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

Kanzow, Christian; Karl, Veronika; Steck, Daniel J.; Wachsmuth, Daniel

doi:10.1137/17m114114x

Cited by 15 publications

(16 citation statements)

References 38 publications

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“…As before, we discretize the domain Ω := (0, 1) 2 by means of a uniform grid with n ∈ N interior points per row or column. The problem in question is a four-player game where f := 1, The results of the iteration are displayed in Figure 3 and agree with those in [33]. The corresponding iteration numbers are given as follows.…”

Section: Generalized Nash Equilibrium Problemssupporting

confidence: 65%

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

Kanzow¹,

Steck²

2019

SIAM J. Optim.

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This paper deals with quasi-variational inequality problems (QVIs) in a generic Banach space setting. We provide a theoretical framework for the analysis of such problems which is based on two key properties: the pseudomonotonicity (in the sense of Brezis) of the variational operator and a Mosco-type continuity of the feasible set mapping. We show that these assumptions can be used to establish the existence of solutions and their computability via suitable approximation techniques. In addition, we provide a practical and easily verifiable sufficient condition for the Mosco-type continuity property in terms of suitable constraint qualifications.Based on the theoretical framework, we construct an algorithm of augmented Lagrangian type which reduces the QVI to a sequence of standard variational inequalities. A full convergence analysis is provided which includes the existence of solutions of the subproblems as well as the attainment of feasibility and optimality. Applications and numerical results are included to demonstrate the practical viability of the method.

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Section: Generalized Nash Equilibrium Problemssupporting

confidence: 65%

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

Kanzow¹,

Steck²

2019

SIAM J. Optim.

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“…This has the benefit that we avoid the computation of projections and distance functions involving H 1 + (Ω). The resulting modifications to Algorithm 4.1 are fairly straightforward (see, for instance, [6,8,38]). Indeed, the augmented subproblems are now (constrained) variational n = 256, β = 1 n = 256, β = 0.01 n = 1024, β = 1 n = 1024, β = 0.01 k ρ k σ k ρ k σ k ρ k σ k ρ k σ k inequalities over the set {q ∈ H 1 (Ω) : q ≥ α}.…”

Section: Parameter Estimation In Elliptic Systemsmentioning

confidence: 99%

“…Hence, the set M is nonconvex in general, and (1) is the natural framework for our setting. Variational inequalities are a well-known and popular class in both finite and infinitedimensional optimization since they unify various problem types such as constrained minimization and equilibrium-type problems, in particular Nash and (certain) generalized Nash equilibrium problems [16,17,21,30,38]. This opens up a broad spectrum of applications including optimal control, parameter estimation, differential games, and problems in mechanics or shape optimization.…”

Section: Introductionmentioning

confidence: 99%

On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces

Kanzow¹,

Steck²

2018

SIAM J. Control Optim.

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This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the (primal-dual) solution set and give a sufficient condition for such an error bound to hold. In the second part of the paper, we consider an algorithm of augmented Lagrangian type for the solution of such variational problems. We give some global convergence properties of the method and then use the error bound theory to provide estimates for the rate of convergence and to deduce boundedness of the sequence of penalty parameters. Finally, numerical results for optimal control, Nash equilibrium problems, and elliptic parameter estimation problems are presented.Lemma 2.1. Assume that g : X → H is concave. If m : H → R is convex and decreasing, then m • g is convex. In particular:by the concavity of g. Applying m on both sides yieldswhere we used the monotonicity and the convexity of m. Hence, m • g is convex. Assertion (a) now follows because d K is decreasing (see above) and convex [4, Cor. 12.12]. Similarly, for (b), the function y → (λ, y) with λ ∈ K • ∞ is obviously a convex function, and it is decreasing because (λ, k) ≤ 0 for all k ∈ K ∞ . Finally, for (c), note that M = {x ∈ X : g(x) ∈ K} = {x ∈ X : d K (g(x)) ≤ 0}.Hence, M is a lower level set of the convex function d K • g and therefore a convex set.

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“…This formulation is often referred to as a generalized Nash equilibrium; see, e.g., [24,28,29]. However, as noted in [36], it is really a standard Nash equilibrium in the sense of (1.1) since functions are allowed to take the value +∞.…”

Section: Example 33 (Minimax)mentioning

confidence: 99%

“…In addition, suppose that G 1 = H 1 0 (Ω) and L 1 = S. Then we recover frameworks investigated in [9,29].…”

Section: Example 33 (Minimax)mentioning

confidence: 99%

Analysis and Numerical Solution of a Modular Convex Nash Equilibrium Problem

Bùi¹,

Combettes²

2021

Preprint

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

Cited by 15 publications

References 38 publications

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces

Analysis and Numerical Solution of a Modular Convex Nash Equilibrium Problem

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