Iterative hard-thresholding applied to optimal control problems with $L^0(Ω)$ control cost
Year: 2018
Type: preprint
Abstract: We investigate the hard-thresholding method applied to optimal control problems with $L^0(Ω)$ control cost, which penalizes the measure of the support of the control. As the underlying measure space is non-atomic, arguments of convergence proofs in $l^2$ or $\mathbb R^n$ cannot be applied. Nevertheless, we prove the surprising property that the values of the objective functional are lower semicontinuous along the iterates. That is, the function value in a weak limit point is less or equal than the lim-inf of the function values along the iterates. Under a compactness assumption, we can prove that weak limit points are strong limit points, which enables us to prove certain stationarity conditions for the limit points. Numerical experiments are carried out, which show the performance of the method. These indicates that the method is robust with respect to discretization. In addition, we show that solutions obtained by the thresholding algorithm are superior to solutions of $L^1(Ω)$-regularized problems.
Source: arXiv (Cornell University)
Author: Wachsmuth, Daniel
Institution: University of Würzburg
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
Award: G2308810257
OpenAlex ID: W4289809688