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

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  • Pedro Merino

    (Escuela Politécnica Nacional)

Abstract

We propose a local regularization of elliptic optimal control problems which involves the nonconvex $$L^q$$ 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.

Suggested Citation

  • Pedro Merino, 2019. "A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs," Computational Optimization and Applications, Springer, vol. 74(1), pages 225-258, September.
    • Handle: RePEc:spr:coopap:v:74:y:2019:i:1:d:10.1007_s10589-019-00101-0
    DOI: 10.1007/s10589-019-00101-0
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    References listed on IDEAS

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    1. F. Flores-BAZÁN & W. Oettli, 2001. "Simplified Optimality Conditions for Minimizing the Difference of Vector-Valued Functions," Journal of Optimization Theory and Applications, Springer, vol. 108(3), pages 571-586, March.
    2. Georg Stadler, 2009. "Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices," Computational Optimization and Applications, Springer, vol. 44(2), pages 159-181, November.
    3. J. C. De Los Reyes & E. Loayza & P. Merino, 2017. "Second-order orthant-based methods with enriched Hessian information for sparse $$\ell _1$$ ℓ 1 -optimization," Computational Optimization and Applications, Springer, vol. 67(2), pages 225-258, June.
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