Projection methods: an annotated bibliography of books and reviews

Abstract

Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here, projection methods are iterative algorithms that use projections onto sets while relying on the general principle that when a family of (usually closed and convex) sets is present, then projections (or approximate projections) onto the given individual sets are easier to perform than projections onto other sets (intersections, image sets under some transformation, etc.) that are derived from the given family of individual sets. Projection methods employ projections (or approximate projections) onto convex sets in various ways. They may use different kinds of projections and, sometimes, even use different projections within the same algorithm. They serve to solve a variety of problems which are either of the feasibility or the optimization types. They have different algorithmic structures, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavioural patterns. This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological and mathematical problems. This annotated bibliography includes books and review papers on, or related to, projection methods that we know about, use and like. If you know of books or review papers that should be added to this list please contact us.

Keywords:

• projection methods
• annotated bibliography
• convex feasibility
• variational inequalities
• von Neumann
• Kaczmarz
• Cimmino
• fixed points
• row-action methods

AMS Subject Classifications:

• 65K10
• 90C25

Acknowledgements

Some of our colleagues to whom we sent an early version of the paper helped us bring it to its present form. The help, ranging from encouragements to typo hunting, included in some cases also suggestions for addition of specific items that we were unaware of. For this help, we are indebted to Heinz Bauschke, Jon Borwein, Charlie Byrne, John Chinneck, Patrick Combettes, Frank Deutsch, Tommy Elfving, Masao Fukushima, Gabor Herman, Ming Jiang, Igor Konnov, Boris Polyak, Constantin Popa, Simeon Reich, Isao Yamada, and Alexander Zaslavski.