Continuous subdifferential approximations and their applications
The author presents continuous subdifferential and quasidifferential approximations which are suitable for numerical application. A continuous approximation to the Clarke subdifferential \(\partial f\) of a Lipschitzian function \(f: \mathbb{R}^n\to \mathbb{R}\) is a set-valued mapping \(Cf: \mathbb{R}^n\times \mathbb{R}_+\rightrightarrows \mathbb{R}^n\) such that (1) \(Cf(x,\varepsilon)\) is Hausdorff continuous with respect to \(x\), (2) \(\partial f(x) = \text{co}\{v\in \mathbb{R}^n\mid \exists x^k\to x,\;\varepsilon_k\to +0,\;v^k\in Cf(x^k, \varepsilon_k): v^k\to v\}\). In a similar manner, uniform and strong continuous approximations of the subdifferential but also of the quasidifferential (especially for composed functions) are introduced. After some remarks on convergence properties of these approximations, the author gives special constructions for theses notions by use of so-called discrete gradients of the function. The discrete gradients are defined with respect to a given direction and thus allows to approximate a directional derivative. In general, the set of all discrete gradients approximates the entire subdifferential, which is of theoretical interest. For the computation of the descent directions, however, only a few discrete gradients are necessary. The results are used to describe numerical methods for solving unconstrained Lipschitz minimization problems. The author studies convergence properties and verifies the efficiency of the presented methods by many numerical experiments.
scientific article; zbMATH DE number 1982006