# American Institute of Mathematical Sciences

December  2011, 31(4): 1097-1113. doi: 10.3934/dcds.2011.31.1097

## On strong Lagrange duality for weighted traffic equilibrium problem

 1 Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, via Cintia, 80126 Naples, Italy 2 Department of Mathematics and Computer Science, University of Catania, viale Andrea Doria n. 6, 95125 CATANIA, Italy, Italy

Received  October 2009 Revised  April 2010 Published  September 2011

The weighted traffic equilibrium problem introduced in [17], in which the equilibrium conditions have been expressed in terms of a weighted variational inequality, studies a transportation network in presence of congestion. For such a problem, existence and regularity theorems have been proved in [8]. In this paper, we analyze the dual problem and characterize the weighted traffic equilibrium solutions by means of Lagrange multipliers, which allow to describe the behavior of the weighted transportation network.
Citation: Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097
##### References:
 [1] J.-P. Aubin, "Analyse Fonctionnelle Appliquée. Tome 2," Translated from the English, Mathématiques, Presses Universitaires de France, Paris, 1987. [2] A. Barbagallo, Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems, J. Global Optim., 40 (2008), 29-39. doi: 10.1007/s10898-007-9194-5. [3] A. Barbagallo, Existence and regularity of solutions to nonlinear degenerate evolutionary variational inequalities with applications to dynamic network equilibrium problems, Appl. Math. Comput., 208 (2009), 1-13. doi: 10.1016/j.amc.2008.10.030. [4] A. Barbagallo, On the regularity of retarded equilibrium in time-dependent traffic equilibrium problems, Nonlinear Anal., 71 (2009), e2406-e2417. doi: 10.1016/j.na.2009.05.054. [5] A. Barbagallo and M.-G. Cojocaru, Continuity of solutions for parametric variational inequalities in Banach space, J. Math. Anal. Appl., 351 (2009), 707-720. doi: 10.1016/j.jmaa.2008.10.052. [6] A. Barbagallo and A. Maugeri, Duality theory for the dynamic oligopolistic market equilibrium problem, Optimization, 60 (2011), 29-52. doi: 10.1080/02331930903578684. [7] A. Barbagallo and S. Pia, Weighted traffic equilibrium problem with delay in non-pivot Hilbert spaces, in "Applied and Industrial Mathematics in Italy: No. III" (eds. E. De Bernardis, R. Spigler and V. Valente), World Scientific Publishing, Singapore, (2009), 51-61. [8] A. Barbagallo and S. Pia, Weighted variational inequalities in non-pivot Hilbert spaces with applications, Comput. Optim. Appl., 48 (2011), 487-514. doi: 10.1007/s10589-009-9259-0. [9] J. M. Borwein and A. S. Lewis, Practical conditions for Fenchel duality in infinite dimensions, in "Fixed Point Theory and Applications" (Marseille, 1989), Pitman Res. Notes Math. Ser., 252, Longman Sci. Tech., Harlow, (1991), 83-89. [10] J. M. Borwein and A. S. Lewis, Partially finite convex programming. I. Quasi relative interiors and duality theory, Math. Programming, 57 (1992), 15-48. doi: 10.1007/BF01581072. [11] P. Daniele and S. Giuffré, General infinite dimensional duality and applications to evolutionary networks and equilibrium problems, Optim. Lett., 1 (2007), 227-243. doi: 10.1007/s11590-006-0028-z. [12] P. Daniele, S. Giuffré and A. Maugeri, Remarks on general infinite dimensional duality with cone and equality constraints, Commun. Appl. Anal., 13 (2009), 567-577. [13] P. Daniele, A. Maugeri and W. Oettli, Variational inequalities and time-dependent traffic equilibria, C. R. Acad. Sci. Paris, 326 (1998), 1059-1062. [14] P. Daniele, A. Maugeri and W. Oettli, Time-dependent traffic equilibria, J. Optim. Theory Appl., 103 (1999), 543-555. doi: 10.1023/A:1021779823196. [15] P. Daniele, S. Giuffré, G. Idone and A. Maugeri, Infinite dimensional duality and applications, Math. Ann., 339 (2007), 221-239. doi: 10.1007/s00208-007-0118-y. [16] M. B. Donato, A. Maugeri, M. Milasi and C. Vitanza, Duality theory for a dynamic Walrasian pure exchange economy, Pac. J. Optim., 4 (2008), 537-547. [17] S. Giuffré and S. Pia, Weighted traffic equilibrium problem in non pivot Hilbert spaces, Nonlinear Anal., 71 (2009), e2054-e2061. [18] J. Jahn, "Introduction to the Theory of Nonlinear Optimization," Second edition, Springer-Verlag, Berlin, 1996. [19] K. Kuratowski, "Topology," Vol. I, Academic Press, New York-London, Państwowe Wydawnictwo Naukowe, Warsaw, 1966. [20] A. Maugeri and F. Raciti, On general infinite dimensional complementarity problems, Optim. Lett., 2 (2008), 71-90. [21] A. Maugeri and F. Raciti, On existence theorems for monotone and nonmonotone variational inequalities, J. Convex Anal., 16 (2009), 899-911. [22] A. Maugeri and F. Raciti, Remarks on infinite dimansional duality, J. Global Optim., 46 (2010), 581-588. doi: 10.1007/s10898-009-9442-y. [23] C. Ratti, R. M. Pulselli, S. Williams and D. Frenchman, Mobile landscapes: Using location data from cell-phones for urban analysis, Environment and Planning B: Planning and Design, 33 (2006), 727-748. doi: 10.1068/b32047. [24] G. Salinetti and R. J.-B. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev., 21 (1979), 18-33. doi: 10.1137/1021002. [25] G. Salinetti and R. J.-B. Wets, Addendum: On the convergence of convex sets in finite dimensions, SIAM Rev., 22 (1980), 86. doi: 10.1137/1022004. [26] E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, in "Contributions to Nonlinear Functional Analysis" (ed. E.H. Zarantonello), Academic Press, New York, (1971), 237-341.

