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Optimality and Complexity for Constrained Optimization Problems with Nonconvex Regularization

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  • Wei Bian

    (Department of Mathematics, Harbin Institute of Technology, Harbin, China 150001)

  • Xiaojun Chen

    (Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China)

Abstract

In this paper, we consider a class of constrained optimization problems where the feasible set is a general closed convex set, and the objective function has a nonsmooth, nonconvex regularizer. Such a regularizer includes widely used SCAD, MCP, logistic, fraction, hard thresholding, and non-Lipschitz L p penalties as special cases. Using the theory of the generalized directional derivative and the tangent cone, we derive a first order necessary optimality condition for local minimizers of the problem, and define the generalized stationary point of it. We show that the generalized stationary point is the Clarke stationary point when the objective function is Lipschitz continuous at this point, and satisfies the existing necessary optimality conditions when the objective function is not Lipschitz continuous at this point. Moreover, we prove the consistency between the generalized directional derivative and the limit of the classic directional derivatives associated with the smoothing function. Finally, we establish a lower bound property for every local minimizer and show that finding a global minimizer is strongly NP-hard when the objective function has a concave regularizer.

Suggested Citation

  • Wei Bian & Xiaojun Chen, 2017. "Optimality and Complexity for Constrained Optimization Problems with Nonconvex Regularization," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1063-1084, November.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:4:p:1063-1084
    DOI: 10.1287/moor.2016.0837
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    References listed on IDEAS

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    2. Jian Huang & Shuange Ma & Huiliang Xie & Cun-Hui Zhang, 2009. "A group bridge approach for variable selection," Biometrika, Biometrika Trust, vol. 96(2), pages 339-355.
    3. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    4. J. V. Burke & A. S. Lewis & M. L. Overton, 2002. "Approximating Subdifferentials by Random Sampling of Gradients," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 567-584, August.
    5. Wenyu Sun & Ya-Xiang Yuan, 2006. "Optimization Theory and Methods," Springer Optimization and Its Applications, Springer, number 978-0-387-24976-6, June.
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    Cited by:

    1. Hao Wang & Fan Zhang & Yuanming Shi & Yaohua Hu, 2021. "Nonconvex and Nonsmooth Sparse Optimization via Adaptively Iterative Reweighted Methods," Journal of Global Optimization, Springer, vol. 81(3), pages 717-748, November.
    2. Xian Zhang & Dingtao Peng, 2022. "Solving constrained nonsmooth group sparse optimization via group Capped- $$\ell _1$$ ℓ 1 relaxation and group smoothing proximal gradient algorithm," Computational Optimization and Applications, Springer, vol. 83(3), pages 801-844, December.
    3. Tianxiang Liu & Ting Kei Pong & Akiko Takeda, 2023. "Doubly majorized algorithm for sparsity-inducing optimization problems with regularizer-compatible constraints," Computational Optimization and Applications, Springer, vol. 86(2), pages 521-553, November.
    4. Fan Wu & Wei Bian, 2020. "Accelerated iterative hard thresholding algorithm for $$l_0$$l0 regularized regression problem," Journal of Global Optimization, Springer, vol. 76(4), pages 819-840, April.
    5. Bo Wen & Xiaojun Chen & Ting Kei Pong, 2018. "A proximal difference-of-convex algorithm with extrapolation," Computational Optimization and Applications, Springer, vol. 69(2), pages 297-324, March.

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