Authors
Tianxiang Liu (RIKEN-AIP) and Bruno F. Lourenço (ISM)
Abstract
We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and Hölderian error bounds and includes, for example, the error bounds obtainable under the theory of amenable cones. One of the main results we prove is that the convergence rate of several algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function. This allows us to show new and old results related to the convergence rate of several projection algorithms and, also, of a damped Douglas-Rachford algorithm. Finally, applications to conic feasibility problems are also given and we show that a number of algorithms have convergence rates depending explicitly on the singularity degree of the problem.