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Modified alternating direction methods for the modified multiple-sets split feasibility problems

Yuning Yang(nk0310145***at***gmail.com)
Su Zhang(zhangsu***at***nankai.edu.cn)
Qingzhi Yang(qz-yang***at***nankai.edu.cn)

Abstract: Inthispaper, weproposetwonewmultiple-setssplitfeasibilityproblem(MSFP)models, where the MSFP requires to find a point closest to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation will be closest to the intersection of another family of closed convex sets in the image space. This problem arises in image restoration, signal processing and intensity-modulated radiation therapy (IMRT). The background of the first new model, called the modified multiple-sets split feasibility problem (MMSFP), comes from IMRT. Comparing with MSFP, the MMSFP has three advantages. At the practical level, it is more able to reflect the real world problem; at the algorithmic level, its structure is more separable and the size of each part is smaller, which enables us to apply a modified alternating direction method (ADM) to solve it, which produces parallel steps in each iteration. This parallel feature fits the development of modern parallel-architecture computers. Then, to overcome the di?culty of computing projections onto the constraint sets, a special version of this method with the strategy of projection onto half-space is given. The second new model is to find a least l2-norm solution of the MSFP (or MMSFP). For this problem, a modified ADM with parallel feature is also provided. The convergence of the three ADMs are established, and the convergence rate of the third method is shown to be O(1/t). Numerical results provide at the last show the e?ciency of our methods.

Keywords: multiple-sets split feasibility problem, alternating direction method, parallel computing, half-space, convergence rate

Category 1: Complementarity and Variational Inequalities


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Entry Submitted: 10/09/2012
Entry Accepted: 10/09/2012
Entry Last Modified: 10/09/2012

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