On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method

Schiro, Dane A.; Pang, Jong Shi; Shanbhag, Uday V.

doi:10.1007/s10107-012-0558-3

Cited by 38 publications

(23 citation statements)

References 46 publications

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“…A third consequence is that any z ∈ SOL (q, M) having |σ (z)| > s, i.e., more than s positive components, will be elusive in the sense that it cannot be computed by LS1 applied to (q, d, M) where d is an extended s-step vector. (The reader is referred to a recent paper [19] where it is demonstrated that LS1 can only compute solutions with a particular property of the LCP formulation of a generalized Nash equilibrium problem; such problems provide a class of realistic LCPs where many solutions are elusive when they are solved by LS1.) A fourth consequence of Theorem 1 is that all matrices M ∈ SP must belong to the class Q, which is to say that for every q, the LCP (q, M) has at least one solution.…”

Section: A New Matrix Class Spmentioning

confidence: 98%

Some LCPs solvable in strongly polynomial time with Lemke’s algorithm

2016

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We identify a class of Linear Complementarity Problems (LCPs) that are solvable in strongly polynomial time by Lemke's Algorithm (Scheme 1) or by the Parametric Principal Pivoting Method (PPPM). This algorithmic feature for the class of problems under consideration here is attributable to the proper selection of the covering vector in Scheme 1 or the parametric direction vector in the PPPM which leads to solutions of limited and monotonically increasing support size; such solutions are sparse. These and other LCPs may very well have multiple solutions, many of which are unattainable by either algorithm and thus are said to be elusive. The initial conditions imposed on the new matrix class identified in Sect. 2 are subsequently relaxed in later sections.

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Section: A New Matrix Class Spmentioning

confidence: 98%

Some LCPs solvable in strongly polynomial time with Lemke’s algorithm

2016

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“…The closest setting is the one of affine generalized Nash equilibrium problems (AGNEPs), where GNEPs with linear constraints and quadratic cost functions are considered, which are convex in the player variables. This setting was introduced explicitly in [6] and also investigated in [7,8]. Various examples of GNEPs where players share affine constraints are summarized in [6].…”

Section: Introductionmentioning

confidence: 98%

“…This setting was introduced explicitly in [6] and also investigated in [7,8]. Various examples of GNEPs where players share affine constraints are summarized in [6]. Furthermore, the very common case of players having finite strategy sets and minimizing their expected losses, allowing mixed strategies, yields special LGNEPs (cf.…”

Section: Introductionmentioning

confidence: 98%

The Cone Condition and Nonsmoothness in Linear Generalized Nash Games

Stein

Sudermann‐Merx

2015

J Optim Theory Appl

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We consider linear generalized Nash games and introduce the so-called cone condition, which characterizes the smoothness of a gap function that arises from a reformulation of the generalized Nash equilibrium problem as a piecewise linear optimization problem based on the Nikaido-Isoda function. Other regularity conditions such as the linear independence constraint qualification or the strict MangasarianFromovitz condition are only sufficient for smoothness, but have the advantage that they can be verified more easily than the cone condition. Therefore, we present special cases, where these conditions are not only sufficient, but also necessary for smoothness of the gap function. Our main tool in the analysis is a global extension of the gap function that allows us to overcome the common difficulty that its domain may not cover the whole space.

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“…We already gave some references of works related to ours with respect to the linear complementarity. The work by Schiro et al [20] is one of them and deals actually with a more general problem as ours. However, the termination criterion for their algorithm is quite technical and it does not seem to be possible to prove that their algorithm terminates for sure on a equilibrium.…”

Section: Introductionmentioning

confidence: 99%

“…In 1965, Lemke [14] designed a pivoting algorithm for solving a linear complementarity problem under a quite general form. This algorithm has been adapted and extended several times -see for instance Adler and Verma [1], Asmuth et al [3], Cottle et al [6], Schiro et al [20] to be able to deal with linear complementarity problems that do not directly fit in the required framework of the original Lemke algorithm.…”

Section: Introductionmentioning

confidence: 99%

A Lemke-Like Algorithm for the Multiclass Network Equilibrium Problem

Meunier

Pradeau

2013

Web and Internet Economics

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Abstract. We consider a nonatomic congestion game on a connected graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origin-destination pair. Under some mild conditions, it is known that a Nash equilibrium exists, but the computation of an equilibrium in the multiclass case is an open problem for general functions. We consider the specific case where the cost functions are affine and propose an extension of Lemke's algorithm able to solve this problem. At the same time, it provides a constructive proof of the existence of an equilibrium in this case.

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On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method

Cited by 38 publications

References 46 publications

Some LCPs solvable in strongly polynomial time with Lemke’s algorithm

Some LCPs solvable in strongly polynomial time with Lemke’s algorithm

The Cone Condition and Nonsmoothness in Linear Generalized Nash Games

A Lemke-Like Algorithm for the Multiclass Network Equilibrium Problem

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