The noncooperative transportation problem and linear generalized Nash games

Stein, Oliver; Sudermann‐Merx, Nathan

doi:10.1016/j.ejor.2017.10.001

Cited by 29 publications

(13 citation statements)

References 37 publications

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“…Let us first focus on player 1's problem. We have that P 2 (Y ) = [2,4], which is obviously included in the interior of A 1 = [1,5], so that (1) holds. Given the symmetry of the problem, we conclude that GNEP is strictly inter-feasible, while it is obviously not fully inter-feasible.…”

Section: Example 33mentioning

confidence: 99%

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Towards Tractable Constraint Qualifications for Parametric Optimisation Problems and Applications to Generalised Nash Games

Aussel

Svensson

2019

J Optim Theory Appl

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A generalised Nash game is a non-cooperative game in which each player is facing an optimisation problem where both the objective function and the feasible set depend on the variables of the other players. A classical way to treat numerically this difficult problem is to solve the nonlinear system composed of the concatenation of the Karush-Kuhn-Tucker optimality conditions of each player's problem. The aim of this work is to provide constraint qualification conditions ensuring that both problems share the same set of solutions. Our main target here is to elaborate tractable conditions, that is, sets of conditions that are as simple as possible to fulfil. This is achieved through the analysis of "minimal" qualification conditions for parametric optimisation problems.

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Section: Example 33mentioning

confidence: 99%

“…Good surveys for the state of the art of GNEPs are [3,4]. In [5,6], some recent applications are discussed. Our present work is related to constraint qualifications, which are essentially a nonlinear aspect of the problem.…”

Section: Introductionmentioning

confidence: 99%

Towards Tractable Constraint Qualifications for Parametric Optimisation Problems and Applications to Generalised Nash Games

Aussel

Svensson

2019

J Optim Theory Appl

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“…Later on, Arrow and Debreu [3] extended this notion to the generalized Nash equilibrium for games, where both the payoff function and the set of feasible strategies depend on others' strategies. Initially motivated by economic applications, the notion of equilibrium in games has received a vivid interest thanks to its various applications in social science [12], biology [44,46] (evolutionary games and replicator dynamics), computer science [2,40], environment modeling [8,10] or energy problems [11,26,27,45] to cite few among others. These applications have motivated the evolution of the Nash equilibrium concept, and its use, to complex games that now require a deep understanding of theoretical and computational mathematics used for identifying, computing and analyzing (all) the equilibrium strategy(ies) of a given game.…”

Section: Introductionmentioning

confidence: 99%

A parametrized variational inequality approach to track the solution set of a generalized nash equilibrium problem

Migot

Cojocaru

2020

European Journal of Operational Research

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In this paper, we present a numerical method to describe the solution set of a generalized Nash equilibrium problem (GNEP). Previous approaches show how to reformulate the GNEP as a family of parametric variational inequalities in the special case where the game has shared constraints. We extend this result to generalized Nash problems by means of an umbrella shared constraint approximation of the game. We show the validity of our approach on numerical examples from the literature, and we provide new results that pinpoint the handling of the algorithm's parameters for its implementation. Last but not least, we extend, solve, and discuss an applied example of a generalized Nash equilibrium problem of environmental accords between countries.

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“…Thus some of the players' constraints are coupled ; these are in addition to the players' private constraints that do not contain the rival players' decision variables. Sources of the GNEP include infrastructure systems such as communication or radio systems [49,50], power grids [57,35], transportation networks [5], modern traffic systems with e-hailing services [4], supply and demand constraints for transportation systems [54], and pollution quota for environmental application [40,7]. We refer the readers to [22,27,30] for more detailed surveys on this active research topic and its many applications.…”

Section: Introductionmentioning

confidence: 99%

Exact Penalization of Generalized Nash Equilibrium Problems

Ba¹,

Pang²

2018

Preprint

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This paper presents an exact penalization theory of the generalized Nash equilibrium problem (GNEP) that has its origin from the renowned Arrow-Debreu general economic equilibrium model. While the latter model is the foundation of much of mathematical economics, the GNEP provides a mathematical model of multi-agent non-cooperative competition that has found many contemporary applications in diverse engineering domains. The most salient feature of the GNEP that distinguishes it from a standard non-cooperative (Nash) game is that each player's optimization problem contains constraints that couple all players' decision variables. Extending results for stand-alone optimization problems, the penalization theory aims to convert the GNEP into a game of the standard kind without the coupled constraints, which is known to be more readily amenable to solution methods and analysis. Starting with an illustrative example to motivate the development, the paper focuses on two kinds of coupled constraints, shared (i.e., common) and finitely representable. Constraint residual functions and the associated error bound theory play an important role throughout the development.

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The noncooperative transportation problem and linear generalized Nash games

Cited by 29 publications

References 37 publications

Towards Tractable Constraint Qualifications for Parametric Optimisation Problems and Applications to Generalised Nash Games

Towards Tractable Constraint Qualifications for Parametric Optimisation Problems and Applications to Generalised Nash Games

A parametrized variational inequality approach to track the solution set of a generalized nash equilibrium problem

Exact Penalization of Generalized Nash Equilibrium Problems

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