XploRe Help : nmBFGS

Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Library:	nummath
See also:	nmgraditer nmgraddiff nmcongrad nmlinminappr

Quantlet:		nmBFGS
Description:		Broyden-Fletcher-Goldfarb-Shanno method to find a minimum of a given function.

Usage:	`min = nmBFGS(func,x0{,fder,linmin,ftol,gtol,maxiter,nowarn}) or min = nmBFGS(func,x0{,fder,opt})`
Input:
	`func`	string, name of the function to minimize. The function should have just one parameter x, which is either a n x 1 vector, if fder is given (see below), or a n x k matrix (for any k >= 1) containing x = (x1,x2,...,xk), if the gradient is to be computed automatically; xi (n x 1 vector; i=1..k) represent points at which the function should be evaluated. As a result, the function should return a scalar.
	`x0`	n x 1 vector, the initial estimate of the minimum
	`fder`	(optional) derivatives of func; there are several possible formats: 1. string, name of the function for computing the gradient of func; its output is a n x 1 vector 2. empty string; in this case the gradient will be computed automatically using the quantlet nmgraddiff with a default step h 3. scalar h; the gradient will be computed automatically using the quantlet nmgraddiff with the given step h
	`opt`	(optional) list containing all or some of the following items: linmin, ftol, maxiter and nowarn as described below
	`linmin`	(optional) string, name of the routine for 1D (line) minimization; default is linmin = "nmlinminappr"
	`ftol`	(optional) scalar, reserved for future usage; convergence tolerance of the function value, default is ftol = 1e-7
	`gtol`	(optional) scalar, convergence tolerance of the value of the function gradient; default is gtol = 1e-9
	`maxiter`	(optional) scalar, maximal number of iterations; default is maxiter = 250
	`nowarn`	(optional) scalar; by default, nowarn = 0. If nowarn is set to a nonzero value, no warnings will be shown and nowarn = 1 will be set to 1 for quantlets called by nmBFGS having this option
Output:
	`min.xmin`	n x 1 vector, minimum of func (isolated to a fractional precision of tol)
	`min.fmin`	scalar, minimal function value f(xmin)
	`min.iter`	scalar, number of performed iterations

Example:

library("nummath") ; ; definition of function ; proc(p)=ftion(x) p = x[1,]^2 + 3*(x[2,]-1)^4 endp ; nmBFGS("ftion",#(1,2)) ; ; search for a minimum of f=x1^2 + 3*(x2-1)^4 with ; initial estimation x0=(1,2)

Result:

Contents of _tmp.xmin
[1,]  2.1573e-21
[2,]  0.99967
Contents of _tmp.fmin
[1,]  3.6672e-14
Contents of _tmp.iter
[1,]       25

Example:

library("nummath") ; ; variables: simulated probit ; randomize(1) n = 1000 x = normal(n,2) eps = normal(n) x1 = matrix(n)~x beta = #(0.5,1,-2) y = x1 * beta + eps yd =(y >= 0) ; ; definition of probit likelihood function ; L = log prod(cdfn(x1*beta)^yd *(1-cdfn(x1*beta))^(1-yd)) ; = sum log(...) ; for numerical reasons, we will define it rather as follows: ; proc(L)=ftion(beta) yd = getglobal("yd") x1 = getglobal("x1") L1 = log((cdfn(x1*beta).*yd) +(1-yd)) L2 = log(((1-cdfn(x1*beta)).*(1-yd)) + yd) L = - sum(L1) - sum(L2) ; maximum of likelihood = minimum of L endp ; min = nmBFGS("ftion",#(0,0,0)) min ; notice that ML(beta=min.xmin) = -min.fmin

Result:

Contents of min.xmin
[1,]  0.53581
[2,]  0.93027
[3,]  -2.0261
Contents of min.fmin
[1,]    288.3
Contents of min.iter
[1,]       12

Example:

; shows inputing optional parameters as a list library("nummath") ; definition of function proc(p)=ftion(x) p = x[1,]^4 + 5*(x[2,]-7)^2 + 8 endp ; gtol = 1e-5 maxiter = 15 opt = list(gtol,maxiter) nmBFGS("ftion",#(1,2),"",opt) ; ; search for a minimum of f=x1^4 + 5*(x2-7)^2 + 8 ; with initial estimation x0=(1,2)

Result:

Contents of _tmp.xmin
[1,] -0.012726
[2,]        7

Contents of _tmp.fmin
[1,]        8

Contents of _tmp.iter
[1,]       13

Author: L. Cizkova, 20021011 license MD*Tech