Double-layered dynamics: A unified theory of projected dynamical systems and evolutionary variational inequalities

Cojocaru, Monica-Gabriela; Daniele, Patrizia; Nagurney, Anna

doi:10.1016/j.ejor.2005.06.008

Cited by 39 publications

(40 citation statements)

References 36 publications

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“…We first solve a test TDEP problem, with known dynamic behaviour, to show that the proposed method finds the known equilibrium solutions of this TDEP. Since the TDEP problem satisfies very strong conditions, the dynamic behaviour of its states is relatively simple, namely, we expect all initial states x.t; 0/ 2 K to evolve, over a finite amount of time steps , to the unique equilibrium curve x 2 K. This example is presented in detail in [14]. Consider a transportation network with one OD pair, with two links; the flow on links is, respectively, x 1 We assume now that the initial condition for problem (5) From the previous dynamic analysis of this problem in [14] we know that the equilibrium curve will be reached in a finite number n of steps .…”

Section: Examples and Discussionmentioning

confidence: 99%

“…In [14] the association between a PrDE and an EVI as above is called double layered dynamics (DLD). Results on existence, uniqueness and stability of solutions to DLD (5) may be found in [8,9,14].…”

Section: Tdepmentioning

confidence: 98%

“…Step 3 may be of interest in some cases, since it detects potential equilibrium states arising from evolution from an initial condition in finite time (see conditions and examples of finite time evolution in [8,9,14]). …”

Section: Assumptionsmentioning

confidence: 99%

“…In order to ensure that point-wise computed equilibria [typically using a formulation based on (4)] are meaningful to interpolate, uniqueness of point-wise states is required (see, for instance, [3,14,16,27], etc.). Consequently, the current literature is concerned with imposing monotonicity type or pseudo-monotonicity conditions (strict or strong) specifically to ensure that a TDEP has a unique curve x of equilibrium states.…”

Section: Evolution Solutions Of Tdepmentioning

confidence: 99%

“…[8][9][10][11]14]. Consequently, any non-stationary solution of the projected equation represents a time-dependent, evolving state of the underlying problem.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Evolution Solutions of Equilibrium Problems: A Computational Approach

Cojocaru

Greenhalgh

2016

Mathematical Analysis, Approximation Theory and Their Applications

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This paper proposes a computational method to describe evolution solutions of known classes of time-dependent equilibrium problems (such as time-dependent traffic network, market equilibrium or oligopoly problems, and dynamic noncooperative games). Equilibrium solutions for these classes have been studied extensively from both a theoretical (regularity, stability behaviour) and a computational point of view.In this paper we highlight a method to further study the solution set of such problems from a dynamical systems perspective, namely we study their behaviour when they are not in an (market, traffic, financial, etc.) equilibrium state. To this end, we define what is meant by an evolution solution for a time-dependent equilibrium problem and we introduce a computational method for tracking and visualizing evolution solutions using a projected dynamical system defined on a carefully chosen L 2 -space. We strengthen our results with various examples.

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Section: Examples and Discussionmentioning

confidence: 99%

Section: Tdepmentioning

confidence: 98%

Section: Assumptionsmentioning

confidence: 99%

Section: Evolution Solutions Of Tdepmentioning

confidence: 99%

“…[8][9][10][11]14]. Consequently, any non-stationary solution of the projected equation represents a time-dependent, evolving state of the underlying problem.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Evolution Solutions of Equilibrium Problems: A Computational Approach

Cojocaru

Greenhalgh

2016

Mathematical Analysis, Approximation Theory and Their Applications

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2009

Fragile Networks

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Network Economics

Nagurney¹

2009

Handbook of Computational Econometrics

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Throughout history, networks have provided the foundations by which humans have conducted their economic activities. Transportation and logistical networks make possible the movement of individuals, goods, and services, whereas communications networks enable the exchange of messages and information. Energy networks provide the fuel to support economic activities.As noted in Nagurney (2003), the emergence and evolution of physical networks over space and time, such as transportation and communications networks, and the effects of human decision making on such networks, have given rise, in turn, to the development of elegant theories and scientific methodologies that are network-based. Networks, as a science, have impacted disciplines in a broad range encompassing economics, engineering, computer science, applied mathematics, physics, sociology, and even biology. The novelty of networks is that they are pervasive, providing the medium for connectivity of our societies and economies, while, methodologically, network theory has developed into a powerful and dynamic mechanism for abstracting complex problems,

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Double-layered dynamics: A unified theory of projected dynamical systems and evolutionary variational inequalities

Cited by 39 publications

References 36 publications

Evolution Solutions of Equilibrium Problems: A Computational Approach

Evolution Solutions of Equilibrium Problems: A Computational Approach

Bibliography

Network Economics

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