# General-purpose optimization

### Description

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

also tries to unify the calling sequence to allow
a number of tools to use the same front-end. These include
`spg`

from the BB package, `ucminf`

, `nlm`

, `nlminb`

,
`Rtnmin`

, `lbfgsb3`

, `lbfgs`

, `nmkb`

and `hjkb`

from the dfoptim package, `uobyqa`

, `newuoa`

and `bobyqa`

from the minqa package.

Note that
optim() itself allows Nelder–Mead, quasi-Newton and
conjugate-gradient algorithms as well as box-constrained optimization
via L-BFGS-B. 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 methods but SANN. Note that this is upper-case.

### Usage

1 2 3 4 |

### 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 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 |

`hess` |
A function to return (as a symmetric matrix) the Hessian of the objective function for those methods that can use this information. |

`lower, upper` |
Bounds on the variables for methods such as |

`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 |

`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 |

`control` |
A list of control parameters. See ‘Details’. |

`...` |
For |

### Details

Note that arguments after `...`

must be matched exactly.

Possible method codes at the time of writing are 'Nelder-Mead', 'BFGS', 'CG', 'L-BFGS-B', 'nlm', 'nlminb', 'spg', 'ucminf', 'newuoa', 'bobyqa', 'nmkb', 'hjkb', 'Rcgmin', or 'Rvmmin'.

The default methods for unconstrained problems (no `lower`

or
`upper`

specified) are an implementation of the Nelder and Mead
(1965) and a Variable Metric method based on the ideas of Fletcher
(1970) as modified by him in conversation with Nash (1979). Nelder-Mead
uses only function values and is robust but relatively slow. It will
work reasonably well for non-differentiable functions. The Variable
Metric method, `"BFGS"`

updates an approximation to the inverse
Hessian using the BFGS update formulas, along with an acceptable point
line search strategy. This method appears to work best with analytic
gradients. (`"Rvmmmin"`

provides a box-constrained version of this
algorithm.

If no `method`

is given, and there are bounds constraints provided,
the method is set to `"L-BFGS-B"`

.

Method `"CG"`

is a conjugate gradients method based on that by
Fletcher and Reeves (1964) (but with the option of Polak–Ribiere or
Beale–Sorenson updates). The particular implementation is now dated,
and improved yet simpler codes are being implemented (as at June 2009),
and furthermore a version with box constraints is being tested.
Conjugate gradient methods will generally be more fragile than the
BFGS method, but as they do not store a matrix they may be successful
in much larger optimization problems.

Method `"L-BFGS-B"`

is that of Byrd *et. al.* (1995) which
allows *box constraints*, that is each variable can be given a lower
and/or upper bound. The initial value must satisfy the constraints.
This uses a limited-memory modification of the BFGS quasi-Newton
method. If non-trivial bounds are supplied, this method will be
selected, with a warning.

Nocedal and Wright (1999) is a comprehensive reference for the previous three methods.

Function `fn`

can return `NA`

or `Inf`

if the function
cannot be evaluated at the supplied value, but the initial value must
have a computable finite value of `fn`

. However, some methods, of
which `"L-BFGS-B"`

is known to be a case, require that the values
returned should always be finite.

While `optim`

can be used recursively, and for a single parameter
as well as many, this may not be true for `optimx`

. `optim`

also accepts a zero-length `par`

, and just evaluates the function
with that argument.

Method `"nlm"`

is from the package of the same name that implements
ideas of Dennis and Schnabel (1983) and Schnabel et al. (1985). See nlm()
for more details.

Method `"nlminb"`

is the package of the same name that uses the
minimization tools of the PORT library. The PORT documentation is at
<URL: http://netlib.bell-labs.com/cm/cs/cstr/153.pdf>. See nlminb()
for details. (Though there is very little information about the methods.)

Method `"spg"`

is from package BB implementing a spectral projected
gradient method for large-scale optimization with simple constraints due
R adaptation, with significant modifications, by Ravi Varadhan,
Johns Hopkins University (Varadhan and Gilbert, 2009), from the original
FORTRAN code of Birgin, Martinez, and Raydan (2001).

Method `"Rcgmin"`

is from the package of that name. It implements a
conjugate gradient algorithm with the Yuan/Dai update (ref??) and also
allows bounds constraints on the parameters. (Rcgmin also allows mask
constraints – fixing individual parameters – but there is no interface
from `"optimx"`

.)

Methods `"bobyqa"`

, `"uobyqa"`

and `"newuoa"`

are from the
package `"minqa"`

which implement optimization by quadratic approximation
routines of the similar names due to M J D Powell (2009). See package minqa
for details. Note that `"uobyqa"`

and `"newuoa"`

are for
unconstrained minimization, while `"bobyqa"`

is for box constrained
problems. While `"uobyqa"`

may be specified, it is NOT part of the
`all.methods = TRUE`

set.

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.)`follow.on`

= TRUE or FALSE. If TRUE, and there are multiple methods, then the last set of parameters from one method is used as the starting set for the next.

`save.failures`

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

`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 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`kkt`

is FALSE but`hessian`

is TRUE, then`KKT1`

is generated, but`KKT2`

is not.`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

`starttests`

= TRUE if we want to run tests of the function and parameters: feasibility relative to bounds, analytic vs numerical gradient, scaling tests, before we try optimization methods. Default is TRUE.

`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

`usenumDeriv`

= TRUE if the

`numDeriv`

function`grad()`

is to be used to compute gradients when the argument`gr`

is NULL or not supplied.

The following `control`

elements apply only to some of the methods. The list
may be incomplete. See individual packages for details.

`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`

.`ndeps`

A vector of step sizes for the finite-difference approximation to the gradient, on

`par/parscale`

scale. Defaults to`1e-3`

. For`optim`

.`maxit`

The maximum number of iterations. Defaults to

`100`

for the derivative-based methods, and`500`

for`"Nelder-Mead"`

.`abstol`

The absolute convergence tolerance. Only useful for non-negative functions, as a tolerance for reaching zero.

`reltol`

Relative convergence tolerance. The algorithm stops if it is unable to reduce the value by a factor of

`reltol * (abs(val) + reltol)`

at a step. Defaults to`sqrt(.Machine$double.eps)`

, typically about`1e-8`

. For`optim`

.`alpha`

,`beta`

,`gamma`

Scaling parameters for the

`"Nelder-Mead"`

method.`alpha`

is the reflection factor (default 1.0),`beta`

the contraction factor (0.5) and`gamma`

the expansion factor (2.0).`REPORT`

The frequency of reports for the

`"BFGS"`

and`"L-BFGS-B"`

methods if`control$trace`

is positive. Defaults to every 10 iterations for`"BFGS"`

and`"L-BFGS-B"`

.`type`

for the conjugate-gradients method. Takes value

`1`

for the Fletcher–Reeves update,`2`

for Polak–Ribiere and`3`

for Beale–Sorenson.`lmm`

is an integer giving the number of BFGS updates retained in the

`"L-BFGS-B"`

method, It defaults to`5`

.`factr`

controls the convergence of the

`"L-BFGS-B"`

method. Convergence occurs when the reduction in the objective is within this factor of the machine tolerance. Default is`1e7`

, that is a tolerance of about`1e-8`

.`pgtol`

helps control the convergence of the

`"L-BFGS-B"`

method. It is a tolerance on the projected gradient in the current search direction. This defaults to zero, when the check is suppressed.

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.

- 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 `par`

s, 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;

`scalecheck()`

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | ```
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
``` |

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