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1854. First- and second-order models for nonsmooth functions based on derivative sampling

Invited abstract in session WB-41: Structured nonconvex optimization , stream Nonsmooth Optimization.

Wednesday, 10:30-12:00
Room: 97 (building: 306)

Authors (first author is the speaker)

1. Bennet Gebken
Department of Mathematics, Technical University of Munich

Abstract

In optimization, many solvers are based on iteratively building a local model of the objective function and then minimizing the model instead of the original function. In the smooth case, such models can be derived from the Taylor expansion based on derivatives of different orders. In the nonsmooth case on the other hand, models are more challenging to construct: The first issue is the lack of a Taylor expansion. While replacing the gradient in smooth first-order methods by the Clarke subdifferential from nonsmooth analysis yields a way to characterize descent directions, there is no simple way to derive higher-order models. The second issue is that generalized derivatives like the Clarke subdiff. are difficult to work with in practice, since they can be unstable and impossible to evaluate in the general case.
In this talk, I will present two models for (unstructured) locally Lipschitz continuous functions. The first model is a simple first-order model, which is based on approximating the Clarke subdiff. by the Goldstein eps-subdiff. which, in turn, can be approximated in practice by a deterministic gradient sampling approach. The second model is based on the idea of sampling Hessian matrices in addition to the gradient. More precisely, it is defined as the maximum of (existing) second-order Taylor expansions in a neighborhood of a point. After introducing each model, I will present ways to generate them in practice and discuss the behavior of the resulting descent methods.

Keywords

Status: accepted

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