Skip to main content
SpringerLink
Log in

Necessary optimality conditions for nonsmooth semi-infinite programming problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, for a nonsmooth semi-infinite programming problem where the objective and constraint functions are locally Lipschitz, analogues of the Guignard, Kuhn-Tucker, and Cottle constraint qualifications are given. Pshenichnyi-Levin-Valadire property is introduced, and Karush-Kuhn-Tucker type necessary optimality conditions are derived.

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. Bot R.I., Grad S.M., Wanka G.: On strong and total Lagrange duality for convex optimization problems. J. Math. Anal. Appl. 337, 1315–1325 (2008)

    Article  Google Scholar 

  2. Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)

    Google Scholar 

  3. DingZhu Du., Panos M., Pardalos P.M., Weili Wu.: Mathematical Theory of Optimization. Kluwer Academic Publishers, Dordrecht (2001)

    Google Scholar 

  4. Dinh N., Goberna M.A., López M.A., Son T.Q.: New farkash-type constraint qualifications in convex infinite programming. ESIAM: Control. Optim. Calc. Var. 13, 580–597 (2007)

    Article  Google Scholar 

  5. Fajardo M.D., López M.A.: Locally farkas-minkowski systems in convex semi-infinite programming. J. Optim. Theory Appl. 103, 313–335 (1999)

    Article  Google Scholar 

  6. Giorgi J., Gwirraggio A., Thierselder J.: Mathematics of Optimization; Smooth and Nonsmooth Cases. Elsivier, Amsterdam (2004)

    Google Scholar 

  7. Goberna M.A., López M.A.: Linear Semi-Infinite Optimazation. Wiley, Chichester (1998)

    Google Scholar 

  8. Gorberna M.A., López M.A., Pastor J.: Farkas-minkowski system in semi-infinite programming. Appl. Math. Optim. 7, 295–308 (1981)

    Article  Google Scholar 

  9. Guerra-Vazquez, F., Jongen, H.Th., Shikhman, V.: General semi-infinite programming: symmetric Mangasarian-Fromovitz constraint qualification and the closure of the feasible set. SIAM J. Optim. (2010, to appear)

  10. Hettich R., Haaren E., Ries M., Still G.: Accurate numerical approximations of eirenfrequencies and eigenfunctions of elliptic membranes. Z. Angew. Math. Mech. 67, 589–597 (1987)

    Article  Google Scholar 

  11. Hiriart- Urruty J.B., Lemarechal C.: Convex Analysis and Minimization Algorithms. Springer, Berlin, Heidelberg (1991)

    Google Scholar 

  12. Horst R., Pardalos P.M.: Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht (1995)

    Google Scholar 

  13. Kanzi N., Nobakhtian S.: Optimality conditions for nonsmooth semi-infinite programming. Optimization 53, 717–727 (2008)

    Google Scholar 

  14. Kanzi N., Nobakhtian S.: Nonsmooth semi-infinite programming problems with mixed constraints. J. Math. Anal. Appl. 351, 170–181 (2008)

    Article  Google Scholar 

  15. Kropat E., Pickl S., Rößler A., Weber G.-W.: A new algorithm from semi-infinite optimization for a problem of time-minimal control. J. Comput Technol 5 4, 43–81 (2000)

    Google Scholar 

  16. Li W., Nahak C., Singer I.: Constraint qualifications in semi-infinite systems of convex inequalities. SIAM J. Optim. 11, 31–52 (2000)

    Article  Google Scholar 

  17. Li Ch., Nǵ K.F., Pong T.K.: Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM J. Optim. 19, 163–187 (2008)

    Article  Google Scholar 

  18. López M.A., Vercher E.: Optimality conditions for nondifferentiable convex semi-infinite programming. Math. Program. 27, 307–319 (1983)

    Article  Google Scholar 

  19. López M.A., Still G.: Semi-infinite programming. Eur. J. Opera. Res. 180, 491–518 (2007)

    Article  Google Scholar 

  20. Puente R., VeraDe Serio V.N.: Locally farkas minkowski linear inequality systems. Top 7, 103–121 (1999)

    Article  Google Scholar 

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

    Book  Google Scholar 

  22. Stein O.: On constraint qualifications in nonsmooth optimization. J. Optim. Theory. Apll. 121, 647–671 (2004)

    Article  Google Scholar 

  23. Zheng X.Y., Yang X.: Lagrange multipliers in nonsmooth semi-infinite optimization problems. J. Oper. Res. 32, 168–181 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Payame noor University (PNU), Rezvanshahr-e-Sadugh, Yazd, Iran

    Nader Kanzi

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

    Nader Kanzi

Authors
  1. Nader Kanzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nader Kanzi.

Additional information

This research was in part supported by a grant from IPM (No. 88900026).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kanzi, N. Necessary optimality conditions for nonsmooth semi-infinite programming problems. J Glob Optim 49, 713–725 (2011). https://doi.org/10.1007/s10898-010-9561-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-010-9561-5

Keywords

Search

Navigation