Noncooperative Games with Elliptic Systems

Roubíček, Tomáš

doi:10.1007/978-3-0348-8691-8_21

Cited by 6 publications

(7 citation statements)

References 11 publications

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“…He provided conditions for existence and uniqueness of normalized Nash equilibriums. Some work has been done for standard NEPs in certain specific problem settings [5,27,28,29,31,32,33]. For GNEPs, we are only aware of the papers [8,13,14], the latter two of which are confined to an optimal control setting.…”

Section: Introductionmentioning

confidence: 99%

The Multiplier-Penalty Method for Generalized Nash Equilibrium Problems in Banach Spaces

Kanzow¹,

Karl²,

Steck³

et al. 2019

SIAM J. Optim.

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This paper deals with generalized Nash equilibrium problems (GNEPs) in Banach spaces. We prove an existence result for normalized equilibria of jointly convex GNEPs and then propose an augmented Lagrangian-type algorithm for their computation. A thorough convergence analysis is conducted which considers the existence of subproblem solutions as well as feasibility and optimality of limit points. We then apply our investigations to a class of multiobjective optimal control problems which are governed by a linear partial differential equation and provide some numerical results to demonstrate the performance of the method.

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Section: Introductionmentioning

confidence: 99%

The Multiplier-Penalty Method for Generalized Nash Equilibrium Problems in Banach Spaces

Kanzow¹,

Karl²,

Steck³

et al. 2019

SIAM J. Optim.

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“…The estimate (29) holds due to the Hölder inequality with Sobolev embeddings provided α is sufficiently large, which here means (25). Note that, in case n = 3,…”

Section: Satisfying (13) and Being Sufficiently Small In The Norms mentioning

confidence: 85%

“…= 1, which explains the last bound in (25). Then, in view of (20), by using the Young inequality and (28), we can estimate:…”

Section: Satisfying (13) and Being Sufficiently Small In The Norms mentioning

confidence: 92%

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Nash Equilibria in Noncooperative Predator-Prey Games

Ramos

Roubíček

2007

Appl Math Optim

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Pod vodárenskou věží 4, CZ-182 08 Praha 8, Czech Republic.Abstract: A non-cooperative game governed by a distributed-parameter predator-prey system is considered, assuming that two players control initial conditions for predator and prey, respectively. Existence of a Nash equlibrium is shown under the condition that the desired population profiles and the environmental carrying capacity for the prey are sufficiently small. A conceptual approximation algorithm is proposed and analyzed. Finally, numerical simulations are performed, too.

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“…A standard reference for single-objective optimal control problems (one player) is [6]. Roubíček [7] examines a two-player game with an elliptic pde as constraint, introduces a relaxation, and shows that the relaxed games are limit points of a sequence of standard Nash equilibrium problems, i.e., problems where the feasible strategies of each player are independent of the strategies of the others. In [1,8] a standard Nash equilibrium problem formulation of some multiobjective optimal control problems, governed by a pde and without further control or state constraints, is treated using a conjugate gradient method for a finite differences discretization.…”

Section: Introductionmentioning

confidence: 99%

Jointly Convex Generalized Nash Equilibria and Elliptic Multiobjective Optimal Control

Dreves

Gwinner

2015

J Optim Theory Appl

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We deal with jointly convex generalized Nash equilibrium problems in infinite-dimensional spaces. For their solution, we extend a finite-dimensional optimization approach and design a convergent algorithm in Hilbert space. Then we apply our investigations to a class of multiobjective optimal control problems with control and state constraints that are governed by elliptic partial differential equations. We present a new reformulation as a jointly convex generalized Nash equilibrium problem. We study a finite element approximation of such a multiobjective optimal control problem, and further we prove convergence in appropriate function spaces. Finally, we provide some numerical results that show the effectiveness of our algorithm for multiobjective optimal control problems.

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Noncooperative Games with Elliptic Systems

Cited by 6 publications

References 11 publications

The Multiplier-Penalty Method for Generalized Nash Equilibrium Problems in Banach Spaces

The Multiplier-Penalty Method for Generalized Nash Equilibrium Problems in Banach Spaces

Nash Equilibria in Noncooperative Predator-Prey Games

Jointly Convex Generalized Nash Equilibria and Elliptic Multiobjective Optimal Control

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