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Computational Optimization and Applications

, Volume 74, Issue 1, pp 225–258 | Cite as

A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs

  • Pedro MerinoEmail author
Article

Abstract

We propose a local regularization of elliptic optimal control problems which involves the nonconvex \(L^q\) quasi-norm penalization in the cost function. The proposed Huber type regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized nonsmooth problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem.

Keywords

Optimal control Nonconvex DC programming DCA Elliptic PDE 

Mathematics Subject Classification

90C26 90C46 49J20 49K20 

Notes

Acknowledgements

I wish to thank the anonymous referees for their helpful advise in the revisions. Also to Prof Eduardo Casas and Prof. Juan Carlos De los Reyes for their suggestions and comments which help me to improve this manuscript.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Research Center of Mathematical Modeling (MODEMAT) and Department of MathematicsEscuela Politécnica NacionalQuitoEcuador

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