R: Davidon-Fletcher-Powell Method

fletcher-powell {pracma} | R Documentation

## Davidon-Fletcher-Powell Method

### Description

Davidon-Fletcher-Powell method 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.

### Usage

    fletcher_powell(x0, f, g = NULL,
                    maxiter = 1000, tol = .Machine$double.eps^(2/3))

### Arguments

`x0` | start value.
`f` | function to be minimized.
`g` | gradient function of `f`; if `NULL`, a numerical gradient will be calculated.
`maxiter` | max. number of iterations.
`tol` | relative tolerance, to be used as stopping rule.

### Details

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.

### Value

List with following components:

`xmin` | minimum solution found.
`fmin` | value of `f` at minimum.
`niter` | number of iterations performed.

### Note

Used some Matlab code as described in the book “Applied Numerical Analysis Using Matlab” by L. V.Fausett.

### References

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.

### See Also

`steep_descent`

### Examples

    ##  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] 14

[Package pracma version 2.0.4 Index]