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 & Continuous Dynamical Systems - A, 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, (1987). Google Scholar

[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. doi: 10.1007/s10898-007-9194-5. Google Scholar

[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. doi: 10.1016/j.amc.2008.10.030. Google Scholar

[4]

A. Barbagallo, On the regularity of retarded equilibrium in time-dependent traffic equilibrium problems,, Nonlinear Anal., 71 (2009). doi: 10.1016/j.na.2009.05.054. Google Scholar

[5]

A. Barbagallo and M.-G. Cojocaru, Continuity of solutions for parametric variational inequalities in Banach space,, J. Math. Anal. Appl., 351 (2009), 707. doi: 10.1016/j.jmaa.2008.10.052. Google Scholar

[6]

A. Barbagallo and A. Maugeri, Duality theory for the dynamic oligopolistic market equilibrium problem,, Optimization, 60 (2011), 29. doi: 10.1080/02331930903578684. Google Scholar

[7]

A. Barbagallo and S. Pia, Weighted traffic equilibrium problem with delay in non-pivot Hilbert spaces,, in, (2009), 51. Google Scholar

[8]

A. Barbagallo and S. Pia, Weighted variational inequalities in non-pivot Hilbert spaces with applications,, Comput. Optim. Appl., 48 (2011), 487. doi: 10.1007/s10589-009-9259-0. Google Scholar

[9]

J. M. Borwein and A. S. Lewis, Practical conditions for Fenchel duality in infinite dimensions,, in, 252 (1991), 83. Google Scholar

[10]

J. M. Borwein and A. S. Lewis, Partially finite convex programming. I. Quasi relative interiors and duality theory,, Math. Programming, 57 (1992), 15. doi: 10.1007/BF01581072. Google Scholar

[11]

P. Daniele and S. Giuffré, General infinite dimensional duality and applications to evolutionary networks and equilibrium problems,, Optim. Lett., 1 (2007), 227. doi: 10.1007/s11590-006-0028-z. Google Scholar

[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. Google Scholar

[13]

P. Daniele, A. Maugeri and W. Oettli, Variational inequalities and time-dependent traffic equilibria,, C. R. Acad. Sci. Paris, 326 (1998), 1059. Google Scholar

[14]

P. Daniele, A. Maugeri and W. Oettli, Time-dependent traffic equilibria,, J. Optim. Theory Appl., 103 (1999), 543. doi: 10.1023/A:1021779823196. Google Scholar

[15]

P. Daniele, S. Giuffré, G. Idone and A. Maugeri, Infinite dimensional duality and applications,, Math. Ann., 339 (2007), 221. doi: 10.1007/s00208-007-0118-y. Google Scholar

[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. Google Scholar

[17]

S. Giuffré and S. Pia, Weighted traffic equilibrium problem in non pivot Hilbert spaces,, Nonlinear Anal., 71 (2009). Google Scholar

[18]

J. Jahn, "Introduction to the Theory of Nonlinear Optimization,", Second edition, (1996). Google Scholar

[19]

K. Kuratowski, "Topology," Vol. I,, Academic Press, (1966). Google Scholar

[20]

A. Maugeri and F. Raciti, On general infinite dimensional complementarity problems,, Optim. Lett., 2 (2008), 71. Google Scholar

[21]

A. Maugeri and F. Raciti, On existence theorems for monotone and nonmonotone variational inequalities,, J. Convex Anal., 16 (2009), 899. Google Scholar

[22]

A. Maugeri and F. Raciti, Remarks on infinite dimansional duality,, J. Global Optim., 46 (2010), 581. doi: 10.1007/s10898-009-9442-y. Google Scholar

[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. doi: 10.1068/b32047. Google Scholar

[24]

G. Salinetti and R. J.-B. Wets, On the convergence of sequences of convex sets in finite dimensions,, SIAM Rev., 21 (1979), 18. doi: 10.1137/1021002. Google Scholar

[25]

G. Salinetti and R. J.-B. Wets, Addendum: On the convergence of convex sets in finite dimensions,, SIAM Rev., 22 (1980). doi: 10.1137/1022004. Google Scholar

[26]

E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets,, in, (1971), 237. Google Scholar

show all references

References:
[1]

