Abstract
This paper introduces a new three-step algorithm to solve the split feasibility problem. The main advantage is that one of the projective operators interferes only in the final step, resulting in less computations at each iteration. An example is provided to support the theoretical approach. The numerical simulation reveals that the newly introduced procedure has increased performance compared to other existing methods, including the classic CQ algorithm. An interesting outcome of the numerical modeling is an approximate visual image of the solution set.
Highlights
The split feasibility problem was first introduced by Censor and Elfving [1]for solving a class of inverse problems
Censor and Elfving produced consistent algorithms to solve the newly introduced type of problem. Their procedure did not benefit from much popularity, as it was requiring matrix inverses at each step, making it less efficient
The CQ algorithm is exactly the gradient projection algorithm applied to the optimization problem
Summary
The split feasibility problem (abbreviated SFP) was first introduced by Censor and Elfving [1]. Censor and Elfving produced consistent algorithms to solve the newly introduced type of problem Their procedure did not benefit from much popularity, as it was requiring matrix inverses at each step, making it less efficient. It is well known that the SFP can be naturally rephrased as a constrained minimization problem From this perspective, the CQ algorithm is exactly the gradient projection algorithm applied to the optimization problem. The SPF could be rephrased as a fixed point searching issue, the involved operator being nonexpansive This opens to the perspective of using alternative iteration procedures for reckoning a solution. In [11], the possibility of seeing the CQ algorithm as a special case of Krasnosel’skii-Mann type algorithm was pointed out This ensures the weak convergence of the procedure. Extensions to SFP constrained by variational inequalities [13], fixed point problems [14,15,16], and zeros of nonlinear operators or equilibrium problems [17,18] could be challenging research topics
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