Skip to main content
SpringerLink
Log in

On Characterizing the Solution Sets of Pseudoinvex Extremum Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study the minimization of a pseudoinvex function over an invex subset and provide several new and simple characterizations of the solution set of pseudoinvex extremum problems. By means of the basic properties of pseudoinvex functions, the solution set of a pseudoinvex program is characterized, for instance, by the equality \(\nabla f(x)^{T}\eta(\bar{x},x)=0\) , for each feasible point x, where \(\bar{x}\) is in the solution set. Our study improves naturally and extends some previously known results in Mangasarian (Oper. Res. Lett. 7: 21–26, 1988) and Jeyakumar and Yang (J. Opt. Theory Appl. 87: 747–755, 1995).

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. Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  2. Jeyakumar, V., Yang, X.Q.: On characterizing the solution sets of pseudolinear programs. J. Optim. Theory Appl. 87, 747–755 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Burke, J.V., Ferris, M.C.: Characterization of solution sets of convex programs. Oper. Res. Lett. 10, 57–60 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Jeyakumar, V.: Infinite-dimensional convex programming with applications to constrained approximation. J. Optim. Theory Appl. 75, 469–586 (1992)

    Article  MathSciNet  Google Scholar 

  5. Jeyakumar, V., Yang, X.Q.: Convex composite multi-objective nonlinear programming. Math. Program. 59, 325–343 (1993)

    Article  MathSciNet  Google Scholar 

  6. Jeyakumar, V., Lee, G.M., Dinh, N.: Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs. J. Optim. Theory Appl. 123, 83–103 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jeyakumar, V., Lee, G.M., Dinh, N.: Characterizations of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 174, 1380–1395 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Wu, Z.L., Wu, S.Y.: Characterizations of the solution sets of convex programs and variational inequality problems. J. Optim. Theory Appl. 130, 339–358 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Weir, T., Mond, B.: Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Yang, X.M., Yang, X.Q., Teo, K.L.: Generalized invexity and generalized invariant monotonicity. J. Optim. Theory Appl. 117, 607–625 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mohan, S.R., Neogy, S.K.: On invex sets and preinvex functions. J. Math. Anal. Appl. 189, 901–908 (1995)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Chongqing Normal University, Chongqing, 400047, People’s Republic of China

    X. M. Yang

  2. Chongqing Key Laboratory of Operations Research and System Engineering, Chongqing, 400047, People’s Republic of China

    X. M. Yang

Authors
  1. X. M. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. M. Yang.

Additional information

Communicated by F. Giannessi.

This research was partially supported by National Natural Science Foundation of China Grants No. 10771228 and 10831009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, X.M. On Characterizing the Solution Sets of Pseudoinvex Extremum Problems. J Optim Theory Appl 140, 537–542 (2009). https://doi.org/10.1007/s10957-008-9470-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-008-9470-7

Keywords

Search

Navigation