General-purpose optimization

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Description

General-purpose optimization wrapper function that calls multiple other R tools for optimization, including the existing optim() function tools.

Because SANN does not return a meaningful convergence code (conv), opm() does not call the SANN method, but it can be invoked in optimr().

There is a pseudo-method "ALL" that runs all available methods. Note that this is upper-case.

Usage

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opm(par, fn, gr=NULL, hess=NULL, lower=-Inf, upper=Inf, 
            method=c("Nelder-Mead","BFGS"), hessian=FALSE,
            control=list(),
             ...)

Arguments

par

a vector of initial values for the parameters for which optimal values are to be found. Names on the elements of this vector are preserved and used in the results data frame.

fn

A function to be minimized (or maximized), with a first argument the vector of parameters over which minimization is to take place. It should return a scalar result.

gr

A function to return (as a vector) the gradient for those methods that can use this information.

If 'gr' is NULL, whatever default actions are supplied by the methods specified will be used. However, some methods REQUIRE a gradient function, so will fail in this case. opm() will generally return with convergence set to 9998 for such methods.

If 'gr' is a character string, this character string will be taken to be the name of an available gradient approximation function. Examples are "grfwd", "grback", "grcentral" and "grnd", with the last name referring to the default method of package numDeriv.

hess

A function to return (as a symmetric matrix) the Hessian of the objective function for those methods that can use this information. At the time of writing, we know of none in the methods which can be called by opm().

lower, upper

Bounds on the variables for methods such as "L-BFGS-B" that can handle box (or bounds) constraints. These are vectors.

method

A vector of the methods to be used, each as a character string. Note that this is an important change from optim() that allows just one method to be specified. See ‘Details’. If method has just one element, "ALL" (capitalized), all available and appropriate methods will be tried.

hessian

A logical control that if TRUE forces the computation of an approximation to the Hessian at the final set of parameters. If FALSE (default), the hessian is calculated if needed to provide the KKT optimality tests (see kkt in ‘Details’ for the control list). This setting is provided primarily for compatibility with optim().

control

A list of control parameters. See ‘Details’.

...

For optimx further arguments to be passed to fn and gr; otherwise, further arguments are not used.

Details

Note that arguments after ... must be matched exactly.

For details of how opm() calls the methods, see the documentation and code for optimr(). The documentation and code for individual methods may also be useful. Note that some simplification of the calls may have been necessary, for example, to provide reasonable default values for method controls.

The control argument is a list that can supply any of the following components:

trace

Non-negative integer. If positive, tracing information on the progress of the optimization is produced. Higher values may produce more tracing information: for method "L-BFGS-B" there are six levels of tracing. trace = 0 gives no output (To understand exactly what these do see the source code: higher levels give more detail.)

fnscale

An overall scaling to be applied to the value of fn and gr during optimization. If negative, turns the problem into a maximization problem. Optimization is performed on fn(par)/fnscale. For methods from the set in optim(). Note potential conflicts with the control maximize.

parscale

A vector of scaling values for the parameters. Optimization is performed on par/parscale and these should be comparable in the sense that a unit change in any element produces about a unit change in the scaled value.For optim.

save.failures

= TRUE (default) if we wish to keep "answers" from runs where the method does not return convcode==0. FALSE otherwise.

maximize

= TRUE if we want to maximize rather than minimize a function. (Default FALSE). Methods nlm, nlminb, ucminf cannot maximize a function, so the user must explicitly minimize and carry out the adjustment externally. However, there is a check to avoid usage of these codes when maximize is TRUE. See fnscale below for the method used in optim that we deprecate.

all.methods

= TRUE if we want to use all available (and suitable) methods. This is equivalent to setting method="ALL"

kkt

=FALSE if we do NOT want to test the Kuhn, Karush, Tucker optimality conditions. The default is generally TRUE. However, because the Hessian computation may be very slow, we set kkt to be FALSE if there are more than than 50 parameters when the gradient function gr is not provided, and more than 500 parameters when such a function is specified. We return logical values KKT1 and KKT2 TRUE if first and second order conditions are satisfied approximately. Note, however, that the tests are sensitive to scaling, and users may need to perform additional verification. If hessian is TRUE, this overrides control kkt.

