Variable metric nonlinear function minimization, driver.

A driver to call the unconstrained and bounds constrained versions of 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. In the present 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.

This set of codes is entirely in R to allow users to explore and understand the method. It also allows bounds (or box) constraints and masks (equality constraints) to be imposed on parameters.

Rvmmin(par, fn, gr, lower, upper, bdmsk, control = list(), ...)

   Rvmminb(par, fn, gr, lower, upper, bdmsk, control = list(), ...)
   
   Rvmminu(par, fn, gr, 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 using auxiliary data in .... The first argument of fn must be par.

gr

A function that returns the gradient of the objective at the supplied set of parameters par using auxiliary data in .... The first argument of fn must be par. This function returns the gradient as a numeric vector.

Note that a gradient function must generally be provided. However, to ensure compatibility with other optimizers, if gr is NULL, the forward gradient approximation from routine grfwd will be used.

The use of numerical gradients for Rvmmin is discouraged. First, the termination test uses a size measure on the gradient, and numerical gradient approximations can sometimes give results that are too large. Second, if there are bounds constraints, the step(s) taken to calculate the approximation to the derivative are NOT checked to see if they are out of bounds, and the function may be undefined at the evaluation point.

There is also the option of using the routines grfwd, grback, grcentral or grnd. The last of these calls the grad() function from package numDeriv. These are called by putting the name of the (numerical) gradient function in quotation marks, e.g.,

gr="grfwd"

to use the standard forward difference numerical approximation.

Note that all but the grnd routine use a stepsize parameter that can be redefined in a special scratchpad storage variable deps. The default is deps = 1e-07. However, redefining this is discouraged unless you understand what you are doing.

lower

A vector of lower bounds on the parameters.

upper

A vector of upper bounds on the parameters.

bdmsk

An indicator vector, having 1 for each parameter that is "free" or unconstrained, and 0 for any parameter that is fixed or MASKED for the duration of the optimization.

control

An optional list of control settings.

...

Further arguments to be passed to fn.

Details

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 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.

Function fn must return a numeric value.

The control argument is a list.

maxit: A limit on the number of iterations (default 500 + 2*n where n is the number of parameters). This is the maximum number of gradient evaluations allowed.
maxfeval: A limit on the number of function evaluations allowed (default 3000 + 10*n).
trace: Set 0 (default) for no output, > 0 for diagnostic output (larger values imply more output).
dowarn: = TRUE if we want warnings generated by optimx. Default is TRUE.
checkgrad: = TRUE if we wish analytic gradient code checked against the approximations computed by numDeriv. Default is FALSE.
checkbounds: = TRUE if we wish parameters and bounds to be checked for an admissible and feasible start. Default is TRUE.
keepinputpar: = TRUE if we want bounds check to stop program when parameters are out of bounds. Else when FALSE, moves parameter values to nearest bound. Default is FALSE.
maximize: To maximize user_function, supply a function that computes (-1)*user_function. An alternative is to call Rvmmin via the package optimx.
eps: a tolerance used for judging small gradient norm (default = 1e-07). a gradient norm smaller than (1 + abs(fmin))*eps*eps is considered small enough that a local optimum has been found, where fmin is the current estimate of the minimal function value.
acctol: To adjust the acceptable point tolerance (default 0.0001) in the test ( f <= fmin + gradproj * steplength * acctol ). This test is used to ensure progress is made at each iteration.
stepredn: Step reduction factor for backtrack line search (default 0.2)
reltest: Additive shift for equality test (default 100.0)
stopbadupdate: A logical flag that if set TRUE will halt the optimization if the Hessian inverse cannot be updated after a steepest descent search. This indicates an ill-conditioned Hessian. A settign of FALSE causes Rvmmin methods to be aggressive in trying to optimize the function, but may waste effort. Default TRUE.

As of 2011-11-21 the following controls have been REMOVED

usenumDeriv: There is now a choice of numerical gradient routines. See argument gr.

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.

Examples