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Optimality Conditions for Group Sparse Constrained Optimization Problems

Author

Listed:
  • Wenying Wu

    (School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China
    These authors contributed equally to this work.)

  • Dingtao Peng

    (School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China
    These authors contributed equally to this work.)

Abstract

In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: T B -stationary point, N B -stationary point, T C -stationary point and N C -stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided.

Suggested Citation

  • Wenying Wu & Dingtao Peng, 2021. "Optimality Conditions for Group Sparse Constrained Optimization Problems," Mathematics, MDPI, vol. 9(1), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:1:p:84-:d:473685
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    References listed on IDEAS

    as
    1. 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.
    2. Lukas Meier & Sara Van De Geer & Peter Bühlmann, 2008. "The group lasso for logistic regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 53-71, February.
    3. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
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