems on M. The gradient method to solve the problem min{f(p) : p M}, where f : M → R is a continuously differentiable convex function is presented with Lipschitz step-size, adaptive step-size and Armijo’s step-size. The first procedure requires that the objective function has Lipschitz continuous gradient, which is not necessary for the other approaches. Convergence of the whole sequence to a minimizer, without any level set boundedness assumption, is proved. Iteration-complexity bound for functions with Lipschitz continuous gradient is also presented. In addition, all these approaches are considered in the multiobjective setting. Here we also consider the subgradient method to solve the problem min{f(p) : p M}, where f : M → R is a convex function. Iteration-complexity bounds of the subgradient method with exogenous step-size and Polyak’s step size are stablished, completing and improving recent results on the subject. Finally, some examples and numerical experiments are presented.