Alternating Projections on Nontangential Manifolds

We consider sequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(B_{k})_{k=0}^{\infty}$\end{document} of points obtained by projecting a given point B=B0 back and forth between two manifolds \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}_{1}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}_{2}$\end{document}, and give conditions guaranteeing that the sequence converges to a limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B_{\infty}\in\mathcal{M}_{1}\cap\mathcal{M}_{2}$\end{document}. Our motivation is the study of algorithms based on finding the limit of such sequences, which have proved useful in a number of areas. The intersection is typically a set with desirable properties but for which there is no efficient method for finding the closest point Bopt in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}_{1}\cap\mathcal{M}_{2}$\end{document}. Under appropriate conditions, we prove not only that the sequence of alternating projections converges, but that the limit point is fairly close to Bopt, in a manner relative to the distance ∥B0−Bopt∥, thereby significantly improving earlier results in the field.