# General-purpose optimization

### Description

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

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`

, and
`nlminb`

. 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), optimz::optim() does not call the SANN method.

### 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 |

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

`method` |
A list of the methods to be used. Note that this is an important change from optim() that allows just one method to be specified. See ‘Details’. The default of NULL causes an appropriate set of methods to be supplied depending on the presence or absence of bounds on the parameters. The default unconstrained set is Rvmminu, Rcgminu, lbfgsb3, newuoa and nmkb. The default bounds constrained set is Rvmminb, Rcgminb, lbfgsb3, bobyqa and nmkb. |

`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.

By default this function performs minimization, but it will maximize
if `control$maximize`

is TRUE. The original optim() function allows
`control$fnscale`

to be set negative to accomplish this. DO NOT
use both methods.

Possible method codes are 'Nelder-Mead', 'BFGS', 'CG', 'L-BFGS-B', 'nlm', 'nlminb', 'spg', 'ucminf', 'newuoa', 'bobyqa', 'nmkb', 'hjkb', 'Rcgmin', 'lbfgsb3' or 'Rvmmin'. These are in base R or in CRAN repositories. From R-forge, method 'Rtnmin' is available. Other methods are likely to be added over time.

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 have been implemented. Furthermore,
`"Rcgmin"`

allows box constraints as well as fixed (masked)
parameters. 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 optimization problems with a large number of parameters.

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 is selected
by the original `optim()`

function, with a warning. Unfortunately,
the authors of the original Fortran version of this method released a
correction for bugs in 2011, but these have not been incorporated into
the distributed R codes, which are a C translation of a version that
appears to be from the mid-1990s. While it seems the errors affect
very few computations, users may wish to use the Fortran codes in
package `lbfgsb3`

.

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 as yet no
interface from `"optimr"`

.)

Method `"Rvmmin"`

is from the package of that name. It implements
the same variable metric method as the base optim() function with method
`"BFGS"`

but allows bounds constraints on the parameters. (Rvmmin
also allows mask constraints – fixing individual parameters – but
there is as yet no interface from `"optimr"`

.)

Method `"Rtnmin"`

is from the package of that name. It implements a
truncated Newton method of Stephen Nash translated from Matlab. It
allows bounds constraints on the parameters.

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.

Methods `"nmkb"`

and `"hjkb"`

are from package dfoptim. They
implement respectively variants of the Nelder-Mead and Hooke and Jeeves
derivative-free methods, but both allow bounds constraints. However, it is
important to note that `"nmkb"`

must NOT have starting parameters on
a lower or upper bound, as a transformation of the paramters is used to
effect the constraints.

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.

`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`

= TRUE if we want warnings generated by optimx. 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

For ‘optim’, a list with components:

`par` |
The best set of parameters found. |

`value` |
The value of ‘fn’ corresponding to ‘par’. |

`counts` |
A two-element integer vector giving the number of calls to ‘fn’ and ‘gr’ respectively. 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. |

`convergence` |
An integer code. ‘0’ indicates successful completion |

` message` |
A character string giving any additional information returned by the optimizer, or ‘NULL’. |

`hessian` |
Always NULL for this routine. |

### 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/.

?? Nocedal

?? Yuan and Dai