Skip to main content
SpringerLink
Log in

Lagrangian duality and saddle points for sparse linear programming

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babu P, Pelckmans K, Stoica P, et al. Linear systems, sparse solutions, and Sudoku. IEEE Signal Process Lett, 2010, 17: 40–42

    Article  Google Scholar 

  2. Beck A, Hallak N. On the minimization over sparse symmetric sets: Projections, optimality conditions and algorithms. Math Oper Res, 2015, 41: 196–223

    Article  MathSciNet  Google Scholar 

  3. Beck A, Yonina C E. Sparsity constrained nonlinear optimization: Optimality conditions and algorithms. SIAM J Optim, 2013, 23: 1480–1509

    Article  MathSciNet  Google Scholar 

  4. Bertsimas D, King A, Mazumder R. Best subset selection via a modern optimization lens. Ann Statist, 2016, 44: 813–852

    Article  MathSciNet  Google Scholar 

  5. Bucher M, Schwartz A. Second-order optimality conditions and improved convergence results for regularization methods for cardinality-constrained optimization problems. J Optim Theory Appl, 2018, 178: 383–410

    Article  MathSciNet  Google Scholar 

  6. Chen A I, Graves S C. Sparsity-constrained transportation problem. ArXiv:1402.2309, 2014

  7. Chen X J, Xiang S H. Sparse solutions of linear complementarity problems. Math Program, 2016, 159: 539–556

    Article  MathSciNet  Google Scholar 

  8. Cui Y, Pang J S. On the finite number of directional stationary values of piecewise programs. ArXiv:1803.00190, 2018

  9. Donoho D L. Compressed sensing. IEEE Trans Inform Theory, 2006, 52: 1289–1306

    Article  MathSciNet  Google Scholar 

  10. Donoho D L, Tanner J. Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc Natl Acad Sci USA, 2005, 102: 9446–9451

    Article  MathSciNet  Google Scholar 

  11. Friedlander M P, Tseng P. Exact regularization of convex programs. SIAM J Optim, 2007, 18: 1326–1350

    Article  MathSciNet  Google Scholar 

  12. Li D, Sun X L, Wang J. Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection. Math Finance, 2006, 16: 83–101

    Article  MathSciNet  Google Scholar 

  13. Li X, Song W. The first-order necessary conditions for sparsity constrained optimization. J Oper Res Soc China, 2015, 3: 521–535

    Article  MathSciNet  Google Scholar 

  14. Liu J, Chen J, Ye J. Large-scale sparse logistic regression. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York: ACM Press, 2009, 547–556

    Chapter  Google Scholar 

  15. Lu Z S, Yong Z. Sparse approximation via penalty decomposition methods. SIAM J Optim, 2013, 23: 2448–2478

    Article  MathSciNet  Google Scholar 

  16. Mordukhovich B S, Nam N M. An Easy Path to Convex Analysis and Applications. Synthesis Lectures on Mathematics and Statistics, vol. 6. Williston: Morgan & Claypool, 2013

    Google Scholar 

  17. Pan L L, Luo Z Y, Xiu N Y. Restricted robinson constraint qualification and optimality for cardinality-constrained cone programming. J Optim Theory Appl, 2017, 175: 104–118

    Article  MathSciNet  Google Scholar 

  18. Pan L L, Xiu N H, Fan J. Optimality conditions for sparse nonlinear programming. Sci China Math, 2017, 60: 759–776

    Article  MathSciNet  Google Scholar 

  19. Pan L L, Xiu N H, Zhou S L. On solutions of sparsity constrained optimization. J Oper Res Soc China, 2015, 3: 421–439

    Article  MathSciNet  Google Scholar 

  20. Rockafellar R T. Convex Analysis. Princeton: Princeton University Press, 2015

    Google Scholar 

  21. Rockafellar R T, Wets R J. Variational Analysis. Berlin: Springer, 1998

    Book  Google Scholar 

  22. Scholtes S. Introduction to Piecewise Differentiable Equations. New York: Springer, 2012

    Book  Google Scholar 

  23. Shang M J, Zhang C, Xiu N H. Minimal zero norm solutions of linear complementarity problems. J Optim Theory Appl, 2014, 163: 795–814

    Article  MathSciNet  Google Scholar 

  24. Sun C C, Dai R, Mesbahi M. Weighted network design with cardinality constraints via alternating direction method of multipliers. IEEE Trans Control Netw Syst, 2018, 5: 2073–2084

    Article  MathSciNet  Google Scholar 

  25. Wei Z H, Link S. Embedded cardinality constraints. In: Proceedings of the International Conference on Advanced Information Systems Engineering. Cham: Springer, 2018, 523–538

    Chapter  Google Scholar 

  26. Zhu W X, Dong Z S, Yu Y L, et al. Lagrange dual method for sparsity constrained optimization. IEEE Access, 2018, 6: 28404–28416

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

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

    Chen Zhao, Weiyue Li & Naihua Xiu

  2. State Key Lab of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, 100044, China

    Ziyan Luo

  3. School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK

    Houduo Qi

Authors
  1. Chen Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ziyan Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weiyue Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Houduo Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Naihua Xiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chen Zhao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-018-9546-9

Keywords

MSC(2010)

Search

Navigation