Southern Ontario Dynamics Day April 07, 2006 1 Dynamic Networks and Evolutionary Variational...

Southern Ontario Dynamics Day ApriSouthern Ontario Dynamics Day April 07, 2006l 07, 2006

11

Dynamic Networks and Dynamic Networks and Evolutionary Variational Evolutionary Variational

InequalitiesInequalities

Patrizia Daniele Patrizia Daniele Department of Mathematics and Computer SciencesDepartment of Mathematics and Computer Sciences

University of CataniaUniversity of CataniaVisiting Scholar at DEAS (Harvard University) Visiting Scholar at DEAS (Harvard University)

Spring Semester 2006Spring Semester 2006


22

1.1. Lions, J. L., and Stampacchia, G. (1967), “Variational Inequalities”, Lions, J. L., and Stampacchia, G. (1967), “Variational Inequalities”, Comm. Pure Appl. Math.Comm. Pure Appl. Math., , 2222, 493-519. , 493-519.

2.2. Brezis, H. (1967), “Inequations d'Evolution AbstraitesBrezis, H. (1967), “Inequations d'Evolution Abstraites””, , Comptes Comptes Rendus de l'Academie des SciencesRendus de l'Academie des Sciences, Paris., Paris.

3.3. Steinbach, J. (1998), Steinbach, J. (1998), ““On a Variational Inequality Containing a Memory On a Variational Inequality Containing a Memory Term with an Application in Electro-Chemical MachiningTerm with an Application in Electro-Chemical Machining””, , Journal of Journal of Convex AnalysisConvex Analysis, , 55, 63-80., 63-80.

4.4. Daniele, P., Maugeri, A., and Oettli, W. (1998), Daniele, P., Maugeri, A., and Oettli, W. (1998), ““Variational Variational Inequalities and Time-Dependent Traffic EquilibriaInequalities and Time-Dependent Traffic Equilibria””, , Comptes Rendus Comptes Rendus de l'Academie des Sciencesde l'Academie des Sciences, Paris, , Paris, 326326, Serie I, 1059-1062., Serie I, 1059-1062.

5.5. Daniele, P., Maugeri, A., and Oettli, W. (1999), Daniele, P., Maugeri, A., and Oettli, W. (1999), ““Time-Dependent Time-Dependent Variational InequalitiesVariational Inequalities””, , Journal of Optimization Theory and Journal of Optimization Theory and ApplicationsApplications, , 104104, 543-555., 543-555.


33

6.6. Raciti, F. (2001), Raciti, F. (2001), ““Equilibrium in Time-Dependent Traffic Networks Equilibrium in Time-Dependent Traffic Networks with Delaywith Delay””, in , in Equilibrium Problems: Nonsmooth Optimization Equilibrium Problems: Nonsmooth Optimization andandVariational Inequality ModelsVariational Inequality Models, F. Giannessi, A. Maugeri and P.M. , F. Giannessi, A. Maugeri and P.M. Pardalos Pardalos ((edseds)), Kluwer Academic Publishers, The Netherlands, 247-253., Kluwer Academic Publishers, The Netherlands, 247-253.

7.7. Gwinner, J. (2003), Gwinner, J. (2003), ““Time Dependent Variational Inequalities - Some Time Dependent Variational Inequalities - Some Recent TrendsRecent Trends””, in , in Equilibrium Problems and Variational ModelsEquilibrium Problems and Variational Models, P. , P. Daniele, F. Giannessi and A. Maugeri Daniele, F. Giannessi and A. Maugeri ((edseds)), Kluwer Academic , Kluwer Academic Publishers, Dordrecht, The Netherlands, 225-264.Publishers, Dordrecht, The Netherlands, 225-264.

8.8. Scrimali, L. (2004), Scrimali, L. (2004), ““Quasi-Variational Inequalities in Transportation Quasi-Variational Inequalities in Transportation NetworksNetworks””, , M3AS: Mathematical Models and Methods in Applied M3AS: Mathematical Models and Methods in Applied SciencesSciences, , 1414, 1541-1560., 1541-1560.

