Proximal Point Algorithms on Hadamard Manifolds: Linear Convergence and Finite Termination

Abstract

Abstract. In the present paper, we consider inexact proximal point algorithms for finding singular points of multivalued vector fields on Hadamard manifolds. The rate of convergence is shown to be linear under the mild assumption of metric subregularity. Furthermore, if the sequence of parameters associated with the iterative scheme converges to 0, then the convergence rate is superlinear. At the same time, the finite termination of the inexact proximal point algorithm is also provided under a weak sharp minima-like condition. Applications to optimization problems are provided. Some of our results are new even in Euclidean spaces, while others improve and/or extend some known results in Euclidean spaces. As a matter of fact, in the case of exact proximal point algorithm, our results improve the corresponding results in [G. C. Bento and J. X. Cruz Neto, Optim., 63 (2014), pp. 1281-1288. Finally, several examples are provided to illustrate that our results are applicable while the corresponding results in the Hilbert space setting are not.

An inexact proximal point method for variational inequality on Hadamard manifolds

“…Note that for θ k = 0 in (7), Algorithm 1 merges into algorithm introduced in [5]. The error criterion (7) was introduced in the celebrated paper [38] to analyze an inexact version of the proximal point method to find zeroes of maximal monotone operators in linear context; see [46,47] for a generalization to Riemannian setting.…”

“…Remark. In [46,47] is presented an inexact version of the proximal point method for to find singularity of a vector field on Hadamard manifolds. These papers differs from the present paper in two ways, namely, [46,47] use only absolute summable error criteria and the enlargement X ǫ of X was not considered.…”

“…Extensions of concepts and techniques of optimization from the Euclidean space to the Riemannian context have been a subject of intense research in recent years. An special attention has been given to methods of Riemannian mathematical programming; papers published on this topic involving proximal point methods include, but are not limited to, [2,3,6,7,29,35,42,46,47]. It is well known that one of the reasons for this extension is the possibility of transforming nonconvex or non-monotone problems in the Euclidean context into Riemannian convex or monotone problems, by introducing a suitable metric, which enables modified numerical methods to find solutions for these problems; see [7,8,12,15,20,37].…”

“…For instance, it generalize the algorithm studied in [38] to Riemannian setting, by considering two of the four errors considered there. Considering that the Riemannian algorithm studied in [46,47] does not use the enlargement of the vector field in consideration, then in this sense our algorithm has it as particular instance. Moreover, our algorithm also merges into algorithms studied in [5,42].…”

A Riemannian derivative-free Polak–Ribiére–Polyak method for tangent vector field

“…Thus, S m ++ is a Hadamard manifold manifold of nonpositive curvature everywhere [22,36]. The dimension of S m ++ is equal to m(m + 1)/2 [21, Proposition 2.1].…”