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Solving Non-Smooth Semi-Linear Optimal Control Problems with Abs-Linearization

Olga Ebel(ebelo***at***math.upb.de)
Andrea Walther(awalther***at***math.uni-paderborn.de)
Stephan Schmidt(stephan.schmidt***at***mathematik.uni-wuerzburg.de)

Abstract: We investigate optimization problems with a non-smooth partial differential equation as constraint, where non-smoothness is assumed to be caused by the functions abs(), min() and max(). For the efficient as well as robust solution of such problems, we propose a new optimization method based on abs-linearisation, i.e., a special handling of the non-smoothness without regularization. The key idea of this approach is the determination of stationary points by an appropriate decomposition of the original non-smooth problem into several smooth so-called branch problems. Each of these branch problems can be solved by classical means. The exploitation of corresponding optimality conditions for the smooth case identifies the next branch and thus yields a successive reduction of the objective value. This approach is able to solve the considered class of non-smooth optimization problems without any regularization of the non-smoothness and additionally maintains reasonable convergence properties. Numerical results for non-smooth optimization problems illustrate the proposed approach and its performance.

Keywords: Non-Smooth Optimization, SALi, Abs-Linearization, PDE Constrained Optimization, Non-Smooth PDE

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

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Entry Submitted: 10/26/2018
Entry Accepted: 10/26/2018
Entry Last Modified: 10/26/2018

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