NR {lazy.mat}

R Documentation

Newton-Raphson Method for Minimization

Description

Newton-Raphson Method for Minimization

Usage

NR(
  x,
  obj,
  gradient = NULL,
  hessian = NULL,
  sd = 0,
  flipd = 0,
  maxiter = 100,
  eps = 1e-07,
  epsg = 1e-09,
  epsx = 1e-09,
  maxiter2 = 10,
  normalize = NULL,
  check_pm = 0,
  print = 1,
  ...
)

Arguments

x	initial parameter vector
obj	function to be minimized, the first arg of which must be x.
gradient	function which returns the gradient vector or NULL
hessian	function which returns the Hessian matrix or NULL
sd	= 1 to use steepest descent
flipd	change the sign of d if t(gradient)%*%Hessian%*%gradient < 0
maxiter	max # of iterations
eps	eps for the relative change of the function value
epsg	eps for the mazimum absolute value of the gradient
epsx	eps for the maximum absolute change of the parameter value
maxiter2	max # of stepsize halving
normalize	a function to normalize the solution after each iteration
check_pm	= 1 to verify partial match of the ... argument
print	= 1 to print the result, = 2 to print the history
...	additional parameters to obj, gradient and hessian

Details

parameter values will be updated as

  x_k+1 = x_k - alpha ginv(Hessian) grad

where alpha is the stepsize to be halved, from 1, when obj increases.

When gradient or hessian are NULL, JacobianMat of this package will be used to calculate them numerically.
See the examples in here ExamplesOfGNNR.

Warning
When the formal arguments of this function partially match the names given as ..., this function does NOT work. In order to avoid this, if any of ... matches the formal arguments, specify those formal arguments by their complete names when calling this function.

Value

A list of:
par the parameter value at minimum
objective the function value minimized
grad the gradient vector
hist a matrix containing the iteration history
iterations number of iterations performed

Examples

 # function to be minimized
 #  x is the variable w.r.t. which f is minimized.
 #  a and b are the optional parameters of f.
 #
 symbolic_f <- expression( exp(a*x)/(sin(b*x^2)+x) )

 # convert symbolic expression to numeric function with optional parameters
 # eval(parse(text=paste("f <- function(x) ", symbolic_f)))
 f <- function(x, a=1, b=1){eval(symbolic_f)}

 # symbolic differentiation of f with optional parameters
 symbolic_df2 <- deriv(symbolic_f,"x", hessian=TRUE
                       , function(x, a=0, b=0){} )

 # convert symbolic expression to numeric function with optional parameters
 df <- function(x, a=1, b=1){
   attributes(symbolic_df2(x, a=a, b=b))$gradient[,1, drop=0] }
 df2 <- function(x, a=1, b=1){
   matrix(attributes(symbolic_df2(x, a=a, b=b))$hessian[,1,1]
          , length(x),length(x)) }


 # optional parameter to f
 a=2; b=5

## Not run: 
 # shape of f
  x=seq(1,3,0.01)
  plot( x, f(x,a,b), type="l",main=paste(symbolic_f, " with a=",a,", b=",b) )

## End(Not run)

 # initial value
 xinit=2.65


 #
 # minimization of f by NR
 #

 # numerical gradient and Hessian
 resNR1=NR( xinit, f, flipd=1, a=a,b=b, print=2 )
 xNR1=resNR1$par
 gNR1=df( xNR1, a=a, b=b )
 # analytic gradient and numerical Hessian
 resNR2=NR( xinit, f, flipd=1, gradient=df, a=a,b=b, print=2 )
 xNR2=resNR2$par
 gNR2=df( xNR2, a=a, b=b )
 # analytic gradient and Hessian
 resNR3=NR( xinit, f, flipd=1, gradient=df, hessian=df2, a=a,b=b, print=2 )
 xNR3=resNR3$par
 gNR3=df( xNR3, a=a, b=b )
 Print(xNR1,xNR2,xNR3)
 Print(gNR1,gNR2,gNR3)

## Not run: 
 #
 # minimization of f by native nlminb
 #


 res1=nlminb( xinit, f, control=list(trace=0), a=a, b=b )
 x1=res1$par
 g1=df( x1, a=a, b=b )
 res2=nlminb( xinit, f, control=list(trace=0), a=a, b=b, gradient=df )
 x2=res2$par
 g2=df( x2, a=a, b=b )
 res3=nlminb( xinit, f, control=list(trace=0), a=a, b=b
              , gradient=df, hessian=df2 )
 x3=res3$par
 g3=df( x3, a=a, b=b )
 Print(x1,x2,x3)
 Print(g1,g2,g3)

## End(Not run)

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