Skip to main content
SpringerLink
Log in

A natural extension of the classical envelope theorem in vector differential programming

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

Abstract

The aim of this paper is to extend the classical envelope theorem from scalar to vector differential programming. The obtained result allows us to measure the quantitative behaviour of a certain set of optimal values (not necessarily a singleton) characterized to become minimum when the objective function is composed with a positive function, according to changes of any of the parameters which appear in the constraints. We show that the sensitivity of the program depends on a Lagrange multiplier and its sensitivity.

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. Hotelling, H.: Edgeworth’s taxation paradox and the nature of demand and supply functions. J. Polit. Econ. 40, 557–616 (1932)

    Article  Google Scholar 

  2. Viner, J.: Cost curves and supply curves. Zeitschrift fur Nation-alokonomie 3 (1931). Reprinted in Readings in price theory. Homewood, Il. Richard D. Irwin (1951)

  3. Samuelson, P.: Foundations of Economic Analysis. Harvard University Press, Cambridge (1947). http://www.worldcat.org/title/foundations-of-economic-analysis/oclc/609188111

  4. Afriat, S.N.: Theory of maxima and the method of Lagrange. SIAM J. Appl. Math. 20, 343–357 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  5. Epstein, L.G.: The Le Châtelier principle in optimal control problems. J. Econ. Theory 19, 103–122 (1978)

    Article  MATH  Google Scholar 

  6. Caputo, M.R.: The envelope theorem and comparative statics of Nash equilibria. Games Econ. Behav. 13, 201–224 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Rincon-Zapatero, J., Santos, M.: Differentiability of the value function without interiority assumptions. J. Econ. Theory 144, 1948–1964 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Silberherg, E.: The Le Chatelier principle as a corollary to a generalized envelope theorem. J. Econ. Theory 3, 146–155 (1971)

    Article  Google Scholar 

  9. Silherberg, E.: A revision of comparative statics methodology. or, How to do economics on the back of an envelope. J. Econ. Theory 7, 159–172 (1974)

    Article  Google Scholar 

  10. Rockafellar, R.T.: Directional differentiability of the optimal value function in a nonlinear programming problem. Math. Progr. Study 21, 213–226 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  11. Benveniste, L., Scheinkman, J.: On the differentiability of the value function in dynamic models of economics. Econometrica 47, 727–32 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Balbás, A., Fernández, F.J., Jiménez Guerra, P.: On the envolvent theorem in multiobjective programming. Indian J. Pure Appl. Math. 26(11), 1035–1047 (1995)

    MATH  MathSciNet  Google Scholar 

  13. Balbás, A., Jiménez Guerra, P.: Sensitivity analysis for convex multiobjective programming in abstract spaces. J. Math. Anal. Appl. 202, 645–648 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Balbás, A., Ballvé, M., Jiménez Guerra, P.: Sensitivity and optimality conditions in the multiobjective differential programming. Indian J. Pure Appl. Math. 29(7), 6711-680 (1998)

    Google Scholar 

  15. Balbás, A., Ballvé, M., Jiménez Guerra, P.: Sensitivity in multiobjective programming under homogeneity assumptions. J. Multicrit. Decis. Anal. 8, 133–138 (1999)

    Article  MATH  Google Scholar 

  16. Balbás, A., Jiménez Guerra, P.: Sensitivity in multiobjective programming by differential equations methods. The case of homogeneous functions. In: Lecture Notes in Economics and Mathematical Systems, vol. 455. Springer, Berlin, pp. 188–196 (1997)

  17. Balbás, A., Ballvé, M.: Density theorems for ideal points in vector optimization. Eur. J. Oper. Res. 133, 260–266 (2001)

    Article  MATH  Google Scholar 

  18. Jiménez Guerra, P., Melguizo, M.A., Muñoz, M.J.: The envelope theorem for multiobjective convex programming via contingent derivatives. J. Math. Anal. Appl. 372, 197–207 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jiménez Guerra, P., Melguizo Padial, M.A., Muñoz Bouzo, M.J.: Sensitivity analysis in multiobjective differential programming. Comput. Math. Appl. 52, 109–120 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jiménez Guerra, P., Melguizo Padial, M.A., Muñoz Bouzo, M.J.: Sensitivity analysis in convex programming. Comput. Math. Appl. 58, 1239–1246 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Jiménez Guerra, P., Melguizo, M.A.: Sensitivity analysis in differential programming through the Clarke Derivative. Mediterr. J. Math. 9, 537–550 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. García, F., Melguizo Padial M. A.: Sensitivity analysis in convex optimization through the circatangent derivative. J. Optim. Theory Appl. (2014). doi:10.1007/s10957-014-0609-4

  23. Rudin, W.: Functional Analysis. McGraw-Hill, New Delhi (1977)

    Google Scholar 

  24. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank the referee for providing useful comments which have served to improve the readability of the paper. This work has been partially supported by the Generalitat Valenciana project GV/2014/072 and Universidad de Alicante project GRE11-08.

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad de Alicante, Carretera San Vicente del Raspeig, s/n, 03690, San Vicente del Raspeig, Alicante, Spain

    F. García Castaño & M. A. Melguizo Padial

Authors
  1. F. García Castaño
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Melguizo Padial
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. Melguizo Padial.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

García Castaño, F., Melguizo Padial, M.A. A natural extension of the classical envelope theorem in vector differential programming. J Glob Optim 63, 757–775 (2015). https://doi.org/10.1007/s10898-015-0307-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-015-0307-2

Keywords

Search

Navigation