9.9. Daniele, P., and Maugeri, A. (2001), Daniele, P., and Maugeri, A. (2001), ““On Dynamical Equilibrium On Dynamical Equilibrium Problems and Variational InequalitiesProblems and Variational Inequalities””, in , in Equilibrium Problems: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality ModelsNonsmooth Optimization and Variational Inequality Models , F. , F. Giannessi, A. Maugeri, and P.M. Pardalos (eds), Kluwer Academic Giannessi, A. Maugeri, and P.M. Pardalos (eds), Kluwer Academic Publishers, Dordrecht, Netherlands, 59-69.Publishers, Dordrecht, Netherlands, 59-69.


44

10.10. Daniele, P. (2003), Daniele, P. (2003), ““Evolutionary Variational Inequalities and Evolutionary Variational Inequalities and Economic Models for Demand Supply MarketsEconomic Models for Demand Supply Markets””, , M3AS: Mathematical M3AS: Mathematical Models and Methods in Applied SciencesModels and Methods in Applied Sciences, , 44, 471-489., 471-489.

11.11. Daniele, P. (2004), Daniele, P. (2004), ““Time-Dependent Spatial Price Equilibrium Time-Dependent Spatial Price Equilibrium Problem: Existence and Stability Results for the Quantity Formulation Problem: Existence and Stability Results for the Quantity Formulation ModelModel””, , Journal of Global OptimizationJournal of Global Optimization, , 2828, 283-295., 283-295.

12.12. Daniele, P. (2003), “Variational Inequalities for Evolutionary Financial Daniele, P. (2003), “Variational Inequalities for Evolutionary Financial EquilibriumEquilibrium””, in , in Innovations in Financial and Economic NetworksInnovations in Financial and Economic Networks, , A. Nagurney (ed), Edward Elgar Publishing, Cheltenham, England, 84-A. Nagurney (ed), Edward Elgar Publishing, Cheltenham, England, 84-109.109.

13.13. Cojocaru, M.G., Daniele, P., and Nagurney, A. (2005), “Projected Cojocaru, M.G., Daniele, P., and Nagurney, A. (2005), “Projected Dynamical Systems and Evolutionary Dynamical Systems and Evolutionary ((Time-DependentTime-Dependent)) Variational Variational Inequalities Via Hilbert Spaces with ApplicationsInequalities Via Hilbert Spaces with Applications””, , Journal of Journal of Optimization Theory and ApplicationsOptimization Theory and Applications, , 2727, no.3, 1-15., no.3, 1-15.


55

14.14. Cojocaru, M.G., Daniele P., and Nagurney, A. (2005),Cojocaru, M.G., Daniele P., and Nagurney, A. (2005), “ “Double-Layered Double-Layered Dynamics: A Unified Theory of Projected Dynamical Systems and Dynamics: A Unified Theory of Projected Dynamical Systems and Evolutionary Variational InequalitiesEvolutionary Variational Inequalities””, European Journal of Operational , European Journal of Operational Research.Research.

15.15. Cojocaru, M.G., Daniele P., and Nagurney, A. (2005), “Projected Cojocaru, M.G., Daniele P., and Nagurney, A. (2005), “Projected Dynamical Systems, Evolutionary Variational Inequalities, Dynamical Systems, Evolutionary Variational Inequalities, Applications, and a Computational ProcedureApplications, and a Computational Procedure””, in , in Pareto Optimality, Pareto Optimality, Game Theory and EquilibriaGame Theory and Equilibria, A. Migdalas, P. Pardalos and L. , A. Migdalas, P. Pardalos and L. Pitsoulis Pitsoulis ((edseds)), NOIA Series, Springer, Berlin, Germany. , NOIA Series, Springer, Berlin, Germany.

16.16. Nagurney, A., Liu, Z., Cojocaru, M.G., Daniele, P. (2006), “Static and Nagurney, A., Liu, Z., Cojocaru, M.G., Daniele, P. (2006), “Static and Dynamic Transportation Network Equilibrium Reformulations of Dynamic Transportation Network Equilibrium Reformulations of Electric Power Supply Chain Networks with Known Demands”, Electric Power Supply Chain Networks with Known Demands”, Transportation Research E, to appear.Transportation Research E, to appear.

17.17. Daniele P. (2006), “Dynamic Networks and Evolutionary Variational Daniele P. (2006), “Dynamic Networks and Evolutionary Variational Inequalities”, Edward Elgar Publishing.Inequalities”, Edward Elgar Publishing.


