Let C and Q be nonempty closed convex sets in R N and R M , respectively, and A is an M by N real matrix. The split feasibility problem (SFP) is to find x ∈ C with Ax ∈ Q, if such x exists. Byrne proposed an iterative method called the CQ algorithm that involves only the orthogonal projections onto C and Q and does not need to compute the matrix inverse, which may be the main advantage compared with other algorithms. The CQ algorithm is as follows:where γ ∈ (0, 2/L), with L being the largest eigenvalue of the matrix A T A and P C and P Q denote the orthogonal projections onto C and Q, respectively. Byrne (2002 Inverse Problems 18 2004 Inverse Problems 20 103-20) proved the convergence of the CQ algorithm for arbitrary nonzero matrix A.In his algorithm, Byrne assumed that the projections P C and P Q are easily calculated. In this paper, a modification of the CQ algorithm, called the relaxed CQ algorithm, is given, in which we replace P C and P Q by P C k and P Q k , respectively, and the latter are easy to implement. Under mild assumptions, the convergence of the proposed algorithm is established. Then another algorithm for SFP is given; with the help of the CQ algorithm and its relaxed version, it is easy to obtain the convergence of this algorithm and corresponding relaxed scheme.
Let C and Q be nonempty, closed convex sets in R n and R m respectively, and A be an m × n real matrix. The split feasibility problem (SFP) is to find x ∈ C with Ax ∈ Q, if such points exist. Byrne proposed the following CQ algorithm to solve the SFP:where γ ∈ (0, 2/ρ(A T A)) with ρ(A T A) the spectral radius of the matrix A T A and P C and P Q denote the orthogonal projections onto C and Q, respectively. However, in some cases, it is difficult or even impossible to compute P C and P Q exactly. In this paper, based on the CQ algorithm, we present several algorithms to solve the SFP. Compared with the CQ algorithm, our algorithms are more practical and easier to implement. They can be regarded as improvements of the CQ algorithm.
Background: Nonalcoholic fatty liver disease (NAFLD) has become prevalent in recent decades, especially in developed countries, and approaches for the prevention and treatment of NAFLD are not clear. The aim of this research was to analyze and summarize randomized controlled trials that investigated the effects of probiotics on NAFLD. Methods: Seven databases (PubMed, Embase, the Web of Science, the Cochrane Library, China National Knowledge Infrastructure, Wan Fang Data, and VIP Database) were searched. Then, eligible studies were identified. Finally, proper data extraction, synthesis and analysis were performed by trained researchers. Results: Anthropometric parameters: with use of probiotics weight was reduced by 2.31 kg, and body mass index (BMI) was reduced by 1.08 kg/m2. Liver function: probiotic treatment reduced the alanine aminotransferase level by 7.22 U/l, the aspartate aminotransferase level by 7.22 U/l, the alkaline phosphatase level by 25.87 U/l, and the glutamyl transpeptidase level by −5.76 U/l. Lipid profiles: total cholesterol, low-density lipoprotein cholesterol, and triglycerides were significantly decreased after probiotic treatment. Their overall effects (shown as standard mean difference) were −0.73, −0.54, and −0.36, respectively. Plasma glucose: probiotics reduced the plasma glucose level by 4.45 mg/dl and the insulin level by 0.63. Cytokines: probiotic treatment decreased tumor necrosis factor alpha by 0.62 and leptin by 1.14. Degree of liver fat infiltration (DFI): the related risk of probiotics for restoring DFI was 2.47 (95% confidence interval, 1.61–3.81, p < 0.001). Conclusion: Probiotic treatment or supplementation is a promising therapeutic method for NAFLD.
Projection algorithms are practically useful for solving variational inequalities (VI). However some among them require the knowledge related to VI in advance, such as Lipschitz constant. Usually it is impossible in practice. This paper studies the variable-step basic projection algorithm and its relaxed version under weakly co-coercive condition. The algorithms discussed need not know constant/function associated with the co-coercivity or weak co-coercivity and the step-size is varied from one iteration to the next. Under certain conditions the convergence of the variable-step basic projection algorithm is established. For the practical consideration, we also give the relaxed version of this algorithm, in which the projection onto a closed convex set is replaced by another projection at each iteration and latter is easy to calculate. The convergence of relaxed scheme is also obtained under certain assumptions. Finally we apply these two algorithms to the Split Feasibility Problem (SFP).
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