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Optimality conditions for sparse nonlinear programming

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Abstract

The sparse nonlinear programming (SNP) is to minimize a general continuously differentiable function subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Fréchet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn-Tucker (KKT) conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11431002) and Shandong Province Natural Science Foundation (Grant No. ZR2016AM07). The authors thank two anonymous referees whose insightful comments helped us a lot to improve the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    LiLi Pan, NaiHua Xiu & Jun Fan

  2. Department of Mathematics, Shandong University of Technology, Zibo, 255049, China

    LiLi Pan

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  1. LiLi Pan
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  2. NaiHua Xiu
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  3. Jun Fan
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Correspondence to LiLi Pan.

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Pan, L., Xiu, N. & Fan, J. Optimality conditions for sparse nonlinear programming. Sci. China Math. 60, 759–776 (2017). https://doi.org/10.1007/s11425-016-9010-x

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  • DOI: https://doi.org/10.1007/s11425-016-9010-x

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