Solving linear generalized Nash equilibrium problems numerically

Dreves, Axel; Sudermann‐Merx, Nathan

doi:10.1080/10556788.2016.1165676

Cited by 12 publications

(12 citation statements)

References 26 publications

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“…of the form Ax ≤ b for some suitable matrix A and vector b." In fact, many existing generalized Nash game models adopt affine coupling constraints, as well documented in Schiro, Pang & Shanbhag (2013) and Dreves & Sudermann (2016).…”

Section: Game Formulationmentioning

confidence: 99%

A distributed primal-dual algorithm for computation of generalized Nash equilibria via operator splitting methods

Pavel

2017

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

106

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In this paper, we propose a distributed primal-dual algorithm for computation of a generalized Nash equilibrium (GNE) in noncooperative games over network systems. In the considered game, not only each player's local objective function depends on other players' decisions, but also the feasible decision sets of all the players are coupled together with a globally shared affine inequality constraint. Adopting the variational GNE, that is the solution of a variational inequality, as a refinement of GNE, we introduce a primal-dual algorithm that players can use to seek it in a distributed manner. Each player only needs to know its local objective function, local feasible set, and a local block of the affine constraint. Meanwhile, each player only needs to observe the decisions on which its local objective function explicitly depends through the interference graph and share information related to multipliers with its neighbors through a multiplier graph. Through a primal-dual analysis and an augmentation of variables, we reformulate the problem as finding the zeros of a sum of monotone operators. Our distributed primal-dual algorithm is based on forward-backward operator splitting methods. We prove its convergence to the variational GNE for fixed step-sizes under some mild assumptions. Then a distributed algorithm with inertia is also introduced and analyzed for variational GNE seeking. Finally, numerical simulations for network Cournot competition are given to illustrate the algorithm efficiency and performance.Engineering network systems, like power grids, communication networks, transportation networks and sensor networks, play a foundation role in modern society. The efficient and secure operation of various network systems relies on efficiently solving decision and control problems arising in those large scale network systems. In many decision problems, the nodes can be regarded as agents that need to make local decisions possibly limited by the shared network resources within local feasible sets. Meanwhile, each agent has a local cost/utility function to be optimized, which depends on the decisions of other agents. The traditional manner for solving such decision problems over networks is the centralized optimization approach, which relies on a control center to gather the data of the problem and to optimize the social welfare (usually taking the form of the sum of local objective functions) within the local and global constraints. The centralized optimization approach may not be suitable for decision problems over large scale networks, since it needs bidirectional communication between all the network nodes and the control center, it is not robust to the failure of the center node, and the computational burden for the center is unbearable. It is also not preferable because the privacy of each agent might be compromised when the data is transferred $

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Section: Game Formulationmentioning

confidence: 99%

A distributed primal-dual algorithm for computation of generalized Nash equilibria via operator splitting methods

Pavel

2017

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

106

View full text Add to dashboard Cite

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“…In that, this is a direct consequence of the definition of Pareto optimality. Moreover, we can prove that any solution of problem (9) is a Pareto optimum of problem (15).…”

Section: Methods Based On Potential Functionsmentioning

confidence: 99%

“…Theorem 3 Let x be a 0-approximate global solution of problem (9), then x is a Pareto optimum of problem (15).…”

Section: Methods Based On Potential Functionsmentioning

confidence: 99%

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Algorithms for generalized potential games with mixed-integer variables

Sagratella

2017

Comput Optim Appl

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We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixedinteger variables, i.e., games in which some variable are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss-Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches. Keywords Generalized Nash equilibrium problem • Generalized potential game • Mixed-integer nonlinear problem • Parametric optimization The work of the author has been partially supported by Avvio alla Ricerca 2015 Sapienza University of Rome, under grant 488.

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“…This class of somewhat separable GNEPs contains important subclasses such as the linear GNEPs that were studied in [17] or even the NEPs.…”

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confidence: 99%

Optimality Conditions and Constraint Qualifications for Generalized Nash Equilibrium Problems and Their Practical Implications

Bueno¹,

Haeser²,

Rojas³

2019

SIAM J. Optim.

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Generalized Nash Equilibrium Problems (GNEPs) are a generalization of the classic Nash Equilibrium Problems (NEPs), where each player's strategy set depends on the choices of the other players. In this work we study constraint qualifications and optimality conditions tailored for GNEPs and we discuss their relations and implications for global convergence of algorithms. Surprisingly, differently from the case of nonlinear programming, we show that, in general, the KKT residual can not be made arbitrarily small near a solution of a GNEP. We then discuss some important practical consequences of this fact. We also prove that this phenomenon is not present in an important class of GNEPs, including NEPs. Finally, under a weak constraint qualification introduced, we prove global convergence to a KKT point of an Augmented Lagrangian algorithm for GNEPs and under the quasinormality constraint qualification for GNEPs, we prove boundedness of the dual sequence.

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Solving linear generalized Nash equilibrium problems numerically

Cited by 12 publications

References 26 publications

A distributed primal-dual algorithm for computation of generalized Nash equilibria via operator splitting methods

A distributed primal-dual algorithm for computation of generalized Nash equilibria via operator splitting methods

Algorithms for generalized potential games with mixed-integer variables

Optimality Conditions and Constraint Qualifications for Generalized Nash Equilibrium Problems and Their Practical Implications

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