A Class of Variable Metric Decomposition Methods for Monotone Variational Inclusions

Abstract

We extend the general decomposition scheme of [32], which is based on the hybrid inexact proximal point method of [38], to allow the use of variable metric in subproblems, along the lines of [23]. We show that the new general scheme includes as special cases the splitting method for composite mappings [25] and the proximal alternating directions method [13, 17] (in addition to the decomposition methods of [10, 42] that were already covered by [32]). Apart from giving a unified insight into the decomposition methods in question and openning the possibility of using variable metric, which is a computationally important issue, this development also provides linear rate of convergence results not previously available for splitting of composite mappings and for the proximal alternating directions methods. [10] X. Chen and M. Teboulle. A proximal-based decomposition method for convex minimization problems. Mathematical Programming, 64:81–101, 1994.Mathematical Programming, 64:81–101, 1994. [13] J. Eckstein. Some saddle-function splitting methods for convex programming. Optimization Methods and Software, 4:75–83, 1994. [17] B. He, L.Z. Liao, D. Han and H. Yang. A new inexact alternating directions method for monotone variational inequalities. Mathematical Programming, 92:103–118, 2002.Mathematical Programming, 92:103–118, 2002. [23] L.A. Parente, P.A. Lotito and M.V. Solodov. A class of inexact variable metric proximal point algorithms. SIAM Journal on Optimization, 19:240–260, 2008. [25] T. Pennanen. A splitting method for composite mappings. [25] T. Pennanen. A splitting method for composite mappings. Numerical Functional Analysis and Optimization, 23:875–890, 2002. [25] T. Pennanen. A splitting method for composite mappings. [25] T. Pennanen. A splitting method for composite mappings. Numerical Functional Analysis and Optimization, 23:875–890, 2002. SIAM Journal on Optimization, 19:240–260, 2008. [25] T. Pennanen. A splitting method for composite mappings. [25] T. Pennanen. A splitting method for composite mappings. Numerical Functional Analysis and Optimization, 23:875–890, 2002. [25] T. Pennanen. A splitting method for composite mappings. Numerical Functional Analysis and Optimization, 23:875–890, 2002. [32] M.V. Solodov. A class of decomposition methods for convex optimization and monotone variational inclusions via the hybrid inexact proximal point framework. Optimization Methods and Software, 19:557–575, 2004.Optimization Methods and Software, 19:557–575, 2004. [38] M.V. Solodov and B.F. Svaiter. A unified framework for some inexact proximal point algorithms. Numerical Functional Analysis and Optimization, 22:1013–1035, 2001.Numerical Functional Analysis and Optimization, 22:1013–1035, 2001. [42] P. Tseng. Alternating projection-proximal methods for convex programming and variational inequalities. SIAM Journal on Optimization, 7:951–965, 1997.SIAM Journal on Optimization, 7:951–965, 1997.