J.-P. Aubin, "Analyse Fonctionnelle Appliquée. Tome 2,", Translated from the English, (1987). Google Scholar

[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. doi: 10.1007/s10898-007-9194-5. Google Scholar

[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. doi: 10.1016/j.amc.2008.10.030. Google Scholar

[4]

A. Barbagallo, On the regularity of retarded equilibrium in time-dependent traffic equilibrium problems,, Nonlinear Anal., 71 (2009). doi: 10.1016/j.na.2009.05.054. Google Scholar

[5]

A. Barbagallo and M.-G. Cojocaru, Continuity of solutions for parametric variational inequalities in Banach space,, J. Math. Anal. Appl., 351 (2009), 707. doi: 10.1016/j.jmaa.2008.10.052. Google Scholar

[6]

A. Barbagallo and A. Maugeri, Duality theory for the dynamic oligopolistic market equilibrium problem,, Optimization, 60 (2011), 29. doi: 10.1080/02331930903578684. Google Scholar

[7]

A. Barbagallo and S. Pia, Weighted traffic equilibrium problem with delay in non-pivot Hilbert spaces,, in, (2009), 51. Google Scholar

[8]

A. Barbagallo and S. Pia, Weighted variational inequalities in non-pivot Hilbert spaces with applications,, Comput. Optim. Appl., 48 (2011), 487. doi: 10.1007/s10589-009-9259-0. Google Scholar

[9]

J. M. Borwein and A. S. Lewis, Practical conditions for Fenchel duality in infinite dimensions,, in, 252 (1991), 83. Google Scholar

[10]

J. M. Borwein and A. S. Lewis, Partially finite convex programming. I. Quasi relative interiors and duality theory,, Math. Programming, 57 (1992), 15. doi: 10.1007/BF01581072. Google Scholar

[11]

P. Daniele and S. Giuffré, General infinite dimensional duality and applications to evolutionary networks and equilibrium problems,, Optim. Lett., 1 (2007), 227. doi: 10.1007/s11590-006-0028-z. Google Scholar

[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. Google Scholar

[13]

P. Daniele, A. Maugeri and W. Oettli, Variational inequalities and time-dependent traffic equilibria,, C. R. Acad. Sci. Paris, 326 (1998), 1059. Google Scholar

[14]

P. Daniele, A. Maugeri and W. Oettli, Time-dependent traffic equilibria,, J. Optim. Theory Appl., 103 (1999), 543. doi: 10.1023/A:1021779823196. Google Scholar

[15]

P. Daniele, S. Giuffré, G. Idone and A. Maugeri, Infinite dimensional duality and applications,, Math. Ann., 339 (2007), 221. doi: 10.1007/s00208-007-0118-y. Google Scholar

[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. Google Scholar

[17]

S. Giuffré and S. Pia, Weighted traffic equilibrium problem in non pivot Hilbert spaces,, Nonlinear Anal., 71 (2009). Google Scholar

[18]

J. Jahn, "Introduction to the Theory of Nonlinear Optimization,", Second edition, (1996). Google Scholar

[19]

K. Kuratowski, "Topology," Vol. I,, Academic Press, (1966). Google Scholar

[20]

A. Maugeri and F. Raciti, On general infinite dimensional complementarity problems,, Optim. Lett., 2 (2008), 71. Google Scholar

[21]

A. Maugeri and F. Raciti, On existence theorems for monotone and nonmonotone variational inequalities,, J. Convex Anal., 16 (2009), 899. Google Scholar

[22]

A. Maugeri and F. Raciti, Remarks on infinite dimansional duality,, J. Global Optim., 46 (2010), 581. doi: 10.1007/s10898-009-9442-y. Google Scholar

[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. doi: 10.1068/b32047. Google Scholar

[24]

G. Salinetti and R. J.-B. Wets, On the convergence of sequences of convex sets in finite dimensions,, SIAM Rev., 21 (1979), 18. doi: 10.1137/1021002. Google Scholar

[25]

G. Salinetti and R. J.-B. Wets, Addendum: On the convergence of convex sets in finite dimensions,, SIAM Rev., 22 (1980). doi: 10.1137/1022004. Google Scholar

[26]

E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets,, in, (1971), 237. Google Scholar

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