Hooke-Jeeves Derivative-Free Minimization R (working for MPFR)
An implementation of the Hooke-Jeeves algorithm for derivative-free optimization.
hjkMpfr(par, fn, control = list(), ...)
par |
Starting vector of parameter values. The initial vector may lie on the boundary. If |
fn |
Nonlinear objective function that is to be optimized. A scalar function that takes a real vector as argument and returns a scalar that is the value of the function at that point. |
control |
|
... |
Additional arguments passed to |
Argument control is a list specifing changes to default values of
algorithm control parameters.
Note that parameter names may be abbreviated as long as they are unique.
The list items are as follows:
tolConvergence tolerance. Iteration is terminated when the
step length of the main loop becomes smaller than tol. This does
not imply that the optimum is found with the same accuracy.
Default is 1.e-06.
maxfevalMaximum number of objective function evaluations allowed. Default is Inf, that is no restriction at all.
maximizeA logical indicating whether the objective function is to be maximized (TRUE) or minimized (FALSE). Default is FALSE.
targetA real number restricting the absolute function value. The procedure stops if this value is exceeded. Default is Inf, that is no restriction.
infoA logical variable indicating whether the step number, number of function calls, best function value, and the first component of the solution vector will be printed to the console. Default is FALSE.
If the minimization process threatens to go into an infinite loop, set
either maxfeval or target.
A list with the following components:
par |
Best estimate of the parameter vector found by the algorithm. |
value |
value of the objective function at termination. |
convergence |
indicates convergence ( |
feval |
number of times the objective |
niter |
number of iterations (“steps”) in the main loop. |
This algorithm is based on the Matlab code of Prof. C. T. Kelley, given in his book “Iterative methods for optimization”. It has been implemented for package dfoptim with the permission of Prof. Kelley.
This version does not (yet) implement a cache for storing function values that have already been computed as searching the cache makes it slower.
Hans W Borchers hwborchers@googlemail.com; for Rmpfr: John Nash, June 2012. Modifications by Martin Maechler.
C.T. Kelley (1999), Iterative Methods for Optimization, SIAM.
Quarteroni, Sacco, and Saleri (2007), Numerical Mathematics, Springer.
## simple smooth example:
ff <- function(x) sum((x - c(2:4))^2)
str(rr <- hjkMpfr(rep(mpfr(0,128), 3), ff, control=list(info=TRUE)))
## Hooke-Jeeves solves high-dim. Rosenbrock function {but slowly!}
rosenbrock <- function(x) {
n <- length(x)
sum (100*((x1 <- x[1:(n-1)])^2 - x[2:n])^2 + (x1 - 1)^2)
}
par0 <- rep(0, 10)
str(rb.db <- hjkMpfr(rep(0, 10), rosenbrock, control=list(info=TRUE)))
## rosenbrook() is quite slow with mpfr-numbers:
str(rb.M. <- hjkMpfr(mpfr(numeric(10), prec=128), rosenbrock,
control = list(tol = 1e-8, info=TRUE)))
## Hooke-Jeeves does not work well on non-smooth functions
nsf <- function(x) {
f1 <- x[1]^2 + x[2]^2
f2 <- x[1]^2 + x[2]^2 + 10 * (-4*x[1] - x[2] + 4)
f3 <- x[1]^2 + x[2]^2 + 10 * (-x[1] - 2*x[2] + 6)
max(f1, f2, f3)
}
par0 <- c(1, 1) # true min 7.2 at (1.2, 2.4)
h.d <- hjkMpfr(par0, nsf) # fmin=8 at xmin=(2,2)
## and this is not at all better (but slower!)
h.M <- hjkMpfr(mpfr(c(1,1), 128), nsf, control = list(tol = 1e-15))
## --> demo(hjkMpfr) # -> Fletcher's chebyquad function m = n -- residualsPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.