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A New Approach to Solve Convex Infinite-Dimensional Bilevel Problems: Application to the Pollution Emission Price Problem

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Abstract

This paper presents a new approach to studying bilevel programming problems in infinite-dimensional spaces, based on infinite-dimensional duality and evolutionary variational inequalities. The result is applied to the evolutionary emission price problem. In our model, the government chooses the optimal price of pollution emissions, with consideration to firms’ response to the price. Moreover, firms choose the optimal quantities of production to maximize their profits, given the price of pollution emissions. Finally, we provide a numerical example to show the feasibility of our approach.

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References

  1. Bot, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Global Optim. 139, 67–84 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Daniele, P., Giuffrè, S., Maugeri, A., Raciti, F.: Duality theory and applications to unilateral problems. J. Optim. Theory Appl. 162(3), 718–734 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems Volumes I and II. Springer, New York (2003)

    MATH  Google Scholar 

  7. Mordukhovich, B.: Variational Analysis and Generalized Differentiation, vol. 1 and 2. Springer, Berlin (2006)

    Google Scholar 

  8. Nagurney, A.: Network Economics: A Variational Inequality Approach, Revised second edn. Kluwer Academic Publishers, Boston (1999)

    Book  Google Scholar 

  9. Dempe, S.: Foundations of Bilevel Programming. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

  10. Dempe, S., Dutta, J.: Is bilevel programming a special case of mathematical program with complementarity constraints? Math. Program. Ser. A 131, 37–48 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dempe, S., Zemkoho, A.B.: KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization. SIAM J. Optim. 24(4), 1639–1669 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kalashnikov, V.V., Dempe, S., Pérez-Valdés, G.A., Kalashnykova, N.I., Camancho-Vallejo, J.-F.: Bilevel programming and applications. Math. Probl. Eng. 2015(310301), 16 (2015). doi:10.1155/2015/310301

    MathSciNet  MATH  Google Scholar 

  13. Dempe, S., Mordukhovich, B., Zemkoho, A.B.: Necessary optimality conditions in pessimistic bilevel programming. Optimization 63(4), 505–533 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dewez, S., Labbè, M., Marcotte, P., Savard, G.: New formulations and valid inequalities for a bilevel pricing problem. Oper. Res. Lett. 36(2), 141–149 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Vicente, L.N., Calamai, P.H.: Bilevel and multilevel programming: a bibliography review. J. Global Optim. 5(3), 291–306 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mordukhovich, B., Mau Nam, N., Phan, H.M.: Variational analysis of marginal functions with applications to bilevel programming. J. Optim. Theory Appl. 152, 557–586 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ye, J.J.: Optimal strategies for bilevel dynamic problems. SIAM J. Control Optim. 35(2), 512–531 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ye, J.J.: Necessary conditions for bilevel dynamic optimization problems. SIAM J. Control Optim. 33, 1208–1223 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7(2), 481–507 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  22. Amouzegar, M.A., Moshirvaziri, K.: Determining optimal pollution control policies: an application of bilevel programming. Eur. J. Oper. Res. 119(1), 100–120 (1999)

    Article  MATH  Google Scholar 

  23. Wang, G.-M., Ma, L.-M., Li, L.-L.: An application of bilevel programming problem in optimal pollution emission price. J. Serv. Sci. Mang. 4, 334–338 (2011)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Barbagallo, A.: Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems. J. Global Optim. 40(1–3), 29–39 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Giuffrè, S., Maugeri, A.: New results on infinite dimensional duality in elastic–plastic torsion. Filomat 26(5), 1029–1036 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Giuffrè, S., Idone, G., Maugeri, A.: Duality theory and optimality conditions for generalized complementarity problems. Nonlinear Anal. Theor. 63(5–7), 1655–1664 (2005)

    Article  MATH  Google Scholar 

  32. Scrimali, L.: Infinite dimensional duality theory applied to the study of investment strategies in Kyoto Protocol. J. Optim. Theory Appl. 154, 258–277 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Catania, Catania, Italy

    Antonino Maugeri & Laura Scrimali

Authors
  1. Antonino Maugeri
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  2. Laura Scrimali
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Correspondence to Antonino Maugeri.

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Communicated by Aris Daniilidis.

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Maugeri, A., Scrimali, L. A New Approach to Solve Convex Infinite-Dimensional Bilevel Problems: Application to the Pollution Emission Price Problem. J Optim Theory Appl 169, 370–387 (2016). https://doi.org/10.1007/s10957-016-0894-1

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  • DOI: https://doi.org/10.1007/s10957-016-0894-1

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