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##### References:
 [1] J.-P. Aubin, "Analyse Fonctionnelle Appliquée. Tome 2," Translated from the English, Mathématiques, Presses Universitaires de France, Paris, 1987. [2] A. Barbagallo, Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems, J. Global Optim., 40 (2008), 29-39. doi: 10.1007/s10898-007-9194-5. [3] A. Barbagallo, Existence and regularity of solutions to nonlinear degenerate evolutionary variational inequalities with applications to dynamic network equilibrium problems, Appl. Math. Comput., 208 (2009), 1-13. doi: 10.1016/j.amc.2008.10.030. [4] A. Barbagallo, On the regularity of retarded equilibrium in time-dependent traffic equilibrium problems, Nonlinear Anal., 71 (2009), e2406-e2417. doi: 10.1016/j.na.2009.05.054. [5] A. Barbagallo and M.-G. Cojocaru, Continuity of solutions for parametric variational inequalities in Banach space, J. Math. Anal. Appl., 351 (2009), 707-720. doi: 10.1016/j.jmaa.2008.10.052. [6] A. Barbagallo and A. Maugeri, Duality theory for the dynamic oligopolistic market equilibrium problem, Optimization, 60 (2011), 29-52. doi: 10.1080/02331930903578684. [7] A. Barbagallo and S. Pia, Weighted traffic equilibrium problem with delay in non-pivot Hilbert spaces, in "Applied and Industrial Mathematics in Italy: No. III" (eds. E. De Bernardis, R. Spigler and V. Valente), World Scientific Publishing, Singapore, (2009), 51-61. [8] A. Barbagallo and S. Pia, Weighted variational inequalities in non-pivot Hilbert spaces with applications, Comput. Optim. Appl., 48 (2011), 487-514. doi: 10.1007/s10589-009-9259-0. [9] J. M. Borwein and A. S. Lewis, Practical conditions for Fenchel duality in infinite dimensions, in "Fixed Point Theory and Applications" (Marseille, 1989), Pitman Res. Notes Math. Ser., 252, Longman Sci. Tech., Harlow, (1991), 83-89. [10] J. M. Borwein and A. S. Lewis, Partially finite convex programming. I. Quasi relative interiors and duality theory, Math. Programming, 57 (1992), 15-48. doi: 10.1007/BF01581072. [11] P. Daniele and S. Giuffré, General infinite dimensional duality and applications to evolutionary networks and equilibrium problems, Optim. Lett., 1 (2007), 227-243. doi: 10.1007/s11590-006-0028-z. [12] P. Daniele, S. Giuffré and A. Maugeri, Remarks on general infinite dimensional duality with cone and equality constraints, Commun. Appl. Anal., 13 (2009), 567-577. [13] P. Daniele, A. Maugeri and W. Oettli, Variational inequalities and time-dependent traffic equilibria, C. R. Acad. Sci. Paris, 326 (1998), 1059-1062. [14] P. Daniele, A. Maugeri and W. Oettli, Time-dependent traffic equilibria, J. Optim. Theory Appl., 103 (1999), 543-555. doi: 10.1023/A:1021779823196. [15] P. Daniele, S. Giuffré, G. Idone and A. Maugeri, Infinite dimensional duality and applications, Math. Ann., 339 (2007), 221-239. doi: 10.1007/s00208-007-0118-y. [16] M. B. Donato, A. Maugeri, M. Milasi and C. Vitanza, Duality theory for a dynamic Walrasian pure exchange economy, Pac. J. Optim., 4 (2008), 537-547. [17] S. Giuffré and S. Pia, Weighted traffic equilibrium problem in non pivot Hilbert spaces, Nonlinear Anal., 71 (2009), e2054-e2061. [18] J. Jahn, "Introduction to the Theory of Nonlinear Optimization," Second edition, Springer-Verlag, Berlin, 1996. [19] K. Kuratowski, "Topology," Vol. I, Academic Press, New York-London, Państwowe Wydawnictwo Naukowe, Warsaw, 1966. [20] A. Maugeri and F. Raciti, On general infinite dimensional complementarity problems, Optim. Lett., 2 (2008), 71-90. [21] A. Maugeri and F. Raciti, On existence theorems for monotone and nonmonotone variational inequalities, J. Convex Anal., 16 (2009), 899-911. [22] A. Maugeri and F. Raciti, Remarks on infinite dimansional duality, J. Global Optim., 46 (2010), 581-588. doi: 10.1007/s10898-009-9442-y. [23] C. Ratti, R. M. Pulselli, S. Williams and D. Frenchman, Mobile landscapes: Using location data from cell-phones for urban analysis, Environment and Planning B: Planning and Design, 33 (2006), 727-748. doi: 10.1068/b32047. [24] G. Salinetti and R. J.-B. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev., 21 (1979), 18-33. doi: 10.1137/1021002. [25] G. Salinetti and R. J.-B. Wets, Addendum: On the convergence of convex sets in finite dimensions, SIAM Rev., 22 (1980), 86. doi: 10.1137/1022004. [26] E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, in "Contributions to Nonlinear Functional Analysis" (ed. E.H. Zarantonello), Academic Press, New York, (1971), 237-341.
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