Some modified relaxed alternating projection methods for solving the two-sets convex feasibility problem

Abstract

Let A, B be nonempty, closed convex sets in an n-dimensional Euclid space ℛ ⁿ . One problem is to find a* ∈ A and b* ∈ B such that ‖a* − b*‖ = inf _a∈A,b∈B ‖a − b‖. Relaxed alternating projection (RAP) Cegielski and Suchocka [A. Cegielski and A. Suchocka, Relaxed alternating projection methods, SIAM J. Optim. 19 (2008), pp. 1093–1106.] algorithm is a method to solve this problem. This method directly calculates a self-adapt variable step-size which is larger than a positive number at every iteration, but it needs to compute the orthogonal projections P _A and P _B . However, in some cases, it is difficult or even impossible to compute P _A and P _B exactly. In this article, based on this method, we present some modified RAP algorithms, in which we replace P _A and P _B by and , where A _k and B _k are the halfspaces containing the original sets A and B, respectively. Then we employ three search rules to choose the step-size. In the first rule, we also directly calculate a step-size which is proved to be larger than a positive number. The last two come from the Armijo search rule. We also establish the convergence of the modified algorithms under some conditions. Some numerical results are presented with the given algorithms.

Keywords:

• alternating projection method
• convex feasibility problem
• relaxed technology
• halfspace

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 10871105), Academic Scholarship for Doctoral Candidates, Ministry of Education of China (Grant No. (190)H0511009) and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.