Projected Dynamical Systems and Variational Inequalities with Applications

Nagurney, Anna; Zhang, Ding

doi:10.1007/978-1-4615-2301-7

Cited by 424 publications

(401 citation statements)

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“…Linear time-invariant systems together with static relations described by set-valued mappings have been used extensively. An incomplete inventory includes electrical networks with switching elements as in power converters [12][13][14][15][16], linear relay systems [17,18], piecewise linear systems [19], and projected dynamical systems [20,21]; see also [22][23][24] for further examples and [25,26] for numerical analysis of maximal monotone differential inclusions.…”

Section: Introductionmentioning

confidence: 99%

Linear passive systems and maximal monotone mappings

Camlibel

Schumacher

2015

Math. Program.

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This paper deals with a class of dynamical systems obtained from interconnecting linear systems with static set-valued relations. We first show that such an interconnection can be described by a differential inclusions with a maximal monotone set-valued mappings when the underlying linear system is passive and the static relation is maximal monotone. Based on the classical results on such differential inclusions, we conclude that such interconnections are well-posed in the sense of existence and uniqueness of solutions. Finally, we investigate conditions which guarantee well-posedness but are weaker than passivity. Mathematics Subject Classification

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Section: Introductionmentioning

confidence: 99%

Linear passive systems and maximal monotone mappings

Camlibel

Schumacher

2015

Math. Program.

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“…We first suggest a simple projected Euler method, see [17], for the solution of equation (12). Given a solution ρ(t) at time t, we calculate an approximation of the solution at time t + ∆t.…”

Section: Projected Euler Methodsmentioning

confidence: 99%

“…An indirect but from several point of views very useful approach to finding a ρ that satisfies (10) is to consider a dynamical system (gradient flow, dynamical system, ODE, neurodynamical) reformulation of the problem. Here we give the most direct such formulation for (7), see, e.g., Nagurney and Zhang [17], based on local projections. Let ρ be a function of a time-like variable t and solve, for some initial condition, the ODĖ…”

Section: Dynamical Systems Approachmentioning

confidence: 99%

“…This second dynamical system will turn out to be connected to the classical optimality criteria method of SIMP topology optimization. We do not discuss convergence of algorithms but refer to [17,8].…”

Section: Algorithmsmentioning

confidence: 99%

“…By this procedure algorithms for solving the original optimization problem can be obtained by utilizing standard discretization schemes for ODEs. This approach to optimization is known in wide circuits as neurodynamical optimization [15], projected dynamical system approach [17], differential variational inequalities [19], gradient flow or ODE approach to optimization. However, it seems largely unexploited in structural optimization.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical systems and topology optimization

Klarbring

Torstenfelt

2010

Struct Multidisc Optim

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This paper uses a dynamical systems approach for studying the material distribution (density or SIMP) formulation of topology optimization of structures. Such an approach means that an ordinary differential equation, such that the objective function is decreasing along a solution trajectory of this equation, is constructed. For stiffness optimization two differential equations with this property are considered. By simple explicit Euler approximations of these equations, together with projection techniques to satisfy box constraints, we obtain different iteration formulas. One of these formulas turns out to be the classical optimality criteria algorithm, which, thus, is receiving a new interpretation and framework. Based on this finding we suggest extensions of the optimality criteria algorithm.A second important feature of the dynamical systems approach, besides the purely algorithmic one, is that it points at a connection between optimization problems and natural evolution problems such as bone remodeling and damage evolution. This connection has been hinted at previously but, in the opinion of the authors, not been clearly stated since the dynamical systems concept was missing. To give an explicit example of an evolution problem that is in this way connected to an optimization problem, we study a model of bone remodeling.Numerical examples, related to both the algorithmic issue and the issue of natural evolution represented as bone remodeling, are presented.

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Dynamic multi-sector, multi-instrument financial networks with futures: Modeling and computation

Nagurney

Siokos

1999

Networks

Self Cite

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In this paper, we develop a dynamic model of financial behavior in the case of multiple sectors and multiple instruments in the presence of financial futures, which demonstrates the evolution of the underlying networks through time. The dynamic model is formulated as a projected dynamical system whose set of stationary points coincides with the set of solutions to a variational inequality problem. We identify the network structure of the individual sectors' portfolio optimization problems out of equilibrium and then prove that the equilibrium solution can be reformulated as the solution to a network optimization problem, in which the network represents a merger of the individual networks. We subsequently provide a discrete time algorithm for the solution of the continuous time financial model, which exploits the network structure, and provide convergence results. The model and algorithm are then illustrated through numerical examples. © 1999 John Wiley & Sons, Inc. Networks 33: 93–108, 1999

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Projected Dynamical Systems and Variational Inequalities with Applications

Cited by 424 publications

References 0 publications

Linear passive systems and maximal monotone mappings

Linear passive systems and maximal monotone mappings

Dynamical systems and topology optimization

Dynamic multi-sector, multi-instrument financial networks with futures: Modeling and computation

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