Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1369–1399 | Cite as

A new globally convergent algorithm for non-Lipschitz p-q minimization

  • Zhifang Liu
  • Chunlin WuEmail author
  • Yanan Zhao


We consider the non-Lipschitz p-q (0 < p < 1 ≤ q < ) minimization problem, which has many applications and is a great challenge for optimization. The problem contains a non-Lipschitz regularization term and a possibly nonsmooth fidelity. In this paper, we present a new globally convergent algorithm, which gradually shrinks the variable support and uses linearization and proximal approximations. The subproblem at each iteration is then convex with increasingly fewer unknowns. By showing a lower bound theory for the sequence generated by our algorithm, we prove that the sequence globally converges to a stationary point of the p-q objective function. Our method can be extended to the p-regularized elastic net model. Numerical experiments demonstrate the performances and flexibilities of the proposed algorithm, such as the applicability to measurements with either Gaussian or heavy-tailed noise.


Nonconvex nonsmooth regularization Non-Lipschitz optimization Support shrinking Lower bound theory ADMM (alternating direction method of multipliers) Gaussian noise Heavy-tailed noise 

Mathematics Subject Classification (2010)

49M05 49K30 90C26 94A12 90C30 


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Funding information

This work was supported by the National Natural Science Foundation of China (Grants 11301289, 11531013, and 11871035), Recruitment Program of Global Young Expert, and the Fundamental Research Funds for the Central Universities.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesNankai UniversityTianjinChina

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