On the Convergence Rate of a Proximal Point Algorithm for Vector Function on Hadamard Manifolds
- PDF / 450,730 Bytes
- 13 Pages / 439.37 x 666.142 pts Page_size
- 42 Downloads / 239 Views
On the Convergence Rate of a Proximal Point Algorithm for Vector Function on Hadamard Manifolds Feng-Mei Tang1 · Ping-Liang Huang1
Received: 17 April 2016 / Revised: 11 October 2016 / Accepted: 2 December 2016 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2017
Abstract The proximal point algorithm has many interesting applications, such as signal recovery, signal processing and others. In recent years, the proximal point method has been extended to Riemannian manifolds. The main advantages of these extensions are that nonconvex problems in classic sense may become geodesic convex by introducing an appropriate Riemannian metric, constrained optimization problems may be seen as unconstrained ones. In this paper, we propose an inexact proximal point algorithm for geodesic convex vector function on Hadamard manifolds. Under the assumption that the objective function is coercive, the sequence generated by this algorithm converges to a Pareto critical point. When the objective function is coercive and strictly geodesic convex, the sequence generated by this algorithm converges to a Pareto optimal point. Furthermore, under the weaker growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate. Keywords Inexact proximal point algorithm · Hadamard manifolds · Convergence rate · Pareto critical point · Pareto optimal point Mathematics Subject Classification 26B25 · 65K05 · 53C21
B
Feng-Mei Tang [email protected] Ping-Liang Huang [email protected]
1
College of Science, Shanghai University, Shanghai 200444, China
123
F.-M. Tang, P.-L. Huang
1 Introduction The proximal algorithm has been used in signal recovery, signal processing and others [1, 2], which can be viewed as a standard tool for solving nonsmooth problems with large-scale. This method was first introduced by Martinet [3] for convex minimization and further generalized by Rockafellar [4] to get today’s version. Ferreira and Oliverria [5] proved that if f is geodesic convex, then the sequence generated from proximal point algorithm converges to an optimal solution of the problem on Riemannian manifolds. Quiroz and Oliverria [6, 7] considered the proximal point algorithm for solving the problem of finding an optimal point of a geodesic quasiconvex function on Hadamard manifolds. Assuming that the function is local Lipschitz, the sequence by this algorithm converges to a stationary point. When the objective function is continuous and the proximal parameters is bounded, the sequence converges to a generalized critical point. In addition to Ferreira and Oliveria, other authors considered extensions of the proximal point method to the vectorial setting; see for instance, Ceng and Yao [8], Choung et al. [9], Ceng et al. [10], Villacorta and Oliveria [11] and references therein. Bento et al. [12] introduced the proximal point-type method, assuming that G : Rn → Rm is quasiconvex and continuously differentiable, and
Data Loading...
