Skip to main content
SpringerLink
Log in

Infinite Dimensional Duality Theory Applied to Investment Strategies in Environmental Policy

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

Abstract

In this paper, we develop an infinite dimensional Lagrangian duality framework for modeling and analyzing the evolutionary pollution control problem. Specifically, we examine the situation in which different countries aim at determining the optimal investment allocation in environmental projects and the tolerable pollutant emissions, so as to maximize their welfare. We state the equilibrium conditions underlying the model, and provide an equivalent formulation in terms of an evolutionary variational inequality. Moreover, by means of infinite dimensional duality tools, we prove the existence of Lagrange multipliers that play a fundamental role in order to describe countries’ decision-making processes.

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. UNFCCC: Kyoto Protocol to the United Nations Framework Convention on Climate Change. United Nations, New York (1997)

    Google Scholar 

  2. Kinderleher, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    Google Scholar 

  3. Stampacchia, G.: Variational inequalities, theory and applications of monotone operators. In: Proceedings of a NATO Advanced Study Institute (Venice, 1968), Oderisi, Gubbio, pp. 101–192 (1969)

  4. Mordukhovich, B.: Variational Analysis and Generalized Differentiation. Springer, Berlin (2006)

    Google Scholar 

  5. Nagurney, A.: Network Economics: a Variational Inequality Approach, Revised Second Edition. Kluwer Academic, Boston (1999)

    Google Scholar 

  6. Scrimali, L.: Pollution control quasi-equilibrium problems with joint implementation of environmental projects. Appl. Math. Lett. 25, 385–392 (2012)

    Article  MathSciNet  Google Scholar 

  7. Breton, M., Zaccour, G., Zahaf, M.: A game-theoretic formulation of joint implementation of environmental projects. Eur. J. Oper. Res. 168, 221–239 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Facchinei, F., Fisher, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35, 159–164 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5, 173–210 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)

    Article  MATH  Google Scholar 

  11. Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36, 48–49 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nash, J.F.: Non-cooperative games. Ann. Math. 54, 286–295 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  15. Chan, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7(2), 211–222 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bernard, A., Reilly, J., Vielle, M., Viguier, L.: Russia’s role in the Kyoto Protocol. MIT Program on the Science and Policy of Global Change, Report No. 98 (2002)

  17. Breton, M., Zaccour, G., Zahaf, M.: A differential game of joint implementation of environmental projects. Automatica 41, 1737–1749 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Breton, M., Martín-Herrán, G., Zaccour, G.: Equilibrium investment strategies in foreign environmental projects. J. Optim. Theory Appl. 130, 23–40 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Beckman, M.J., Wallace, J.P.: Continuous lags and the stability of market equilibrium. Economica, New Ser. 36, 58–68 (1969)

    Article  Google Scholar 

  20. Daniele, P.: Dynamic Networks and Evolutionary Variational Inequalities. Edward Elgar, Cheltenham (2006)

    MATH  Google Scholar 

  21. Daniele, P., Maugeri, A., Oettli, W.: Time-dependent traffic equilibria. J. Optim. Theory Appl. 103, 543–554 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Donato, M.B., Milasi, M., Vitanza, C.: Dynamic Walrasian price equilibrium problem: evolutionary variational approach with sensitivity analysis. Optim. Lett. 2, 113–126 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Friesz, T.L., Bernstein, D., Smith, T.E., Tobin, R.L., Wie, B.W.: A variational inequality formulation of the dynamic network user equilibrium problem. Oper. Res. 41, 179–191 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Raciti, F., Scrimali, L.: Time-dependent variational inequalities and applications to equilibrium problems. J. Glob. Optim. 28, 387–400 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Scrimali, L.: Quasi-variational inequalities in transportation networks. Math. Models Methods Appl. Sci. 14, 1541–1560 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Scrimali, L.: The financial equilibrium problem with implicit budget constraints. Cent. Eur. J. Oper. Res. 16, 191–203 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Scrimali, L.: A solution differentiability result for evolutionary quasi-variational inequalities. J. Glob. Optim. 40, 417–425 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Scrimali, L.: Mixed behavior network equilibria and quasi-variational inequalities. J. Ind. Manag. Optim. 5(2), 363–379 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Scrimali, L.: A variational inequality formulation of the environmental pollution control problem. Optim. Lett. 4(2), 259–274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Cojocaru, M.G., Daniele, P., Nagurney, A.: Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications. J. Optim. Theory Appl. 127(3), 549–563 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Donato, M.B., Milasi, M., Scrimali, L.: Walrasian equilibrium problem with memory term. J. Optim. Theory Appl. 151(1), 64–80 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Barbagallo, A., Maugeri, A.: Duality theory for the dynamic oligopolistic market equilibrium problem. Optimization 60(1–2), 29–52 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Boç, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Glob. Optim. 139, 67–84 (2008)

    Google Scholar 

  34. Daniele, P., Giuffrè, S., Maugeri, A.: Remarks on general infinite dimensional duality with cone and equality constraints. Commun. Appl. Anal. 13(4), 567–578 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Daniele, P., Giuffrè, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339, 221–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Daniele, P., Giuffrè, S.: General infinite dimensional duality theory and applications to evolutionary network equilibrium problems. Optim. Lett. 1(3), 227–243 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Maugeri, A., Raciti, F.: Remarks on infinite dimensional duality. J. Glob. Optim. 46(4), 581–588 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (1996)

    MATH  Google Scholar 

  41. Idone, G., Maugeri, A.: Generalized constraints qualification and infinite dimensional duality. Taiwan. J. Math. 13, 1711–1722 (2009)

    MathSciNet  MATH  Google Scholar 

  42. Maugeri, A., Scrimali, L.: Global Lipschitz continuity of solutions to parameterized variational inequalities. Boll. Unione Mat. Ital. (9) 2(1), 45–69 (2009)

    MathSciNet  MATH  Google Scholar 

  43. Maugeri, A., Raciti, F.: On existence theorems for monotone and nonmonotone variational inequalities. J. Convex Anal. 16(3–4), 899–911 (2009)

    MathSciNet  MATH  Google Scholar 

  44. Nesterov, Yu., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. Discrete Contin. Dyn. Syst., Ser. B 31(4), 1383–1396 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy

    Laura Scrimali

Authors
  1. Laura Scrimali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laura Scrimali.

Additional information

Communicated by Michael Patriksson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Scrimali, L. Infinite Dimensional Duality Theory Applied to Investment Strategies in Environmental Policy. J Optim Theory Appl 154, 258–277 (2012). https://doi.org/10.1007/s10957-012-9995-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-9995-7

Keywords

Search

Navigation