On descent-projection method for solving the split feasibility problems

Let Ω and C be nonempty, closed and convex sets in R <Superscript> n </Superscript> and R <Superscript> m </Superscript> respectively and A be an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${m \times n}$$</EquationSource> </InlineEquation> real matrix. The split feasibility problem is to find <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${u \in \Omega}$$</EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${Au \in C.}$$</EquationSource> </InlineEquation> Many problems arising in the image reconstruction can be formulated in this form. In this paper, we propose a descent-projection method for solving the split feasibility problems. The method generates the new iterate by searching the optimal step size along the descent direction. Under certain conditions, the global convergence of the proposed method is proved. In order to demonstrate the efficiency of the proposed method, we provide some numerical results. Copyright Springer Science+Business Media, LLC. 2012