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

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.

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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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Correspondence to Pedro Merino.

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This research has been supported by Research Project PIJ-15-26 funded by Escuela Politécnica Nacional, Quito–Ecuador. Moreover, we acknowledge partial support of SENESCYT-MATHAmSud project SOCDE “Sparse Optimal Control of Differential Equations”.

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Merino, P. A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs. Comput Optim Appl 74, 225–258 (2019). https://doi.org/10.1007/s10589-019-00101-0

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Keywords

  • Optimal control
  • Nonconvex
  • DC programming
  • DCA
  • Elliptic PDE

Mathematics Subject Classification

  • 90C26
  • 90C46
  • 49J20
  • 49K20