nvm: Variable metric nonlinear function minimization, driver.

In optimx: Expanded Replacement and Extension of the 'optim' Function

Variable metric nonlinear function minimization, driver.

Description

A driver to call an R implementation of a variable metric method for minimization of nonlinear functions, possibly subject to bounds (box) constraints and masks (fixed parameters). The algorithm is based on Nash (1979) Algorithm 21 for main structure, which is itself drawn from Fletcher's (1970) variable metric code. This is also the basis of optim() method 'BFGS' which, however, does not deal with bounds or masks, or Rvmmin. In this method, an approximation to the inverse Hessian (B) is used to generate a search direction t = - B %*% g, a simple backtracking line search is used until an acceptable point is found, and the matrix B is updated using a BFGS formula. If no acceptable point can be found, we reset B to the identity i.e., the search direction becomes the negative gradient. If the search along the negative gradient is unsuccessful, the method terminates.

The code is entirely in R to allow users to explore and understand the method. However, nvm() is intended to be called via optimx::optimr() and NOT called directly, as it has limited sanity checks on the problem provided, since such checks are in the optimr() code.

The earlier Rvmmin() function does have such checks, and was originally part of a separate package of the same name. Rvmmin() can also be called via optimr(). It may give slightly different results due to minor internal coding changes, and is kept available for backward compatibility with other packages.

Usage

   nvm(par, fn, gr, bds, control = list())

Arguments

par	A numeric vector of starting estimates.
fn	A function that returns the value of the objective at the supplied set of parameters par. This function is created within optimr() and subsumes auxiliary data in ... supplied to that wrapper.
gr	A function that returns the gradient of the objective at the supplied set of parameters par. Note that this is usually generated within the optimr() wrapper, where a gradient function or a reference to one of the derivative approximation routines must be provided. See the documentation for optimr() for details.
bds	A list of information resulting from function bmchk giving information on the status of the parameters and bounds and masks.
control	An optional list of control settings.

Details

Function fn must return a numeric value.

Note that nvm is to be called from optimr and does NOT allow dot arguments. It is intended to use the internal functions efn and egr generated inside optimr() along with bounds information from bmchk() available there.

The control argument is a list. See the documentation of ctrldefault().

The source codes Rvmmin and nvm for R are still a work in progress, so users should watch the console output. The routine nvm attempts to use minimal checking and works only with a bounds constrained version of the algorithm, which may work as fast as a specific routine for unconstrained problems. This is an open question, and the author welcomes feedback.

Value

A list with components:

par	The best set of parameters found.
value	The value of the objective at the best set of parameters found.
counts	A vector of two integers giving the number of function and gradient evaluations.
convergence	An integer indicating the situation on termination of the function. 0 indicates that the method believes it has succeeded. Other values: 0 indicates successful termination to an acceptable solution 1 indicates that the iteration limit maxit had been reached. 2 indicates that a point with a small gradient norm has been found, which is likely a solution. 20 indicates that the initial set of parameters is inadmissible, that is, that the function cannot be computed or returns an infinite, NULL, or NA value. 21 indicates that an intermediate set of parameters is inadmissible.
message	A description of the situation on termination of the function.
bdmsk	Returned index describing the status of bounds and masks at the proposed solution. Parameters for which bdmsk are 1 are unconstrained or "free", those with bdmsk 0 are masked i.e., fixed. For historical reasons, we indicate a parameter is at a lower bound using -3 or upper bound using -1.

References

Fletcher, R (1970) A New Approach to Variable Metric Algorithms, Computer Journal, 13(3), pp. 317-322.

Nash, J C (1979, 1990) Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, Bristol: Adam Hilger. Second Edition, Bristol: Institute of Physics Publications.

See Also

optim

Examples

# library(optimx)
## Rosenbrock Banana function
fr <- function(x) {
  x1 <- x[1]
  x2 <- x[2]
  100 * (x2 - x1 * x1)^2 + (1 - x1)^2
}
grr <- function(x) { ## Gradient of 'fr'
  x1 <- x[1]
  x2 <- x[2]
  c(-400 * x1 * (x2 - x1 * x1) - 2 * (1 - x1),
    200 *      (x2 - x1 * x1))
}
# Call is from optimr(). In this case, small final gradient
ansrosenbrock0 <- optimr(fn=fr,gr=grr, par=c(1,2), method="nvm")
print(ansrosenbrock0) 
#
# Test if 1-parameter problem is possible
#
cat("test n=1 problem using simple squares of parameter\n")

sqtst<-function(xx) {
  res<-sum((xx-2)*(xx-2))
}

nn<-1
startx<-rep(0,nn)
onepar<-optimr(startx,sqtst, gr="grfwd", method="nvm", control=list(trace=1)) 
print(onepar)

optimx documentation built on April 11, 2025, 5:43 p.m.