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

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Abstract

In this paper, elliptic optimal control problems involving the \(L^1\)-control cost (\(L^1\)-EOCP) is considered. To numerically discretize \(L^1\)-EOCP, the standard piecewise linear finite element is employed. However, different from the finite dimensional \(l^1\)-regularization optimization, the resulting discrete \(L^1\)-norm does not have a decoupled form. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the \(L^1\)-norm. It is clear that this technique will incur an additional error. To avoid the additional error, solving \(L^1\)-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 \(L^1\)-EOCP. Hence, an efficient inexact ABCD method is introduced for solving \(L^1\)-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.

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Notes

  1. For more details about the iFEM software package, we refer to the website http://www.math.uci.edu/~chenlong/programming.html

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Acknowledgements

The authors would like to thank Prof. Defeng Sun and Prof. Kim-Chuan Toh at National University of Singapore for their valuable suggestions that led to improvement in this paper and also would like to thank Prof. Long Chen for the FEM package iFEM [44] in Matlab.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116025, Liaoning, China

    Xiaoliang Song & Bo Yu

  2. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Bo Chen

  3. School of Mathematical and Physics Science, Dalian University of Technology, Panjin, 124200, Liaoning, China

    Bo Yu

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  1. Xiaoliang Song
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  2. Bo Chen
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Correspondence to Xiaoliang Song.

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The research of this author was supported by China Scholarship Council while visiting the National University of Singapore and the National Natural Science Foundation of China (Grant Nos. 91230103, 11571061, 11401075).

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Song, X., Chen, B. & Yu, B. An efficient duality-based approach for PDE-constrained sparse optimization. Comput Optim Appl 69, 461–500 (2018). https://doi.org/10.1007/s10589-017-9951-4

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