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Asymptotic convergence analysis of a new class of proximal point methods. (English) Zbl 1154.90009
Local convergence results for self-adaptive proximal point methods and nonlinear functions with multiple minimizers which have been developed for finite dimensional optimization problems are properly adapted and generalized to an infinite dimensional Hilbert space setting. The main assumption for this is a local error bound condition which, in a neighborhood of a minimizer, relates the distance of an arbitrary function value from the minimum value to the distance of the respective argument of the function from the set of its minimizers. For a twice continuously Fréchet differentiable function this assumption is shown to be equivalent to a similar condition for its first Fréchet derivative. Then a result on the local convergence of the proximal point method is established for almost exact minimizers. The acceptance criterion related to the latter points, however, normally cannot be realized in practice. For this reason the method is studied also in connection with two implementable acceptance criteria for the iterates, described in terms of the subdifferential of the function, and results on the local convergence and rate of convergence of the sequence of iterates are proven for these under different assumptions.

MSC:
90C26Nonconvex programming, global optimization
90C48Programming in abstract spaces
90C06Large-scale problems (mathematical programming)
65Y20Complexity and performance of numerical algorithms