Skip to main content
SpringerLink
Log in

Characterizing Efficiency on Infinite-dimensional Commodity Spaces with Ordering Cones Having Possibly Empty Interior

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

Abstract

Some production models in finance require infinite-dimensional commodity spaces, where efficiency is defined in terms of an ordering cone having possibly empty interior. Since weak efficiency is more tractable than efficiency from a mathematical point of view, this paper characterizes the equality between efficiency and weak efficiency in infinite-dimensional spaces without further assumptions, like closedness or free disposability. This is obtained as an application of our main result that characterizes the solutions to a unified vector optimization problem in terms of its scalarization. Standard models as efficiency, weak efficiency (defined in terms of quasi-relative interior), weak strict efficiency, strict efficiency, or strong solutions are carefully described. In addition, we exhibit two particular instances and compute the efficient and weak efficient solution set in Lebesgue spaces.

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. Zame, William R.: Competitive equilibria in production economies with an infinite-dimensional commodity space. Econometrica 55, 1075–1108 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bonnisseau, J.-M., Crettez, B.: On the characterization of efficient production vectors. Econ. Theor. 31, 213–223 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bonnisseau, J.-M., Cornet, B.: Existence of equilibria when firms follows bounded losses pricing rules. J. Math. Econ. 17, 119–147 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 2. Springer, Berlin (1989)

    Google Scholar 

  5. Jahn, J.: Vector optimization. Theory, Applications and Extensions. Springer, Berlin (2004)

    MATH  Google Scholar 

  6. Flores-Bazán, F., Hernández, E., Novo, V.: Characterizing efficiency without linear structure: a unified approach. J. Global Optim. 41(1), 43–60 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60, 1399–1419 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Eichfelder, G.: Adaptive Scalarization Methods in Multiobjective Optimization. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  10. Bot, R.I., Grad, S.M.: Duality for vector optimization via a general scalarization. Optimization 60, 1269–1290 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers for \(\varepsilon \)-Pareto solutions in vector optimization with nonsolid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Luenberger, D.G.: New optimality principles for economics efficiency and equilibrium. J. Optim. Theory Appl. 75, 221–264 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gerstewitz, C.H., Iwanow, E.: Dualit\(\ddot{\text{ a }}\)t f\(\ddot{\text{ u }}\)r nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau 31, 61–81 (1985)

    MATH  MathSciNet  Google Scholar 

  17. Rubinov, A.M.: Sublinear operator and theirs applications. Rus. Math. Surveys 32, 113–175 (1977)

    MATH  MathSciNet  Google Scholar 

  18. Rubinov, A.M., Gasimov, R.N.: Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a preorder relation. J. Global Optim. 29, 455–477 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Tammer, C., Zalinescu, C.: Lipschitz properties of the scalarization function and applications. Optimization 59, 305–319 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Göpfert, A., Riahi, H., Tammer, Ch., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics. Springer, New york (2003)

    Google Scholar 

  22. Chen, G-Y.; Huang, X.; Yang, X., “Vector optimization. Set-valued and variational analysis”. Lecture Notes in Economics and Mathematical Systems, 541. Springer-Verlag, Berlin, 2005.

  23. Gutiérrez, C., Jiménez, B., Novo, V.: A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17, 688–710 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gutiérrez, C., Jiménez, B., Novo, V.: Optimality conditions via scalarization for a new \(\varepsilon \)-efficiency concept in vector optimization problems. Eur. J. Oper. Res. 201, 11–22 (2010)

    Article  MATH  Google Scholar 

  25. Gutiérrez, C., Jiménez, B., Novo, V.: Improvements sets and vector optimization. Eur. J. Oper. Res. 223, 304–311 (2012)

    Article  Google Scholar 

  26. Weidner, P.: On the characterization of efficient points by means of monotone functionals. Optimization 19, 53–69 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  27. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, Part I: quasi relative interiors and duality theory. Math. Program. A 57, 15–48 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  28. Limber, M.A., Goodrich, R.K.: Quasi interiors, Lagrange multipliers, and \(L^p\) spectral estimation with lattice bounds. J. Optim. Theory Appl. 78, 143–161 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Grad, A.: Quasi-relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints. J. Math. Anal. Appl. 361, 86–95 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Li, S.J., Yang, X.Q., Chen, G.Y.: Nonconvex vector optimization of set-valued mappings. J. Math. Anal. Appl. 283, 337–350 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Follmer, H., Schied, A.: Convex measures of risk and trading constraints. Financ. Stoch. 6, 429–447 (2002)

    Article  MathSciNet  Google Scholar 

  32. Jeyakumar, V., Oettli, W., Natividad, M.: A solvability theorem for a class of quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179, 537–546 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  33. Drummond, L.D.G., Maculan, N., Svaiter, B.F.: On the choice of parameter for the weighting method in vector optimization. Math. Program. Ser. B 111, 201–216 (2008)

    Article  MATH  Google Scholar 

  34. Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvements sets. J. Optim. Theory Appl. 150, 516–529 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  35. Flores-Bazán, F., Hernández, E.: Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. J. Global Optim. 56, 299–325 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  36. Breckner, W.W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4, 109–127 (1997)

    MATH  MathSciNet  Google Scholar 

  37. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. A 122, 301–347 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Aliprantis, C.D., Brown, D.J., Burkinshaw, O.: Existence and optimality of competitive equilibria. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  39. Flores-Bazán, F., Laengle, S., Loyola, G.: Characterizing the efficient points without closedness or free-disposability. Cent. Eur. J. Oper. Res. 21, 401–410 (2013)

    Article  Google Scholar 

  40. Debreu, G.: Theory of Value. Wiley, New York (1959)

    MATH  Google Scholar 

  41. Magill, M., Quinzii, M.: Theory of incomplete markets, vol. 1. MIT Press, Cambridge (1996)

    Google Scholar 

Download references

Acknowledgments

This research, for the first author, was supported in part by CONICYT-Chile through FONDECYT 112-0980 and BASAL projects, CMM, Universidad de Chile. Support by Ecos-Conicyt CE08-2010 and MATH-AmSud 13math-01 are also greatly acknowledged. The authors also wants to express their great gratitude to Nicolas Hadjisavvas for very stimulating discussions during the last stage of this final version.

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Matemática, CI²MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Fabián Flores-Bazán

  2. Departamento de Matemática, Facultad de Ciencias, Universidad del Bío Bío, Av. Collao 1202 Casilla 5-C, Concepción, Chile

    Fernando Flores-Bazán

  3. Departamento de Control de Gestión, Universidad de Chile, Diagonal Paraguay 257, Santiago de Chile, Chile

    Sigifredo Laengle

Authors
  1. Fabián Flores-Bazán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernando Flores-Bazán
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sigifredo Laengle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabián Flores-Bazán.

Additional information

Communicated by Johannes Jahn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Flores-Bazán, F., Flores-Bazán, F. & Laengle, S. Characterizing Efficiency on Infinite-dimensional Commodity Spaces with Ordering Cones Having Possibly Empty Interior. J Optim Theory Appl 164, 455–478 (2015). https://doi.org/10.1007/s10957-014-0558-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0558-y

Keywords

Mathematics Subject Classication (2000)

Search

Navigation