66

General Formulation of the set General Formulation of the set KK for traffic for traffic network problems, spatial price equilibrium network problems, spatial price equilibrium problems, financial equilibrium problemsproblems, financial equilibrium problems

2

1

0, , : ( ) ( ) ( ) a.e.in 0, ;

( ) ( ) a.e. in 0 , 0,1 , 1,..., , 1,...,

, , 0, , : given functions

q

q

ji i j jii

p q

u t L T t u t t T

u t t ,T i q j l

L T

K R

R


77

Discretization ProcedureDiscretization Procedure

1

0 1 0 1

1

,

Partitions of 0, :

, , , ,0

max : 1, , , lim 0.

0, , 0, , : ,

1, ,

n n

j jn n

N Nn n n n n n n

j jn n n n n

n

m m mn jt t

n

T

t t t t t t T

t t j N

P T v L T v v

j N

R R R


88

1 1

2

1,

2

2

1:

0, , : , a.e. in 0, ,

, , 0

, ,

0, , , 0, ,

jn

j j jn n n

t

n j jt t tn n

m

m m

v v s dst t

F t L T F t T

F t

C t F t A t F t B t

A t L T B t L T

K R

R R


99

1

1

0

1

,

,

:

, 0,

jnn

jn

jn

jn

T

Nt

tj

nj

t n n nj j jt

nj

A t H t B t F t H t dt


H t


F t

K

K


1010

1 11 1

1

1

Finite-dimensional Variational Inequality:

: , 0,

where

1 1; .

, piecewise constant

approximation to th

j jn n

j jn n

n

n n n n n n nj m j j j j j j m

t tn nj jj j j jt t

n n n n

Nj j n

n n n jj

H A H B F H F

A A t dt B B t dtt t t t

H t t t H

K K

e solution of EVI


1111

TheoremTheorem positive definite a.e. in 0, the set

(weakly) compact, with feasible cluster points solutions

to the EVI.

n nA t T U H

2

, , 1

, 1

0, , : ( ) ( ) ( ) a.e.in 0, ;

, 0, ( ) a.e. in 0

: : a.e.in , ;

a.e. in ,

m

n mj n j nn j j j j j

j nj j j

F t L T t F t t T

t t F t t ,T

F F t t

F t t

K R

K R


1212

LemmaLemma nn j

j K K K

0

Initial problem:

, , 0T

H t t

C t H t F t H t dt F t t

K

K

TheoremTheorem

1

1

positive definitea.e. in 0, ,

has (weakly) cluster points solutions to the initial problem.

nNj j n

n n n jj

A t T H t t t H


1313

0

strictly monotone

0,T , uniquesolution to the EVI:

, 0, 0,

q

F

u t C

F u t v t u t t T

R

•Discretization the time interval [0,Discretization the time interval [0,TT];];•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.

Continuity assumptionContinuity assumption


1414

Traffic NetworkTraffic Network

2 4 2 2 20 1 1 2j j n

n n n n n

D

O

1

2

2 21 2

1 2

32 0

, , ,20 1

1

0,2 , :0 ; 0 3;

a.e.in 0,2

F ttA t F t B t

F t

F L F t t F t

F t F t t

K R


1515

21 2 1 2

1 1 1 2 2 2

2 1

1 1 1

, 0

2 1 2 1:0 , 0 3,

2 1 32 1 0

2

2 1

2 1 3 2 12 1

2

n n n n n n nj j j j j j j

n n n n n nj j j j j j

n n n n n n n nj j j j j j j j

n nj j

n nj j j

A H B F H F

j jF F F F F

n n

jH F H H F H F

n

jF F

n

j jH H F

n n

K

K R

K

1 0n njH


1616

1

2 22

1

2 1 1if 1 1

2 2 6

4 2 1if 1

2 3 2 1 2 2 6

10 if 1 1

2 2 6

4 2 1 2 2 1 1if 1

2 3 2 1 2 2 6

2 21 ,

nj

nj

nn

n jj

j nj

nH

j n nj

n j

nj

Hn j n j n

jn n j

H t j j Hn n


1717

Direct MethodDirect Method

1 1 1 2 2 2

2 21

2 1 2

22 1 1 1

1 1 1 1

32 1 0

2

0,2 , : 0 ;

