An improved method for solving multiple-sets split feasibility problem
© Du and Chen; licensee Springer 2012
Received: 23 March 2012
Accepted: 11 September 2012
Published: 3 October 2012
The multiple-sets split feasibility problem (MSSFP) has a variety of applications in the real world such as medical care, image reconstruction and signal processing. Censor et al. proposed solving the MSSFP by a proximity function, and then developed a class of simultaneous methods for solving split feasibility. In our paper, we improve a simultaneous method for solving the MSSFP and prove its convergence.
In this section, we present some concepts and properties of the MSSFP.
Let M be a positive definite matrix. We denote the M-norm by . In particular, is the Euclidean norm of the vector .
, and ;
, and ;
- (a)F is called monotone on S if
- (b)F is called strongly monotone on S if there is a such that
- (c)F is called co-coercive (or ν-inverse strongly monotone) on S if there is a such that
- (d)F is called pseudo-monotone on S if
- (e)F is called Lipschitz continuous on S if there exists a constant such that
and F is called nonexpansive iff .
Remark 1 From Lemma 1 and the above definition, we can infer that a monotone mapping is a pseudo-monotone mapping. An inverse strongly monotone mapping is monotone and Lipschitz continuous. A Lipschitz continuous and strongly monotone mapping is a strongly monotone mapping. The projection operator is 1-ism and nonexpansive.
Lemma 2 A mapping F is 1-ism if and only if the mapping is 1-ism, where I is the identity operator.
Proof See [, Lemma 2.3]. □
Remark 2 If F is an inverse strongly monotone mapping, then F is a nonexpansive mapping.
Lemma 3 Let and be defined in (6)-(7), then and are both Lipschitz continuous and inverse strongly monotone on X and Y, respectively.
therefore, is Lipschitz continuous on X, and the Lipschitz constant is . It also follows from [, Corollary 10] that is -ism. Similarly, we can prove that is Lipschitz continuous on Y, and the Lipschitz constant is , furthermore, it is -ism. □
3 Main results
In this section, we will present our method for solving the MSSFP and prove its convergence. Our algorithm is defined as follows:
Algorithm 3.1 Step 1. Give arbitrary , , , , , and , , . Let be the error tolerance for an approximate solution and set .
- (1)Find the smallest nonnegative integer such that and(13)
If , stop. Otherwise, set and go to Step 2.
Next, we analyze the convergence of Algorithm 3.1:
Proof First, we prove .
where the first inequality follows from (24), the second equality follows from the definition of , and .
where the last inequality follows from (25).
where the first inequality follows from (13′) and (14′).
This completes the proof. □
Next, we prove the sequence is Fejér monotone.
where the inequalities follow from Lemma 4 and (17).
Let , then we can get .
Theorem 2 The sequence generated by Algorithm 3.1 converges to a solution of (8).
which means that the sequence is bounded. Thus, it has at least a cluster point.
It then follows from  that is a solution of (12).
Therefore, the whole sequence converges to . This completes the proof. □
Remark 4 Our iteration method is simpler in the form and is an improvement of the corresponding result of .
The multiple-sets split feasibility problem (MSSFP) requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It serves as a model for real-word inverse problems where constraints are imposed on the solution in the domain of a linear operator as well as in the operator’s range.
In this paper, our algorithm converges to a solution of the multiple-sets split feasibility problem (MSSFP), for any starting vector , whenever the MSSFP has a solution. In the inconsistent case, it finds a point which is least violating the feasibility by being ‘closest’ to all sets as ‘measured’ by a proximity function.
where and are convex functions, respectively. Here he uses the subgradient projections instead of the orthogonal projections. This is a huge achievement and it enables the split feasibility problem to achieve computer operation.
Lastly, we want to say that our work is related to significant real-world applications. The multiple-sets split feasibility problem was applied to the inverse problem of intensity-modulated radiation therapy (IMRT). In this field, beams of penetrating radiation are directed at the lesion (tumor) from external sources in order to eradicate the tumor without causing irreparable damage to surrounding healthy tissues; see, e.g., .
We wish to thank the referees for their helpful comments and suggestions. This research was supported by the National Natural Science Foundation of China, under the Grant No.11071279.
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