Generalized Nash games and quasi-variational inequalities

Harker, Patrick T.

doi:10.1016/0377-2217(91)90325-p

Cited by 367 publications

(244 citation statements)

References 16 publications

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“…Therefore, by Lemma 1, the error bound (9) is valid. According to the definition of G and H , the latter is exactly (22) with Φ defined in (20).…”

Section: Error Bounds and Upper Lipschitz Stability For Gnepmentioning

confidence: 99%

“…tends toū, where Φ is given by (20). Assuming now that u(σ ) solves (28) for a given σ , first note that for the part F of Φ, defined in (19), (28) implies that F (u(σ )) = (a 1 , a 2 ).…”

Section: Corollary 1 Under the Assumptions Of Theorem 1 The Estimatementioning

confidence: 99%

“…Then for any starting point u 0 close enough toū = (x 1 ,x 2 ,μ 1 ,μ 2 ) the iterative scheme (31) for Φ given by (20) defines a sequence {u k } which converges quadratically to some u * ∈Ū , whereŪ is the solution set of system (2).…”

Section: Theoremmentioning

confidence: 99%

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On error bounds and Newton-type methods for generalized Nash equilibrium problems

Izmailov

Solodov

2013

Comput Optim Appl

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Error bounds (estimates for the distance to the solution set of a given problem) are key to analyzing convergence rates of computational methods for solving the problem in question, or sometimes even to justifying convergence itself. That said, for the generalized Nash equilibrium problems (GNEP), the theory of error bounds had not been developed in depth comparable to the fields of optimization and variational problems. In this paper, we provide a systematic approach which should be useful for verifying error bounds for both specific instances of GNEPs and for classes of GNEPs. These error bounds for GNEPs are based on more general results for constraints that involve complementarity relations and cover those (few) GNEP error bounds that existed previously, and go beyond. In addition, they readily imply a Lipschitzian stability result for solutions of GNEPs, a subject where again very little had been known. As a specific application of error bounds, we discuss Newtonian methods for solving GNEPs. While we do not propose any significantly new methods in this respect, some new insights into applicability to GNEPs of various approaches and into their convergence properties are presented.

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“…Therefore, by Lemma 1, the error bound (9) is valid. According to the definition of G and H , the latter is exactly (22) with Φ defined in (20).…”

Section: Error Bounds and Upper Lipschitz Stability For Gnepmentioning

confidence: 99%

Section: Corollary 1 Under the Assumptions Of Theorem 1 The Estimatementioning

confidence: 99%

See 1 more Smart Citation

On error bounds and Newton-type methods for generalized Nash equilibrium problems

Izmailov

Solodov

2013

Comput Optim Appl

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“…We denote the feasible strategy space of each player by the point to set mapping: C i : X −i → X i , i ∈ Γ that emphasises the ability of other players to influence the strategies available to player i [39,51,112]. The distinction between a conventional Nash game and a GNEP can be viewed as analogous to the distinction between unconstrained and constrained optimisation.…”

Section: The Generalised Nash Equilibrium Problemmentioning

confidence: 99%

“…Players are also assumed not to cooperate and in this situation each is doing the best she can given what her competitors are doing [45,62,116]. For a GNEP, the strategy profile x * is a Generalised Nash Equilibrium (GNE) if it is both feasible with respect to the mapping C i and if it is a maximizer of each player's utility over the constrained feasible set [51].…”

Section: Definitionmentioning

confidence: 99%

A Metaheuristic Framework for Bi-level Programming Problems with Multi-disciplinary Applications

Koh

2013

Metaheuristics for Bi-Level Optimization

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Bi-level programming problems arise in situations when the decision maker has to take into account the responses of the users to his decisions. Several problems arising in engineering and economics can be cast within the bi-level programming framework. The bi-level programming model is also known as a Stackleberg or leader-follower game in which the leader chooses his variables so as to optimise his objective function, taking into account the response of the follower(s) who separately optimise their own objectives, treating the leader's decisions as exogenous. In this chapter, we present a unified framework fully consistent with the Stackleberg paradigm of bi-level programming that allows for the integration of meta-heuristic algorithms with traditional gradient based optimisation algorithms for the solution of bi-level programming problems. In particular we employ Differential Evolution as the main meta-heuristic in our proposal. We subsequently apply the proposed method (DEBLP) to a range of problems from many fields such as transportation systems management, parameter estimation and game theory. It is demonstrated that DEBLP is a robust and powerful search heuristic for this class of problems characterised by non smoothness and non convexity.

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Hybrid Bertrand‐Cournot Models of Electricity Markets with Multiple Strategic Subnetworks and Common Knowledge Constraints

Zhang

2010

Restructured Electric Power Systems

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Most existing Nash-Cournot models of competition among electricity generators assume that firms behave purely Cournot or Bertrand with respect to transmission decisions by the independent system operator. Such models are unrealistic for markets in which interfaces connecting subnetworks are frequently saturated but the congestion pattern within individual subnetworks is less predictable. We propose two approaches for dealing with such situation. The first is a hybrid Bertrand-Cournot model of these markets in which firms are assumed to behave a la Cournot regarding inter-subnetwork transmission quantities, but a la Bertrand regarding intra-subnetwork transmission prices. A second approach is a Bertrand type model where transmission lines that are congested most of the time are designated as "common knowledge constraint" and treated as equality constraints by all market participants including the ISO and all generation firms. Under affine demand functions and quadratic costs, the market equilibrium of these models becomes mixed linear complementarity problems with bisymmetric positive semi-definite matrices. Numerical examples demonstrate that dividing the network into strategic subnetwork leads to prices higher than those predicted by the pure Bertrand model, but lower than those from the pure Cournot model. When public knowledge constraints are recognized in a Bertrand type model the resulting equilibrium does not show a uniform change in prices relative to the pure Bertrand model but we observe a shift in output from lower cost to higher cost generators, lower prices at the high cost nodes and higher prices at the low cost nodes.

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Generalized Nash games and quasi-variational inequalities

Cited by 367 publications

References 16 publications

On error bounds and Newton-type methods for generalized Nash equilibrium problems

On error bounds and Newton-type methods for generalized Nash equilibrium problems

A Metaheuristic Framework for Bi-level Programming Problems with Multi-disciplinary Applications

Hybrid Bertrand‐Cournot Models of Electricity Markets with Multiple Strategic Subnetworks and Common Knowledge Constraints

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