Subgradient algorithms on Riemannian manifolds of lower bounded curvatures

Iteration-complexity of the subgradient method on Riemannian manifolds with lower bounded curvature

“…Our results increase the range of applicability of the method compared to the respective results obtained in [2,31,32]. Moreover, in the asymptotic analysis with exogenous step-size, we do not assume that the solution set is nonempty, completing the result of [28,Theorem 3.1]. It should be noted that our analysis use a recently inequality stablished in [27,29].…”

“…The step-size in Strategy 1 have been used in several paper for analyzing subgradient method; see, for example, [6,12,28].…”

“…We remark that in the first part of the next theorem we do not assume that Ω * = ∅. Additionally, It is worth to point out that the second part was first obtained in [28]. Since it is an immediate consequence of the first part and Lemma 3, we decide to include its proof here.…”

Convergence Analysis of Gradient Algorithms on Riemannian Manifolds Without Curvature Constraints and Application to Riemannian Mass

“…We show in the following lemma some inequalities, which play important roles in our study. For this purpose, we define the function : [0, +∞) → R as in [31] by…”

Subgradient method with feasible inexact projections for constrained convex optimization problems

“…By using this inequality, we establish asymptotic convergence results and iteration-complexity bounds for the sequence generated by our method employing the well known exogenous stepsizes, Polyak's stepsizes, and dynamic stepsizes. We point out that these stepsizes have been discussed extensively in the related literature, including [1,16,24,25,26,36], where many of our results were inspired. Let us describe the results in the present and their relationship with the literature on the subject.…”