Abstract
The sparse linear programming (SLP) is a linear programming problem equipped with a sparsity constraint, which is nonconvex, discontinuous and generally NP-hard due to the combinatorial property involved. In this paper, by rewriting the sparsity constraint into a disjunctive form, we present an explicit formula of the Lagrangian dual problem for the SLP, in terms of an unconstrained piecewise-linear convex programming problem which admits a strong duality under bi-dual sparsity consistency. Furthermore, we show a saddle point theorem based on the strong duality and analyze two classes of stationary points for the saddle point problem. At last, we extend these results to SLP with the lower bound zero replaced by a certain negative constant.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11431002, 11771038 and 11728101), the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (Grant No. RCS2017ZJ001) and China Scholarship Council (Grant No. 201707090019). The authors sincerely appreciate the suggestions and comments from two anonymous referees for the improvement of the paper.
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Zhao, C., Luo, Z., Li, W. et al. Lagrangian duality and saddle points for sparse linear programming. Sci. China Math. 62, 2015–2032 (2019). https://doi.org/10.1007/s11425-018-9546-9
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DOI: https://doi.org/10.1007/s11425-018-9546-9
Keywords
- sparse linear programming
- Lagrangian dual problem
- strong duality
- saddle point theorem
- optimality condition