Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 17 (2022), 139 -- 179

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This work is licensed under a Creative Commons Attribution 4.0 International License.

A STRUCTURE EXPLOITING ALGORITHM FOR NON-SMOOTH SEMI-LINEAR ELLIPTIC OPTIMAL CONTROL PROBLEMS

Olga Weiß and Andrea Walther

Abstract. We investigate optimization problems with a non-smooth partial differential equation as constraint, where the non-smoothness is assumed to be caused by Nemytzkii operators generated 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-linearization, i.e., a special handling of the non-smoothness with proficient exploitation of the non-smooth structure. The exploitation of the given data allows a targeted and optimal decomposition of the optimization problem in order to compute stationary points. This approach is able to solve the considered class of non-smooth optimization problems in very few Newton steps and additionally maintains reasonable convergence properties. Numerical results for non-smooth optimization problems illustrate the proposed approach and its performance.

2020 Mathematics Subject Classification: 49J52; 37N30; 46N10; 65N30; 49M15; 35J61;
Keywords: Non-Smooth Optimization; Constant Abs-Linearization; PDE Constrained Optimization; Non-Smooth PDE; Elliptic Optimal Control Problem

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Olga Weiß
Institut für Mathematik, Humboldt-Universität zu Berlin
Unter den Linden 6, 10117 Berlin, Germany
e-mail: olga.weiss@hu-berlin.de


Andrea Walther
Institut für Mathematik, Humboldt-Universität zu Berlin
Unter den Linden 6, 10117 Berlin, Germany
e-mail: andrea.walther@math.hu-berlin.de

http://www.utgjiu.ro/math/sma