Projected Dynamical Systems, Evolutionary Variational Inequalities, Applications, and a Computational Procedure

In this paper, we establish the equivalence between the solutions to an evolutionary variational inequality and the critical points of a projected dynamical system in infinite–dimensional spaces. We then present an algorithm, with convergence results, for the computation of solutions to evolutionary variational inequalities based on a discretization method and with the aid of projected dynamical systems theory. A numerical traffic network example is given for illustrative purposes.

[1]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[2]  Po-Lung Yu,et al.  An appreciation of professor Rufus Isaacs , 1979 .

[3]  E. H. Zarantonello.  Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory , 1971 .

[4]  Patrizia Daniele.  Time—Dependent Spatial Price Equilibrium Problem: Existence and Stability Results for the Quantity Formulation Model , 2004, J. Glob. Optim..

[5]  F. Raciti.  Bicriterion Weight Varying Spatial Price Networks , 2004 .

[6]  A. Nagurney,et al.  Oligopolistic and competitive behavior of spatially separated markets , 1987 .

[7]  A. Nagurney.  Network Economics: A Variational Inequality Approach , 1992 .

[8]  Antonino Maugeri,et al.  Equilibrium problems and variational models , 2003 .

[9]  P. Daniele,et al.  Variational Inequalities for General Evolutionary Financial Equilibrium , 2005 .

[10]  A. Nagurney,et al.  Projected Dynamical Systems and Evolutionary Variational Inequalities via Hilbert Spaces with Applications1 , 2005 .

[11]  B. Cornet,et al.  Existence of slow solutions for a class of differential inclusions , 1983 .

[12]  中嶋 博.  Convex Programming の新しい方法 (開学記念号) , 1966 .

[13]  Antonino Maugeri,et al.  Time-Dependent Traffic Equilibria , 1999 .

[14]  Panos M. Pardalos,et al.  Network optimization in supply chain management and financial engineering: An annotated bibliography , 2003, Networks.

[15]  A. Nagurney,et al.  On the stability of projected dynamical systems , 1995 .

[16]  M. Cojocaru.  Monotonicity and existence of periodic orbits for projected dynamical systems on Hilbert spaces , 2005 .

[17]  L. Jonker,et al.  Existence of solutions to projected differential equations in Hilbert spaces , 2003 .

[18]  Viale A. Doria,et al.  EVOLUTIONARY VARIATIONAL INEQUALITIES AND ECONOMIC MODELS FOR DEMAND–SUPPLY MARKETS , 2003 .

[19]  F. Cammaroto,et al.  Separation Theorem Based on the Quasirelative Interior and Application to Duality Theory , 2005 .

[20]  A. Nagurney,et al.  Projected Dynamical Systems and Variational Inequalities with Applications , 1995 .

[21]  Anna Nagurney,et al.  Double-layered dynamics: A unified theory of projected dynamical systems and evolutionary variational inequalities , 2006, Eur. J. Oper. Res..

[22]  Anna Nagurney,et al.  Supernetworks: Decision-Making for the Information Age , 2002 .

[23]  Antonino Maugeri,et al.  On Dynamical Equilibrium Problems and Variational Inequalities , 2001 .

[24]  Stella Dafermos,et al.  Traffic Equilibrium and Variational Inequalities , 1980 .

[25]  A. Nagurney.  Migration equilibrium and variational inequalities. , 1989, Economics letters.

[26]  George Isac,et al.  VARIATIONAL INEQUALITIES, COMPLEMENTARITY PROBLEMS AND PSEUDO-MONOTONICITY. DYNAMICAL ASPECTS , 2002 .

[27]  P. Dupuis.  Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets , 1987 .

[28]  P. Pardalos,et al.  Equilibrium problems : nonsmooth optimization and variational inequality models , 2004 .

[29]  A. Nagurney.  Innovations in Financial and Economic Networks , 2003 .

[30]  Anna Nagurney,et al.  Variational Inequalities , 2009, Encyclopedia of Optimization.

[31]  A. Nagurney.  Financial Networks: Statics and Dynamics , 1997 .

[32]  Annamaria Barbagallo,et al.  Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems , 2008, J. Glob. Optim..

[33]  Annamaria Barbagallo,et al.  Regularity Results for Time-Dependent Variational and Quasi-VariationalInequalities and Application to Calculation of Dynamic Traffic Network , 2007 .

[34]  Fabio Raciti.  Equilibria trajectories as stationary solutions of infinite-dimensional projected dynamical systems , 2004, Appl. Math. Lett..

[35]  Antonino Maugeri,et al.  Convex programming, variational inequalities, and applications to the traffic equilibrium problem , 1987 .

[36]  Richard Mounce.  Existence, uniqueness and stability of equilibrium in dynamic traffic networks , 2007 .

[37]  J. Aubin,et al.  Existence of Solutions to Differential Inclusions , 1984 .

[38]  P. Dupuis,et al.  On Lipschitz continuity of the solution mapping to the Skorokhod problem , 1991 .

[39]  J. Aubin,et al.  Differential inclusions set-valued maps and viability theory , 1984 .

[40]  Stella Dafermos,et al.  The general multimodal network equilibrium problem with elastic demand , 1982, Networks.

[41]  Antonino Maugeri,et al.  Variational inequalities and time dependent traffic-equilibria , 1998 .

[42]  George Isac,et al.  The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems , 2004 .

[43]  Anna Nagurney,et al.  Dynamical systems and variational inequalities , 1993, Ann. Oper. Res..

[44]  Joachim Gwinner,et al.  TIME DEPENDENT VARIATIONAL INEQUALITIES - SOME RECENT TRENDS , 2003 .

[45]  C. Henry.  An existence theorem for a class of di erential equations with multival-ued right-hand side , 1973 .

[46]  Jörg Steinbach.  On a variational inequality containing a memory term with an application in electro-chemical machining. , 1998 .

[47]  Carsten Carstensen,et al.  A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems , 1999 .