1. 1 Static and Dynamic Networks Patrizia Daniele Department of Mathematics and Computer Sciences University of Catania, ITALY Visiting Scholar at DEAS (Harvard University) Spring Semester 2006

2. 2 Edward Elgar Publishing August 2006.

3. 3 Acknowledgements 1 Introduction 2 The Traffic Equilibrium Problem 2.1 Static Case 2.2 Dynamic Case 2.3 Existence Theorems 2.4 Additional Constraints 2.5 Calculation of the Solution 2.6 Delay and Elastic Model 2.7 Sources and Remarks 3 Evolutionary Spatial Price Equilibrium 3.1 The Price Formulation 3.2 The Quantity Formulation 3.3 Economic Model for Demand- Supply Markets 3.4 Sources and Remarks 4 The Evolutionary Financial Model 4.1 Quadratic Utility Function 4.2 General Utility Function 4.3 Policy Intervention 4.4 A Numerical Financial Example 4.5 Sources and Remarks 5 Projected Dynamical Systems 5.1 Finite-Dimensional PDS 5.2 Infinite-Dimensional PDS 5.3 Common Formulation 5.4 General Uniqueness Results 5.5 The Relation Between the Two Timeframes 5.6 Computational Procedure 5.7 Numerical Dynamic Traffic Examples 5.8 Sources and Remarks A Definitions and Properties B Weak Convergence C Generalized Derivatives D Variational Inequalities E Quasi-Variational Inequalities F Infinite Dimensional Duality BibliographyIndex

4. 4 Wardrop J.G. (1952), Some theoretical aspects of road traffic research, Proceedings of the Institute of Civil Engineers, Part II, pp. 325- 378. Beckmann M.J., McGuire C.B. and Winsten C.B. (1956), Studies in the Economics of Transportation, Yale University Press, New Haven, Connecticut. Dafermos S.C. and Sparrow F.T. (1969), The traffic assignment problem for a general nerwork, Journal of Research of the National Bureau of Standards 73B, 91-118. Smith M.J. (1979), The Existence, Uniqueness and Stability of Traffic Equilibrium, Transportation Research, 138, 1979, 295-304. Dafermos S. (1980), Traffic equilibria and variational inequalities, Transportation Science 114, 42-54. Kinderlehrer D. and Stampacchia G. (1980), An Introduction to Variational Inequalities and Their Applications, Academic Press, New York.

5. 5 Static Traffic Network

6. 6 Example P4P4 P1P1 P3P3 P2P2

7. 7

8. 8

9. 9

10. 10 Wardrops Principle (1952): Wardrops Principle (1952):

11. 11 Theorem (Smith, 1979): Existence Theorem Theorem 1:

12. 12 Existence and Uniqueness Theorems Theorem 2: Theorem 3:

13. 13 Direct Method Initial VI: Initial VI: New feasible set: New feasible set: New VI: New VI:

14. 14 Step 1: Step 1: Step 2 (First Type Face): Step 2 (First Type Face):

15. 15 New VI: Step 3 (Second Type Face): Step 3 (Second Type Face):

16. 16 New VI: Step 4: mixture of steps 2 and 3 Step 4: mixture of steps 2 and 3

17. 17 Numerical Example P4P4 P1P1 P3P3 P2P2

18. 18 VI:

19. 19

20. 20 Dynamic Traffic Network Ran B. and Boyce D.E. (1996), Modeling Dynamic Transportation Networks: an Intelligent System Oriented Approach, Second Edition, Springer-Verlag, Berlin, Germany. Daniele P., Maugeri A., Oettli W. (1999), Time-Dependent Variational Inequalities, Journal of Optimization Theory and Applications, 104, 543-555. Raciti F. (2001), Equilibrium in Time-Dependent Traffic Networks with Delay, in Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, F. Giannessi, A. Maugeri and P.M. Pardalos (eds), Kluwer Academic Publishers, The Netherlands, 247-253. Scrimali L. (2004), Quasi-Variational Inequalities in Transportation Networks, M3AS: Mathematical Models and Methods in Applied Sciences, 14, 1541-1560.

21. 21 Generalized Wardrops Principle:

22. 22 Canonical Bilinear form on L * x L Canonical Bilinear form on L * x L : Theorem (Variational Formulation)

23. 23 Preliminary Definitions E real topological vector space, K E convex, C: K E * : C hemicontinuous if upper semicontinuous on K; C hemicontinuous if upper semicontinuous on K; C hemicontinuous along line segments if upper semicontinuous on the line segment [x, y]; C hemicontinuous along line segments if upper semicontinuous on the line segment [x, y]; C pseudomonotone if C pseudomonotone if

24. 24 Existence Theorems Theorem 1 (Daniele, Maugeri and Oettli, 1999) X real topological vector space, K X nonempty and convex; F: K X *: 1. 1. 2.F pseudomonotone and hemicontinuous along line segments. Then,

25. 25 Theorem 2 (Daniele, Maugeri and Oettli, 1999) X real topological vector space, K X nonempty and convex; F: K X *: 1. 1. 2.F hemicontinuous. Then,

26. 26 Corollary (Daniele, Maugeri and Oettli, 1999) K L and C: K L *. Each of the following conditions is sufficient for the existence of the solution to the EVI: 1.C strongly hemicontinuous on K and A K nonempty and compact and B K compact: 2.C weakly hemicontinuous on K; 3.C pseudomonotone and hemicontinuous along line segments.