all.methods

= TRUE if we want to use all available (and suitable) methods.

kkttol

= value to use to check for small gradient and negative Hessian eigenvalues. Default = .Machine$double.eps^(1/3)

kkt2tol

= Tolerance for eigenvalue ratio in KKT test of positive definite Hessian. Default same as for kkttol

dowarn

= FALSE if we want to suppress warnings generated by opm() or optimr(). Default is TRUE.

badval

= The value to set for the function value when try(fn()) fails. Default is (0.5)*.Machine$double.xmax

There may be control elements that apply only to some of the methods. Using these may or may not "work" with opm(), and errors may occur with methods for which the controls have no meaning. However, it should be possible to call the underlying optimr() function with these method-specific controls.

Any names given to par will be copied to the vectors passed to fn and gr. Note that no other attributes of par are copied over. (We have not verified this as at 2009-07-29.)

Value

If there are npar parameters, then the result is a dataframe having one row for each method for which results are reported, using the method as the row name, with columns

par_1, .., par_npar, value, fevals, gevals, niter, convcode, kkt1, kkt2, xtimes

where

par_1

..

par_npar

The best set of parameters found.

value

The value of fn corresponding to par.

fevals

The number of calls to fn.

gevals

The number of calls to gr. This excludes those calls needed to compute the Hessian, if requested, and any calls to fn to compute a finite-difference approximation to the gradient.

niter

For those methods where it is reported, the number of “iterations”. See the documentation or code for particular methods for the meaning of such counts.

convcode

An integer code. 0 indicates successful convergence. Various methods may or may not return sufficient information to allow all the codes to be specified. An incomplete list of codes includes

1

indicates that the iteration limit maxit had been reached.

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.

10

indicates degeneracy of the Nelder–Mead simplex.

51

indicates a warning from the "L-BFGS-B" method; see component message for further details.

52

indicates an error from the "L-BFGS-B" method; see component message for further details.

9998

indicates that the method has been called with a NULL 'gr' function, and the method requires that such a function be supplied.

9999

indicates the method has failed.

kkt1

A logical value returned TRUE if the solution reported has a “small” gradient.

kkt2

A logical value returned TRUE if the solution reported appears to have a positive-definite Hessian.

xtimes

The reported execution time of the calculations for the particular method.

The attribute "details" to the returned answer object contains information, if computed, on the gradient (ngatend) and Hessian matrix (nhatend) at the supposed optimum, along with the eigenvalues of the Hessian (hev), as well as the message, if any, returned by the computation for each method, which is included for each row of the details. If the returned object from optimx() is ans, this is accessed via the construct attr(ans, "details")

This object is a matrix based on a list so that if ans is the output of optimx then attr(ans, "details")[1, ] gives the first row and attr(ans,"details")["Nelder-Mead", ] gives the Nelder-Mead row. There is one row for each method that has been successful or that has been forcibly saved by save.failures=TRUE.

There are also attributes

maximize

to indicate we have been maximizing the objective

npar

to provide the number of parameters, thereby facilitating easy extraction of the parameters from the results data frame

follow.on

to indicate that the results have been computed sequentially, using the order provided by the user, with the best parameters from one method used to start the next. There is an example (ans9) in the script ox.R in the demo directory of the package.

Note

Most methods in optimx will work with one-dimensional pars, but such use is NOT recommended. Use optimize or other one-dimensional methods instead.

There are a series of demos available. Once the package is loaded (via require(optimx) or library(optimx), you may see available demos via

demo(package="optimx")

The demo 'brown_test' may be run with the command demo(brown_test, package="optimx")

The package source contains several functions that are not exported in the NAMESPACE. These are

optimx.setup()

which establishes the controls for a given run;

optimx.check()

which performs bounds and gradient checks on the supplied parameters and functions;

optimx.run()

which actually performs the optimization and post-solution computations;

scalechk()

which actually carries out a check on the relative scaling of the input parameters.