0 3; , a.e. in 0, 2

0,2 , :0

13 2 0

2

t H t F t H t H t F t H t

F t F t L F t t

F t F F t t

F t t F t F t L F t t

t H t t F t H t F t

K R

K R

K


1818

1

2 2

1 2 1 2

6if 0 1

2

2 1 6if 1 2

2 6 2

60 if 0 1

2

2 4 1 6if 1 2

2 6 2

, ,n n

t t

H tt

tt

t

H tt t

tt

H t H t H t H t


1919


2020

Traffic NetworkTraffic Network

2 21 2

1 2

30,2 , : 0 ( ) ,0 a.e.in 0,2 ;

2

a.e. in 0 2

F t L F t t F t t

F t F t t ,

K R

0

D

1 1

2 2

Cost functions:

1

2

C H t H t

C H t H t


2121

0

2 2 2 2

1 2 1 2

0

1 0 2 0 1 2

0 0 0

Vector field:

: 0,2 , 0,2 ,

, 1, 2 .

: 0, ,8 sequence of PDS defined by :4

, 1, 2 on

30, 0,

2t

F L L

F H t F H t H t H t

kt k

F H t H t H t H t

t t x y t

R R

K


2222

tt00 HH11((tt00)) HH22((tt00))

00 00 00

1/41/4 1/41/4 00

1/21/2 1/21/2 00

3/43/4 3/43/4 00

11 11 00

5/45/4 9/89/8 1/81/8

3/23/2 5/45/4 1/41/4

7/47/4 11/811/8 3/83/8

22 3/23/2 1/21/2

1

2

Explicit formulas:

if 0 1

1if 1 2

2

0 if 0 1

1if 1 2

2

t tH t t

t

tH t t

t

0

0

21 0 2 0

1 0 2 0 1 0 2 0

Unique equilibrium at :

, :

, ,t

t

H t H t

F H t H t N H t H t

K

R


2323


2424

Economic MarketsEconomic Markets

1 1 2 2

1 1

11 11 21 21

; ;

;

; 1;

g p t p t h t g p t p t k t

f q t k t q t

c x t x t c x t x t

P1

Q1

P2

2 2 2 2 2

1 2 1

11 21

1 1 1, , 0, , 0, , 0, , :

2 2 2

0 1,0 1,0 1,

0 1,0 1

u t p t q t x t L L L

p t h t k t p t h t k t q t h t k t

x t h t k t x t h t k t

K R R R


2525

Direct MethodDirect Method

1 11

2 21

1 11 21

1 11 1

2 21 1

2

1 2 1

11 21

0

0

0 0

0

1 0

3 1 00 :

2 3 0

3 1 2 2; 0; ;

5 52 1

;5

p t x t h t

p t x t k t

v u t q t x t x t k t

p t x t q tu t

p t x t q t

u t

h t k tp t

h t k t

h t k t h t k tp t p t q t

h t k t h tx t x t

K

K

K

1 2 1

11 21 2

2 3;

52 1 3 2

; ; ;5 5

2 1 2 2 3 3; ; .

5 5 5

k t

h t k t h t k tg p t g p t k t f q t

h t k t h t k t h t k tc x t c x t s t


2626

1* *2

0, 0,

1 0 0 1 0

0 1 0 0 1

,0 0 1 1 1

1 0 1 1 0 0

0 1 1 0 1 1

A t u t b t u t u t dt u t

h t

k t

A b t k t

K

8,

2 7

th t k t t

Discretization ProcedureDiscretization Procedure


2727

1 1 1 10, :0

2 2 2 2 2

j j

n n n

1 2 1 11 21 1 2

1 11 21

* *

6 3 15 6 3 15, , , , :0 ,0 ,

8 7 8 76 3 15 6 3 15 6 3 15

0 , 0 ,08 7 8 7 8 7

2 1

82 1 8

4 7, 0 where

2 1 8

4 70

1

n n n n n n n n nj j j j j j j j j

n n nj j j

n n n n n n nj j j j j j j

j ju p p q x x p p

n nj j j

q x xn n n

j

nj

nA u b u u u b

j

n

K

K


2828

2

2

1 2 1

11 21

1 2 1

11

0 0; 0 :

0

2 1 3 6 3 6; 0; ;

40 7 40 72 1 3 6 3 1

; ;10 7 40 7

2 1 3 2 1 8 14 7 2; ; ;

10 7 4 7 40 72 1 3

10 7

n n n nj j j jn n

j jn n nj j j

n n nj j j

n nj j

n n nj j j

nj

Au b Au bp

u p

j jp p q

n nj j

x xn n

j j jg p g p f q

n n nj

c xn

KK

21

2

1

6 3 6; ;

40 714 7 6

Excess: ;40 7

1Solution on : , .

