Abstract
The theory of “point estimate” and the concept of “general convergence”, which were put forward by Smale in order to investigate the complexity of algorithms for solving equations, have been producing a deep impact on the research about the local behavior, the semi-local behavior and the global behavior of iteration methods. The criterion of point estimate introduced by him not only provides a tool for quantitative analysis of the local behavior but also motivates the establishing of the unified determination for the semi-local behavior. Studying the global behavior in the view of discrete dynamical system will lead to many profound research subjects and open up a rich and colorful prospect. In this review, we will make a summarization about the research progress and some applications in nonsmooth optimizations.
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