Knowledgeable users may take advantage of these functions if they are carrying out production calculations where the setup and checks could be run once.

Source

See the manual pages for optim() and the packages the DESCRIPTION suggests.

References

See the manual pages for optim() and the packages the DESCRIPTION suggests.

Nash JC, and Varadhan R (2011). Unifying Optimization Algorithms to Aid Software System Users: optimx for R., Journal of Statistical Software, 43(9), 1-14., URL http://www.jstatsoft.org/v43/i09/.

Nash JC (2014). On Best Practice Optimization Methods in R., Journal of Statistical Software, 60(2), 1-14., URL http://www.jstatsoft.org/v60/i02/.

See Also

spg, nlm, nlminb, bobyqa, Rcgmin, Rvmmin, ucminf, nmkb, hjkb. optimize for one-dimensional minimization; constrOptim or spg for linearly constrained optimization.

Examples

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require(graphics)
cat("Note possible demo(ox) for extended examples\n")


## Show multiple outputs of optimx using all.methods
# genrose function code
genrose.f<- function(x, gs=NULL){ # objective function
## One generalization of the Rosenbrock banana valley function (n parameters)
	n <- length(x)
        if(is.null(gs)) { gs=100.0 }
	fval<-1.0 + sum (gs*(x[1:(n-1)]^2 - x[2:n])^2 + (x[2:n] - 1)^2)
        return(fval)
}

genrose.g <- function(x, gs=NULL){
# vectorized gradient for genrose.f
# Ravi Varadhan 2009-04-03
	n <- length(x)
        if(is.null(gs)) { gs=100.0 }
	gg <- as.vector(rep(0, n))
	tn <- 2:n
	tn1 <- tn - 1
	z1 <- x[tn] - x[tn1]^2
	z2 <- 1 - x[tn]
	gg[tn] <- 2 * (gs * z1 - z2)
	gg[tn1] <- gg[tn1] - 4 * gs * x[tn1] * z1
	return(gg)
}

genrose.h <- function(x, gs=NULL) { ## compute Hessian
   if(is.null(gs)) { gs=100.0 }
	n <- length(x)
	hh<-matrix(rep(0, n*n),n,n)
	for (i in 2:n) {
		z1<-x[i]-x[i-1]*x[i-1]
		z2<-1.0-x[i]
                hh[i,i]<-hh[i,i]+2.0*(gs+1.0)
                hh[i-1,i-1]<-hh[i-1,i-1]-4.0*gs*z1-4.0*gs*x[i-1]*(-2.0*x[i-1])
                hh[i,i-1]<-hh[i,i-1]-4.0*gs*x[i-1]
                hh[i-1,i]<-hh[i-1,i]-4.0*gs*x[i-1]
	}
        return(hh)
}

startx<-4*seq(1:10)/3.
ans8<-opm(startx,fn=genrose.f,gr=genrose.g, hess=genrose.h, 
   control=list(all.methods=TRUE, save.failures=TRUE, trace=1), gs=10)
ans8
ans8[, "gevals"]
ans8["spg", ]
summary(ans8, par.select = 1:3)
summary(ans8, order = value)[1, ] # show best value
head(summary(ans8, order = value)) # best few
## head(summary(ans8, order = "value")) # best few -- alternative syntax

## order by value.  Within those values the same to 3 decimals order by fevals.
## summary(ans8, order = list(round(value, 3), fevals), par.select = FALSE)
summary(ans8, order = "list(round(value, 3), fevals)", par.select = FALSE)

## summary(ans8, order = rownames, par.select = FALSE) # order by method name
summary(ans8, order = "rownames", par.select = FALSE) # same

summary(ans8, order = NULL, par.select = FALSE) # use input order
## summary(ans8, par.select = FALSE) # same