nmk: Nelder-Mead optimziation algorithm for derivative-free optimization

Description

An implementation of the Nelder-Mead algorithm for derivative-free optimization. This allows bounds to be placed on parameters. Bounds are enforced by means of a parameter transformation.

Usage

nmk(par, fn, control = list(), ...)
nmkb(par, fn, lower=-Inf, upper=Inf, control = list(), ...)

Value

A list with the following components:

par: Best estimate of the parameter vector found by the algorithm.
value: The value of the objective function at termination.
feval: The number of times the objective fn was evaluated.
restarts: The number of times the algorithm had to be restarted when it stagnated.
convergence: An integer code indicating type of convergence. 0 indicates successful convergence. Positive integer codes indicate failure to converge.
message: Text message indicating the type of convergence or failure.

Arguments

par: A starting vector of parameter values. Must be feasible, i.e. lie strictly between lower and upper bounds.
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 (see details).
lower: Lower bounds on the parameters. A vector of the same length as the parameters. If a single value is specified, it is assumed that the same lower bound applies to all parameters.
upper: Upper bounds on the parameters. A vector of the same length as the parameters. If a single value is specified, it is assumed that the same upper bound applies to all parameters.
control: A list of control parameters. See *Details* for more information.
...: Additional arguments passed to fn

Author

Ravi Varadhan <rvaradhan@jhmi.edu>, Johns Hopkins University URL:http://www.jhsph.edu/agingandhealth/People/Faculty_personal_pages/Varadhan.html

Details

Argument control is a list specifing any changes to default values of algorithm control parameters for the outer loop. Note that the names of these must be specified completely. Partial matching will not work. The list items are as follows:

tol Convergence tolerance. Iteration is terminated when the absolute difference in function value between successive iteration is below tol. Default is 1.e-06.

maxfeval: Maximum number of objective function evaluations allowed. Default is min(5000, max(1500, 20*length(par)^2)).

regsimp A logical variable indicating whether the starting parameter configuration is a regular simplex. Default is TRUE.

maximize A logical variable indicating whether the objective function should be maximized. Default is FALSE.

restarts.max Maximum number of times the algorithm should be restarted before declaring failure. Default is 3.

trace A logical variable indicating whether the starting parameter configuration is a regular simplex. Default is FALSE.

References

C.T. Kelley (1999), Iterative Methods for Optimization, SIAM.

Examples

Run this code

 rosbkext <- function(x){
# Extended Rosenbrock function
 n <- length(x)
 sum (100*(x[1:(n-1)]^2 - x[2:n])^2 + (x[1:(n-1)] - 1)^2)
 }

np <- 10
set.seed(123)

p0 <- rnorm(np)
xm1 <- nmk(fn=rosbkext, par=p0) # maximum `fevals' is not sufficient to find correct minimum
xm1b <- nmkb(fn=rosbkext, par=p0, lower=-2, upper=2)

### A non-smooth problem
hald <- function(x) {
#Hald J & Madsen K (1981), Combined LP and quasi-Newton methods 
#for minimax optimization, Mathematical Programming, 20, p.42-62.
	i <- 1:21
	t <- -1 + (i - 1)/10
	f <- (x[1] + x[2] * t) / ( 1 + x[3]*t + x[4]*t^2 + x[5]*t^3) - exp(t)
	max(abs(f))
	}

p0 <- runif(5)
xm2 <- nmk(fn=hald, par=p0)
xm2b <- nmkb(fn=hald, par=p0, lower=c(0,0,0,0,-2), upper=4)

## Another 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)
nmk(par0, nsf) # fmin=8 at xmin=(2,2)

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