Regularization of proximal point algorithms in Hadamard manifolds

被引：108

作者：

Ansari, Qamrul Hasan ^{[1
,2
]}

Babu, Feeroz ^{[1
]}

Yao, Jen-Chih ^{[3
]}

机构：

[1] Aligarh Muslim Univ, Dept Math, Aligarh 202002, Uttar Pradesh, India

[2] King Fahd Univ Petr & Minerals, Dept Math & Stat, Dhahran, Saudi Arabia

[3] China Med Univ, Res Ctr Int Comp, China Med Univ Hosp, Taichung, Taiwan

来源：

JOURNAL OF FIXED POINT THEORY AND APPLICATIONS | 2019年 / 21卷 / 01期

关键词：

Inclusion problems; regularization method; proximal point algorithms; maximal monotone vector fields; minimization problems; saddle point problems; Hadamard manifolds; MONOTONE VECTOR-FIELDS; STRONG-CONVERGENCE; OPERATORS; SEMIGROUPS; PROJECTION; SETS;

D O I：

10.1007/s11784-019-0658-2

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, we consider the regularization method for exact as well as for inexact proximal point algorithms for finding the singularities of maximal monotone set-valued vector fields. We prove that the sequences generated by these algorithms converge to an element of the set of singularities of a maximal monotone set-valued vector field. A numerical example is provided to illustrate the inexact proximal point algorithm with regularization. Applications of our results to minimization problems and saddle point problems are given in the setting of Hadamard manifolds.

引用

页数：23

共 33 条

[1]

[Anonymous], 1977, Houston J. Math.

[2] The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces [J].

Bacak, Miroslav ;

Reich, Simeon .

JOURNAL OF FIXED POINT THEORY AND APPLICATIONS, 2014, 16 (1-2) :189-202

[3] Projection and proximal point methods:: convergence results and counterexamples [J].

Bauschke, HH ;

Matousková, E ;

Reich, S .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2004, 56 (05) :715-738

[4] A weak-to-strong convergence principle for Fejer-monotone methods in Hilbert spaces [J].

Bauschke, HH ;

Combettes, PL .

MATHEMATICS OF OPERATIONS RESEARCH, 2001, 26 (02) :248-264

[5]

Bridson M, 1999, METRIC SPACES NONPOS

[6]

Bruck R. E., 1980, Nonlinear Analysis Theory, Methods & Applications, V4, P939, DOI 10.1016/0362-546X(80)90006-1

[7] STRONGLY CONVERGENT ITERATIVE SOLUTION OF 0 EPSILON U(X) FOR A MAXIMAL MONOTONE OPERATOR-U IN HILBERT-SPACE [J].

BRUCK, RE .

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 1974, 48 (01) :114-126

[8] Convex- and monotone-transformable mathematical programming problems and a proximal-like point method [J].

Da Cruz Neto, J. X. ;

Ferreira, O. P. ;

Perez, L. R. Lucambio ;

Nemeth, S. Z. .

JOURNAL OF GLOBAL OPTIMIZATION, 2006, 35 (01) :53-69

[9]

Da Cruz Neto JX., 2000, Balk. J. Geom. Appl, V5, P69

[10] Proximal point algorithm on Riemannian manifolds [J].

Ferreira, OP ;

Oliveira, PR .

OPTIMIZATION, 2002, 51 (02) :257-270

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