An alternating direction-based contraction method for linearly constrained separable convex programming problems

Abstract

The classical alternating direction method (ADM) has been well studied in the context of linearly constrained convex programming and variational inequalities where the involved operator is formed as the sum of two individual functions without crossed variables. Recently, ADM has found many novel applications in diversified areas, such as image processing and statistics. However, it is still not clear whether ADM can be extended to the case where the operator is the sum of more than two individual functions. In this article, we extend the spirit of ADM to solve the general case of the linearly constrained separable convex programming problems whose involved operator is separable into finitely many individual functions. As a result, an alternating direction-based contraction-type method is developed. The realization of tackling this class of problems broadens the applicable scope of ADM substantially.

Keywords:

• alternating direction method
• convex programming
• linear constraint
• separable structure
• contraction method

AMS Subject Classifications::

• 90C06
• 90C22
• 90C25

Acknowledgements

The first author was supported by the NSFC Grant 10971095 and the NSF of Province Jiangsu Grant BK2008255 and the last author was supported by the General Research Fund of Hong Kong: HKBU203311.