Fletcher-Powell Conjugate Gradient Minimization
Conjugate Gradient (CG) minimization through the Davidon-Fletcher-Powell approach for function minimization.
The Davidon-Fletcher-Powell (DFP) and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) methods are the first quasi-Newton minimization methods developed. These methods differ only in some details; in general, the BFGS approach is more robust.
fletcher_powell(x0, f, g = NULL,
maxiter = 1000, tol = .Machine$double.eps^(2/3))x0 |
start value. |
f |
function to be minimized. |
g |
gradient function of |
maxiter |
max. number of iterations. |
tol |
relative tolerance, to be used as stopping rule. |
The starting point is Newton's method in the multivariate case, when the estimate of the minimum is updated by the following equation
x_{new} = x - H^{-1}(x) grad(g)(x)
where H is the Hessian and grad the gradient.
The basic idea is to generate a sequence of good approximations to the inverse Hessian matrix, in such a way that the approximations are again positive definite.
List with following components:
xmin |
minimum solution found. |
fmin |
value of |
niter |
number of iterations performed. |
Used some Matlab code as described in the book “Applied Numerical Analysis Using Matlab” by L. V.Fausett.
J. F. Bonnans, J. C. Gilbert, C. Lemarechal, and C. A. Sagastizabal. Numerical Optimization: Theoretical and Practical Aspects. Second Edition, Springer-Verlag, Berlin Heidelberg, 2006.
## Rosenbrock function
rosenbrock <- function(x) {
n <- length(x)
x1 <- x[2:n]
x2 <- x[1:(n-1)]
sum(100*(x1-x2^2)^2 + (1-x2)^2)
}
fletcher_powell(c(0, 0), rosenbrock)
# $xmin
# [1] 1 1
# $fmin
# [1] 1.774148e-27
# $niter
# [1] 14Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.