2 2

nj

nj

nn n n

j n jj

jc x

nj

sn

j ju t u

n n

K K


2929


3030

1.1. Nagurney, A., Dong, J., and Hughes, M. Nagurney, A., Dong, J., and Hughes, M. ((19921992)), “Formulation and , “Formulation and Computation of General Financial Equilibrium”, Computation of General Financial Equilibrium”, OptimizationOptimization 2626, 339-, 339-354.354.

2.2. Nagurney, A.Nagurney, A. ( (19941994)), “Variational Inequalities in the Analysis and , “Variational Inequalities in the Analysis and Computation of Multi-Sector, Multi-Instrument FinancialComputation of Multi-Sector, Multi-Instrument FinancialEquilibria”, Equilibria”, Journal of Economic Dynamics and Control Journal of Economic Dynamics and Control 1818, 161-184., 161-184.

3.3. Nagurney, A., and Siokos, S. Nagurney, A., and Siokos, S. ((1997a1997a)), “Variational Inequalities for , “Variational Inequalities for International General Financial Equilibrium Modeling and International General Financial Equilibrium Modeling and Computation”, Computation”, Mathematical and Computer Modelling Mathematical and Computer Modelling 2525, 31-49., 31-49.

4.4. Nagurney, A., and Siokos, S. Nagurney, A., and Siokos, S. ((1997b1997b)), Financial Networks: Statics and , Financial Networks: Statics and Dynamics, Springer-Verlag, Heidelberg, Germany.Dynamics, Springer-Verlag, Heidelberg, Germany.

5.5. Dong, J., Zhang, D., and Nagurney, A. Dong, J., Zhang, D., and Nagurney, A. ((19961996)), “A Projected Dynamical , “A Projected Dynamical Systems Model of General Financial Equilibrium with Stability Systems Model of General Financial Equilibrium with Stability Analysis”, Analysis”, Mathematical and Computer ModellingMathematical and Computer Modelling 2424, 35-44., 35-44.

Financial MarketsFinancial Markets


3131

m m sectorssectors: households, domestic business, banks, : households, domestic business, banks, financial institutions, state and local governments, financial institutions, state and local governments, ……

nn financial instrumentsfinancial instruments: mortgages, mutual : mortgages, mutual funds, savings deposits, money market funds, …funds, savings deposits, money market funds, …

ssii((tt): ): total financial volume total financial volume theld by sector theld by sector ii at at

time time tt xxijij((tt):): amount of instrumentamount of instrument j j held as an held as an assetasset in in

sectorsector i's i's portfolioportfolio yyijij((tt)): amount of instrument : amount of instrument jj held as a held as a liabilityliability in in

sector sector ii's portfolio's portfolio


3232


3333

11 12

21 22

( ) ( )( ) :

( ) ( )

i ii

i i

Q t Q tQ t

Q t Q t

:)(

)()(

)(

)(

ty

txtQ

ty

tx

i

ii

T

i

i

rrjj((tt): ): price of instrument price of instrument jj at time at time tt

2 2

1

1

( ), ( ) 0, , : ( ) ( ),

( ) ( ), ( ) 0, ( ) 0,a.e.in 0, :

nn

i i i ij ij

n

ij i ij ijj

P x t y t L T x t s t

y t s t x t y t T

R set of set of feasible feasible assets and assets and liabilitiesliabilities

variance-covariance matrixvariance-covariance matrix

aversion to the risk aversion to the risk at time at time tt

Southern Ontario Dynamics Day April 07, 2006 1 Dynamic Networks and Evolutionary Variational...

Documents

Transcript of Southern Ontario Dynamics Day April 07, 2006 1 Dynamic Networks and Evolutionary Variational...