Skip to main content
SpringerLink
Log in

Nonsmooth sparsity constrained optimization problems: optimality conditions

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This paper concerns a nonsmooth sparsity constrained optimization problem. We present first and second-order necessary and sufficient optimality conditions by using the concept of normal and tangent cones to the sparsity constraint set. Moreover, second-order tangent set to the sparsity constraint is described and then a new second-order necessary optimality condition is established. The results are illustrated by several examples.

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. Aussel, D.: Subdifferential properties of quasiconvex and pseudoconvex functions. J. Optim. Theory Appl. 97, 29–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bahmani, S., Raj, B., Boufounos, P.T.: Greedy sparsity-constrained optimization. J. Mach. Learn. Res. 14, 807–841 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (1979)

    MATH  Google Scholar 

  4. Beck, A., Eldar, Y.C.: Sparsity constrained nonlinear optimization: optimality conditions and algorithms. SIAM J. Optim. 23, 1480–1509 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berg, E.V.D., Friedlander, M.P.: Sparse optimization with least-squares constraints. SIAM J. Optim. 21, 1201–1229 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization, Theoty and Examples. Canadian Mathematical Society, Ottawa (2000)

    Book  Google Scholar 

  7. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1991)

    MATH  Google Scholar 

  8. Donoho, D.L.: Denoising by soft-thresholding. IEEE Trans. Inform. Theory. 41, 613–627 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gorodnitsky, I.F., Rao, B.D.: Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans. Signal Process. 45, 600–616 (1997)

    Article  Google Scholar 

  10. Ivanov, V.I.: On the functions with pseudoconvex sublevel sets and optimality conditions. J. Math. Anal. Appl. 345, 964–974 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lu, Z., Zhang, Y.: Sparse approximation via penalty decomposition methods. SIAM J. Optim. 23, 2448–2478 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Makela, M.M., Karmitsa, N., Eronen V.P.: On generalized pseudo- and quasiconvexities for nonsmooth functions. TUCS report, 989, Turku Centre for Computer Science (2011)

  13. Mallat, S.: A Wavelet Tour of Signal Processing: The Sparse Way. Academic, New York (2008)

    MATH  Google Scholar 

  14. Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Penot, J.P.: Generalized convexity in the light of nonsmooth analysis. Recent developments in optimization. Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 429, pp. 269–290 (1995)

  18. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  19. Ruszczynski, A.P.: Nonlinear Optimization. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  20. Studniarski, M.: Second-order necessary conditions for optimality in nonsmooth nonlinear programming. J. Math. Anal. Appli. 154, 303–317 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Taubman, D., Marcellin, M.W.: Image compression fundamentals, standards and practice. Kluwer, Dordrecht (2001)

    Google Scholar 

  22. Tropp, J.A., Wright, S.J.: Computational methods for sparse solution of linear inverse problems. Proc. IEEE. 98, 948–958 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

The second-named author was partially supported by a Grant from IPM (No. 96900422).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, P. O. Box: 81745-163, Isfahan, Iran

    N. Movahedian, S. Nobakhtian & M. Sarabadan

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran

    S. Nobakhtian

Authors
  1. N. Movahedian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Nobakhtian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Sarabadan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Movahedian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Movahedian, N., Nobakhtian, S. & Sarabadan, M. Nonsmooth sparsity constrained optimization problems: optimality conditions. Optim Lett 13, 1027–1038 (2019). https://doi.org/10.1007/s11590-018-1310-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-018-1310-6

Keywords

Search

Navigation