KM-CQ-like Alternating Inertial Algorithm for the Split Feasibility Problem

What the paper says

(Communicated by Fanwen Meng) The split feasibility problem (SFP) has many important and wide applications, such as radiotherapy, image reconstruction, and signal processing. Let $C$ and $Q$ be nonempty closed convex sets in $\Re^{n}$ and $\Re^{m}$, respectively, and $A$ an $m \times n$ real matrix, this kind of problem is about finding $x\in C, ~\mathrm{s.t.}~Ax\in Q,$ (1) if such $x$ exists. Recent researches on the SFP have been deeply integrated with inertial acceleration method, and some scholars have discovered that these techniques can speed up the process of first-order optimization algorithms. However, the most of them limit the range of inertial factors and cause the sequence $\|x^{k}-z\|(z\in S)$ no longer monotonically non-increasing. The alternating inertial method, expressed as $w^{k}=\begin{cases}x^{k},&\text{if ~}k\text{~is even},\\ x^{k}+\theta _{k}( x^{k}-x^{k-1}) ,&{\text{if ~}k\text{~is odd}},\end{cases}$ (2) is a appropriate improvement strategy, which retains the acceleration effect of inertial method while alleviating the degree of oscillation. In this paper, we propose three KM-CQ-like alternating inertial algorithms that all expand the range of inertial factors. We use projections onto general closed convex sets in the first version, which exploring the acceleration effects of alternating inertial technique preliminarily. The second version simplifies the computation of procedure by relaxing projections onto half-spaces. The last version further optimizes the previous methods by incorporating double projection technique. Our specific contributions in this paper are summarized as follows. (i) The proposed three KM-CQ-like algorithms incorporate alternating inertial steps allowing them to improve the convergence of the algorithms without inertial steps, which improve the range of values of inertial factors and simplify the form of limiting condition. (ii) The second algorithm also add relaxation effects that allows it to accelerate the convergence of the first algorithm and simplify the calculation of projections. On the other hand, we can see it is faster in Example 4.2. (iii) The last algorithm use double projections that are generated by half-spaces in the previous iteration and the current iteration. Similar to the expected conclusions, it converges faster than the second algorithm in Example 4.2. (iv) The convergence of the iterative sequences generated by the proposed algorithms is established, with the monotonicity of sequence $\|{x}^{2k}-z\|(z\in S)$, which alleviates the oscillation of the iterative points. (v) The performance and advantages of the algorithms proposed in this paper are confirmed by two applications in a simple the SFP and signal processing. On the other hand, the increase of dimension will make the advantages of our algorithms more obvious. Under simple parameter settings, these algorithms restore the monotonicity of $\|x^{2k}-z\|$ and achieve the convergence of the algorithm. Experiments demonstrate the feasibility of each algorithm in practical applications and the effectiveness of the acceleration procedure.