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An efficient duality-based approach for PDE-constrained sparse optimization

Author

Listed:
  • Xiaoliang Song

    (Dalian University of Technology)

  • Bo Chen

    (National University of Singapore)

  • Bo Yu

    (Dalian University of Technology
    Dalian University of Technology)

Abstract

In this paper, elliptic optimal control problems involving the $$L^1$$ L 1 -control cost ( $$L^1$$ L 1 -EOCP) is considered. To numerically discretize $$L^1$$ L 1 -EOCP, the standard piecewise linear finite element is employed. However, different from the finite dimensional $$l^1$$ l 1 -regularization optimization, the resulting discrete $$L^1$$ L 1 -norm does not have a decoupled form. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the $$L^1$$ L 1 -norm. It is clear that this technique will incur an additional error. To avoid the additional error, solving $$L^1$$ L 1 -EOCP via its dual, which can be reformulated as a multi-block unconstrained convex composite minimization problem, is considered. Motivated by the success of the accelerated block coordinate descent (ABCD) method for solving large scale convex minimization problems in finite dimensional space, we consider extending this method to $$L^1$$ L 1 -EOCP. Hence, an efficient inexact ABCD method is introduced for solving $$L^1$$ L 1 -EOCP. The design of this method combines an inexact 2-block majorized ABCD and the recent advances in the inexact symmetric Gauss–Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block. The proposed algorithm (called sGS-imABCD) is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, but also show that our proposed algorithm is more efficient than (a) the ihADMM (inexact heterogeneous alternating direction method of multipliers), (b) the accelerated proximal gradient method.

Suggested Citation

  • Xiaoliang Song & Bo Chen & Bo Yu, 2018. "An efficient duality-based approach for PDE-constrained sparse optimization," Computational Optimization and Applications, Springer, vol. 69(2), pages 461-500, March.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:2:d:10.1007_s10589-017-9951-4
    DOI: 10.1007/s10589-017-9951-4
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Li, Hongyi & Wang, Chaojie & Zhao, Di, 2020. "Preconditioning for PDE-constrained optimization with total variation regularization," Applied Mathematics and Computation, Elsevier, vol. 386(C).

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