Martin Knossalla, Concepts on generalized varepsilon-subdifferentials for minimizing locally Lipschitz continuous functions

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Volume 1, Issue 2, 1 August 2017, Pages 265-279

Abstract. In this paper, a general concept around converging descent methods for unconstrained nonsmooth optimization problems is introduced. This concept is of constructive nature. Based on the subdifferential according to Clarke or Mordukhovich, respectively, a general approach to $\varepsilon$ -subdifferentials, a kind of continuous outer approximations, is given in an axiomatic way. This leads to a constructive way to create a converging descent method for a given locally Lipschitz continuous objective function. Hence, from a theoretical point of view, convergence is established through the construction of $\varepsilon$ -subdifferentials. This is in contrast to other approaches which are usually based on the assumption of semismoothness of the objective function. The proposed framework generalizes some other approximation concepts for various types of subdifferentials. Furthermore, based on these continuous outer approximations an algorithm is presented and its global convergence to stationary points is proved.

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Martin Knossalla, Concepts on generalized $\varepsilon$ -subdifferentials for minimizing locally Lipschitz continuous functions, J. Nonlinear Var. Anal. 1 (2017), 265-279.