Modified Alternating Direction Methods for the Modified Multiple-Sets Split Feasibility Problems

In this paper, we propose two new multiple-sets split feasibility problem (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 difficulty 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