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 Φ(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.
TODO:
-
write a pdf vignette.
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.