Stable Convergence Behavior Under Summable Perturbations of a Class of Projection Methods for Convex Feasibility and Optimization Problems

• Published in: IEEE journal of selected topics in signal processing Vol. 1; no. 4; pp. 540 - 547
• Main Authors: Butnariu, D., Davidi, R., Herman, G.T., Kazantsev, I.G.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2007; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Cimmino algorithm; Constraint optimization; Convergence; convex feasibility; cyclic projection method; Image processing; Mathematics; Optimization methods; projection method; Radio access networks; Signal processing algorithms; string-averaging; tomographic optimization; Tomography; total variation; Vectors
• ISSN: 1932-4553; 1941-0484
• DOI: 10.1109/JSTSP.2007.910263

We study the convergence behavior of a class of projection methods for solving convex feasibility and optimization problems. We prove that the algorithms in this class converge to solutions of the consistent convex feasibility problem, and that their convergence is stable under summable perturbations. Our class is a subset of the class of string-averaging projection methods, large enough to contain, among many other procedures, a version of the Cimmino algorithm, as well as the cyclic projection method. A variant of our approach is proposed to approximate the minimum of a convex functional subject to convex constraints. This variant is illustrated on a problem in image processing: namely, for optimization in tomography.