Science China Mathematics

, Volume 62, Issue 1, pp 185–204 | Cite as

A new piecewise quadratic approximation approach for L0 norm minimization problem

  • Qian Li
  • Yanqin BaiEmail author
  • Changjun Yu
  • Ya-xiang Yuan


In this paper, we consider the problem of finding sparse solutions for underdetermined systems of linear equations, which can be formulated as a class of L0 norm minimization problem. By using the least absolute residual approximation, we propose a new piecewise quadratic function to approximate the L0 norm. Then, we develop a piecewise quadratic approximation (PQA) model where the objective function is given by the summation of a smooth non-convex component and a non-smooth convex component. To solve the (PQA) model, we present an algorithm based on the idea of the iterative thresholding algorithm and derive the convergence and the convergence rate. Finally, we carry out a series of numerical experiments to demonstrate the performance of the proposed algorithm for (PQA). We also conduct a phase diagram analysis to further show the superiority of (PQA) over L1 and L1/2 regularizations.


sparse optimization non-convex approximation iterative thresholding algorithm 


90C30 90C59 90C90 


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This work was supported by National Natural Science Foundation of China (Grant No. 11771275).


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Copyright information

© Science in China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Qian Li
    • 1
  • Yanqin Bai
    • 1
    Email author
  • Changjun Yu
    • 1
  • Ya-xiang Yuan
    • 2
  1. 1.Department of MathematicsShanghai UniversityShanghaiChina
  2. 2.Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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