Log in

Duality for optimization problems in Banach algebras

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

Abstract

In this paper we consider Mond–Weir type and Wolfe type duals for a general nonsmooth optimization problem in Banach algebras, and establish some duality results in the presence of a new class of functions, which is a generalization of the class of smooth KT-(p, r)-invex functions.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Spain)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Aghezzaf B., Hachimi M.: Generalized invexity and duality in multiobjective programming problems. J. Global Optim. 18, 91–101 (2000)

    Article  Google Scholar 

  2. Antczak T.: (ρ, r)-invex sets and functions. J. Math. Anal. Appl. 263, 355–379 (2001)

    Article  Google Scholar 

  3. Bazaraa M.S., Sherali H.D., Shetty C.M.: Nonlinear programming theory and algorithms, 2nd edn. Wiley, New York (1993)

    Google Scholar 

  4. Cambini A., Martein L.: Generalized concavity in multi-objective optimization, 2nd edn. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, pp. 1175–1180. Springer, Berlin (2009) ISBN 978-0-387-74758-3

    Chapter  Google Scholar 

  5. Clarke F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)

    Book  Google Scholar 

  6. Clarke F.H., Ledyaev Y.S., Stern R.J., Wolenski P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)

    Google Scholar 

  7. Craven B.D.: Invexity and its Applications. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, 2nd edn., pp. 1770–1774. Springer, Berlin (2009) ISBN 978-0-387-74758-3

    Chapter  Google Scholar 

  8. Floudas, C.A., Pardalos, P.M. (eds): Encyclopedia of Optimization, 2nd edn. Springer, Berlin (2009) ISBN 978-0-387-74758-3

    Google Scholar 

  9. Goemans, M.: Semidefinite programming and combinatorial optimization. Technical report (1998)

  10. Gulati T.R., Mehndiratta G.: Nondifferentiable multiobjective Mond–Weir type second-order symmetric duality over cones. Optim. Lett. 4, 293–309 (2010)

    Article  Google Scholar 

  11. Hanson M.A.: On sufficiency of Kuhn Tucker Conditions. J. Math. Anal. Appl. 80, 545–550 (1981)

    Article  Google Scholar 

  12. Jeyakumar V., Mond B.: On generalized convex mathematical programming. J. Austral. Math. Soc. Ser. B. 34, 43–53 (1992)

    Article  Google Scholar 

  13. Leibowitz G.M.: Lectures on Complex Function Algebras. Scott, Foresman and Company, Illinois (1970)

    Google Scholar 

  14. Lim S., Lee S., Park S.: An ε-sensitivity analysis for semidefinite programming. Eur. J. Oper. Res. 164, 417–422 (2005)

    Article  Google Scholar 

  15. Meziat R., Patino D.: Exact relaxations of non-convex variational problems. Optim. Lett. 2, 505–519 (2008)

    Article  Google Scholar 

  16. Mishra S.K.: Second order symmetric duality with F-convexity. Eur. J. Oper. Res. 127, 507–518 (2000)

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. Mond B., Weir T.: Generalized concavity and duality. In: Schaible, S., Ziemba, W.T. (eds) Generalized Concavity in Optimization and Economics, pp. 263–279. Academic Press, New York (1981)

    Google Scholar 

  19. Mordukhovich, B.S.: Variations Analysis and Generalized Differentiation, I: Basic Theory, vol. 330. Springer, Grundlehren Series (Fundamental Principles of Mathematical Sciences) (2006)

  20. Mordukhovich, B.S.: Variations Analysis and Generalized Differentiation, II: Applications, vol. 331. Springer, Grundlehren Series (Fundamental Principles of Mathematical Sciences) (2006)

  21. Noor M.A.: On generalized preinvex functions and monotonicities. J. Inequal. Pure Appl. Math. 5, 1–9 (2004)

    Google Scholar 

  22. Overton M., Wolkowicz H.: Semidefinite programming. Math. Program. 77, 105–109 (1997)

    Google Scholar 

  23. Parpas P., Rustem B.: Duality gaps in nonconvex optimization. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, 2nd edn, Springer, Berlin (2009) ISBN 978-0-387-74758-3

    Google Scholar 

  24. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

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

    Book  Google Scholar 

  26. Rudin W.: Functional Analysis. McGraw-Hill, New York (1973)

    Google Scholar 

  27. Rueda N.G., Hanson M.A.: Optimality criteria in mathematical programming involving generalized invexity. J. Math. Anal. Appl. 130, 375–385 (1988)

    Article  Google Scholar 

  28. Soleimani-damaneh M.: Characterization of nonsmooth quasiconvex and pseudoconvex functions. J. Math. Anal. Appl. 330(2), 2168–2176 (2007)

    Article  Google Scholar 

  29. Soleimani-damaneh M.: The gap function for optimization problems in Banach spaces. Nonlinear Anal. 69, 716–723 (2008)

    Article  Google Scholar 

  30. Soleimani-damaneh M.: Infinite (semi-infinite) problems to characterize the optimality of nonlinear optimization problems. Eur. J. Oper. Res. 188(1), 49–56 (2008)

    Article  Google Scholar 

  31. Soleimani-damaneh M.: Nonsmooth optimization using Mordukhovich’s subdifferential. SIAM J. Control Optim. 48, 3403–3432 (2010)

    Article  Google Scholar 

  32. Wolfe P.: A duality theorem for nonlinear programming. Q. Appl. Math. 19, 239–244 (1961)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Soleimani-damaneh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Soleimani-damaneh, M. Duality for optimization problems in Banach algebras. J Glob Optim 54, 375–388 (2012). https://doi.org/10.1007/s10898-011-9763-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9763-5

Keywords

Navigation