27. 27 Spatial Market Networks Nagurney A. (1993), Network Economics - A Variational Inequality Approach, Kluwer Academic Publishers, Dordrecht, The Netherlands. Nagurney A. and Zhang D. (1996), Stability of Spatial Price Equilibrium Modeled as a Projected Dynamical System, Journal of Economic Dynamics and Control, 20, 43-63. Daniele P. (2001), Variational Inequalities for Static Equilibrium Market: Lagrangean Function and Duality, in Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, F. Giannessi, A. Maugeri and P. Pardalos (eds), Kluwer Academic Publishers, Dordrecht, The Netherlands, 43-58. Daniele P. (2003), Evolutionary Variational Inequalities and Economic Models for Demand-Supply Markets, M3AS: Mathematical Models and Methods in Applied Sciences, 4, no. 13, 471-489.

28. 28 Economic Model P1P1 PnPn P2P2 QmQm Q2Q2 Q1Q1 Supply Markets Demand Markets … … … …

29. 29 g R n : total supply vector g R n : total supply vector p R n : total supply price vector p R n : total supply price vector f R m : total demand vector f R m : total demand vector q R m : total demand price vector q R m : total demand price vector x R nm : flow vector x R nm : flow vector c R nm : flow cost vector c R nm : flow cost vector s R n : supply excess vector s R n : supply excess vector τ R m : demand excess vector τ R m : demand excess vector Static Case

30. 30 Set of feasible vectors: Given functions:

31. 31 Definition (Equilibrium Conditions)

32. 32 Theorem (Variational Formulation)

33. 33 Dynamic Case g(t):[0,T] L 2 ([0,T], R n ): total supply function g(t):[0,T] L 2 ([0,T], R n ): total supply function p(t):[0,T] L 2 ([0,T], R n ): supply price function p(t):[0,T] L 2 ([0,T], R n ): supply price function f (t):[0,T] L 2 ([0,T], R m ): total demand function f (t):[0,T] L 2 ([0,T], R m ): total demand function q(t):[0,T] L 2 ([0,T], R m ): demand price function q(t):[0,T] L 2 ([0,T], R m ): demand price function x(t):[0,T] L 2 ([0,T], R nm ): flow function x(t):[0,T] L 2 ([0,T], R nm ): flow function c(t):[0,T] L 2 ([0,T], R nm ): unit cost function c(t):[0,T] L 2 ([0,T], R nm ): unit cost function s(t):[0,T] L 2 ([0,T], R n ): supply excess function s(t):[0,T] L 2 ([0,T], R n ): supply excess function τ(t):[0,T] L 2 ([0,T], R m ): demand excess function τ(t):[0,T] L 2 ([0,T], R m ): demand excess function

34. 34 Set of feasible vectors: Given functions:

35. 35 Definition (Equilibrium Conditions)

36. 36 Theorem (Variational Formulation)

37. 37 Theorem (Existence) K L and v: K L *. Each of the following conditions is sufficient for the existence of the solution to the EVI: 1. 1. 2. 2. 3. 3.

38. 38 General Formulation of the set K for traffic network problems, spatial price equilibrium problems, financial equilibrium problems

39. 39 Discretization Procedure

40. 40

41. 41

42. 42

43. 43 Theorem

44. 44 Lemma Theorem

45. 45 Continuity assumption Projected Dynamical System (PDS)

46. 46 Dupuis P. and Nagurney A. (1993), Dynamical Systems and Variational Inequalities, Annals of Operations Research, 44, 9-42. Nagurney A. and Zhang D. (1996), Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, Boston, Massachusetts. Cojocaru M.G. (2002), Projected Dynamical Systems on Hilbert Spaces, PhD Thesis, Queens University, Canada. Cojocaru M.G. and Jonker L.B. (2004), Existence of Solutions to Projected Differential Equations in Hilbert Spaces, Proceedings of the American Mathematical Society, 132, 183-193. Cojocaru M.G., Daniele P. and Nagurney A. (2005), Projected Dynamical Systems and Evolutionary (Time-Dependent) Variational Inequalities Via Hilbert Spaces with Applications, Journal of Optimization Theory and Applications, 27, no. 3, 1-15.

47. 47 Discretization the time interval [0,T];Discretization the time interval [0,T]; Sequence of PDS;Sequence of PDS; Calculation of the critical points of the PDS;Calculation of the critical points of the PDS; Interpolation of the sequence of critical points;Interpolation of the sequence of critical points; Approximation of the equilibrium curve.Approximation of the equilibrium curve. New Discretization Procedure

48. 48 Traffic Network D O

49. 49

50. 50

51. 51 Direct Method

52. 52

53. 53

54. 54 Traffic Network 0 D

55. 55

56. 56 t0t0t0t0 H1(t0)H1(t0)H1(t0)H1(t0) H2(t0)H2(t0)H2(t0)H2(t0) 000 1/41/40 1/21/20 3/43/40 110 5/49/81/8 3/25/41/4 7/411/83/8 23/21/2

57. 57