A Regularized Method for the Split Feasibility Fixed Point Problem

求解分裂可行定點問題的正則化演算技術

本篇文章提出了分裂可行解定點問題及其數學式，除了說明了分裂可行解定點問題的重要性外，也由其數學式證明分裂可行解定點問題包含許多有用的實際問題，例如：可用於醫學上的分裂可行解問題、用於最佳化問題的突可行性問題及用於經濟學上的平衡點問題等。本篇文章也由姚等學者在2012年提出的求解可行性分裂問題方法獲得啟發，進而開發出一個正則化演算法，且該演算法可以強收斂到分裂可行解定點問題的解。由於開發的演算法適用於求解分裂可行解定點問題，所以這個演算法也可以視為分裂可行解問題、突可行性問題及平衡點問題等問題的共通求解技術。

The split feasibility fixed point problem (SFFP) and the mathematical formulation of SFFP are introduced in this article. The SFFP covers several important practical problems, e.g., the split feasibility problem in medicine, the convex feasibility problem in optimization, and the equilibrium problem in economics. We modify the regularized method developed by Yao et al. [28] to and the minimum norm solution of SFFP. The sequence generated by our regularized algorithm strongly converges to the solution of SFFP. Since the split feasibility problem, the convex feasibility problem and the equilibrium problem are special cases of SFFP, our regularized algorithm can be considered as a united methodology for solving these problems.