GNE: GNE package

In GNE: Computation of Generalized Nash Equilibria

GNE package

Description

Generalized Nash Equilibrium computational methods.

Usage

GNE(approach = 
	c("non smooth", "fixed point", "minimization", "constrained equation"), 
	method = "default", xinit, control=list(), ...)

Arguments

approach	a character string for the approach: either "non smooth", "fixed point", "minimization" or "constrained equation".
method	a character string for the computation method: either "default" or the name of the method.
xinit	a numeric vector for the initial point.
...	further arguments to be passed to GNE.nseq, GNE.fpeq or GNE.minpb.
control	a list with control parameters.

Details

Computing generalized Nash Equilibrium can be done in three different approaches.

(i) extended KKT system

It consists in solving the non smooth extended Karush-Kuhn-Tucker (KKT) system \Phi(z)=0.

(ii) fixed point approach

It consists in solving equation y(x)=x.

(iii) gap function minimization

It consists in minimizing a gap function min V(x).

(iv) constrained equation

It consists in solving F(x) such that x belongs to a specific set.

The GNE function is a global function calling the appropriate function GNE.nseq, GNE.fpeq, GNE.ceq or GNE.minpb. Benchmark functions comparing all methods for a given reformulation are available: see bench.GNE.

Additionnal utitilty functions are also available: rejection, projector, stepfunc, complementarity and funSSR.

Value

A list with components:

par

The best set of parameters found.

value

The value of the merit function.

counts

A two-element integer vector giving the number of calls to phi and jacphi respectively.

iter

The outer iteration number.

code

The values returned are

1

Function criterion is near zero. Convergence of function values has been achieved.

2

x-values within tolerance. This means that the relative distance between two consecutive x-values is smaller than xtol.

3

No better point found. This means that the algorithm has stalled and cannot find an acceptable new point. This may or may not indicate acceptably small function values.

4

Iteration limit maxit exceeded.

5

Jacobian is too ill-conditioned.

6

Jacobian is singular.

100

an error in the execution.

message

a string describing the termination code

fvec

a vector with function values.

approach

the name of the approach.

Author(s)

Christophe Dutang

References

F. Facchinei, A. Fischer and V. Piccialli (2009), Generalized Nash equilibrium problems and Newton methods, Math. Program.

A. von Heusinger (2009), Numerical Methods for the Solution of the Generalized Nash Equilibrium Problem, Ph. D. Thesis.

A. von Heusinger and C. Kanzow (2009), Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions, Comput Optim Appl .

F. Facchinei and C. Kanzow (2009), Generalized Nash Equilibrium problems. Preprint 290.

C. Dutang (2013), A survey of GNE computation methods: theory and algorithms, preprint on HAL, https://hal.science/hal-00813531.

See Also

See GNE.fpeq, GNE.minpb, GNE.ceq and GNE.nseq for other approaches.

GNE documentation built on Oct. 17, 2024, 5:08 p.m.