# optim

##### General-purpose Optimization

General-purpose optimization based on Nelder--Mead, quasi-Newton and conjugate-gradient algorithms. It includes an option for box-constrained optimization and simulated annealing.

##### Usage

```
optim(par, fn, gr = NULL, ...,
method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN",
"Brent"),
lower = -Inf, upper = Inf,
control = list(), hessian = FALSE)
```optimHess(par, fn, gr = NULL, ..., control = list())

##### Arguments

- par
- Initial values for the parameters to be optimized over.
- 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 the gradient for the
`"BFGS"`

,`"CG"`

and`"L-BFGS-B"`

methods. If it is`NULL`

, a finite-difference approximation will be used.For the

`"SANN"`

method it specifies a function to generate a new candidate point. If it is`NULL`

a default Gaussian Markov kernel is used. - ...
- Further arguments to be passed to
`fn`

and`gr`

. - method
- The method to be used. See
Details . Can be abbreviated. - lower, upper
- Bounds on the variables for the
`"L-BFGS-B"`

method, or bounds in which to*search*for method`"Brent"`

. - control
- A list of control parameters. See
Details . - hessian
- Logical. Should a numerically differentiated Hessian matrix be returned?

##### Details

Note that arguments after `...`

must be matched exactly.

By default `optim`

performs minimization, but it will maximize
if `control$fnscale`

is negative. `optimHess`

is an
auxiliary function to compute the Hessian at a later stage if
`hessian = TRUE`

was forgotten.

The default method is an implementation of that of Nelder and Mead (1965), that uses only function values and is robust but relatively slow. It will work reasonably well for non-differentiable functions.

Method `"BFGS"`

is a quasi-Newton method (also known as a variable
metric algorithm), specifically that published simultaneously in 1970
by Broyden, Fletcher, Goldfarb and Shanno. This uses function values
and gradients to build up a picture of the surface to be optimized.

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). 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.

Method `"SANN"`

is by default a variant of simulated annealing
given in Belisle (1992). Simulated-annealing belongs to the class of
stochastic global optimization methods. It uses only function values
but is relatively slow. It will also work for non-differentiable
functions. This implementation uses the Metropolis function for the
acceptance probability. By default the next candidate point is
generated from a Gaussian Markov kernel with scale proportional to the
actual temperature. If a function to generate a new candidate point is
given, method `"SANN"`

can also be used to solve combinatorial
optimization problems. Temperatures are decreased according to the
logarithmic cooling schedule as given in Belisle (1992, p.`temp / log(((t-1) %/% tmax)*tmax + exp(1))`

, where `t`

is
the current iteration step and `temp`

and `tmax`

are
specifiable via `control`

, see below. Note that the
`"SANN"`

method depends critically on the settings of the control
parameters. It is not a general-purpose method but can be very useful
in getting to a good value on a very rough surface.

Method `"Brent"`

is for one-dimensional problems only, using
`optimize()`

. It can be useful in cases where
`optim()`

is used inside other functions where only `method`

can be specified, such as in `mle`

from package

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`

.
(Except for method `"L-BFGS-B"`

where the values should always be
finite.)

`optim`

can be used recursively, and for a single parameter
as well as many. It also accepts a zero-length `par`

, and just
evaluates the function with that argument.

The `control`

argument is a list that can supply any of the
following components:
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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.

The parameter vector passed to `fn`

has special semantics and may
be shared between calls: the function should not change or copy it.

##### 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 (which is always the case for`"SANN"`

and`"Brent"`

). Possible error codes are [object Object],[object Object],[object Object],[object Object]message A character string giving any additional information returned by the optimizer, or `NULL`

.hessian Only if argument `hessian`

is true. A symmetric matrix giving an estimate of the Hessian at the solution found. Note that this is the Hessian of the unconstrained problem even if the box constraints are active.- For
`optimHess`

, the description of the`hessian`

component applies.

##### Note

`optim`

will work with one-dimensional `par`

s, but the
default method does not work well (and will warn). Method
`"Brent"`

uses `optimize`

and needs bounds to be available;
`"BFGS"`

often works well enough if not.

##### concept

- minimization
- maximization

##### source

The code for methods `"Nelder-Mead"`

, `"BFGS"`

and
`"CG"`

was based originally on Pascal code in Nash (1990) that was
translated by `p2c`

and then hand-optimized. Dr Nash has agreed
that the code can be made freely available.

The code for method `"L-BFGS-B"`

is based on Fortran code by Zhu,
Byrd, Lu-Chen and Nocedal obtained from Netlib (file

The code for method `"SANN"`

was contributed by A. Trapletti.

##### References

Belisle, C. J. P. (1992) Convergence theorems for a class of simulated
annealing algorithms on $R^d$. *J. Applied Probability*,
**29**, 885--895.

Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995) A limited
memory algorithm for bound constrained optimization.
*SIAM J. Scientific Computing*, **16**, 1190--1208.

Fletcher, R. and Reeves, C. M. (1964) Function minimization by
conjugate gradients. *Computer Journal* **7**, 148--154.

Nash, J. C. (1990) *Compact Numerical Methods for
Computers. Linear Algebra and Function Minimisation.* Adam Hilger.

Nelder, J. A. and Mead, R. (1965) A simplex algorithm for function
minimization. *Computer Journal* **7**, 308--313.

Nocedal, J. and Wright, S. J. (1999) *Numerical Optimization*.
Springer.

##### See Also

`optimize`

for one-dimensional minimization and
`constrOptim`

for constrained optimization.

##### Examples

`library(stats)`

```
require(graphics)
fr <- function(x) { ## Rosenbrock Banana function
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))
}
optim(c(-1.2,1), fr)
(res <- optim(c(-1.2,1), fr, grr, method = "BFGS"))
optimHess(res$par, fr, grr)
optim(c(-1.2,1), fr, NULL, method = "BFGS", hessian = TRUE)
## These do not converge in the default number of steps
optim(c(-1.2,1), fr, grr, method = "CG")
optim(c(-1.2,1), fr, grr, method = "CG", control = list(type = 2))
optim(c(-1.2,1), fr, grr, method = "L-BFGS-B")
flb <- function(x)
{ p <- length(x); sum(c(1, rep(4, p-1)) * (x - c(1, x[-p])^2)^2) }
## 25-dimensional box constrained
optim(rep(3, 25), flb, NULL, method = "L-BFGS-B",
lower = rep(2, 25), upper = rep(4, 25)) # par[24] is *not* at boundary
## "wild" function , global minimum at about -15.81515
fw <- function (x)
10*sin(0.3*x)*sin(1.3*x^2) + 0.00001*x^4 + 0.2*x+80
plot(fw, -50, 50, n = 1000, main = "optim() minimising 'wild function'")
res <- optim(50, fw, method = "SANN",
control = list(maxit = 20000, temp = 20, parscale = 20))
res
## Now improve locally {typically only by a small bit}:
(r2 <- optim(res$par, fw, method = "BFGS"))
points(r2$par, r2$value, pch = 8, col = "red", cex = 2)
## Combinatorial optimization: Traveling salesman problem
library(stats) # normally loaded
eurodistmat <- as.matrix(eurodist)
distance <- function(sq) { # Target function
sq2 <- embed(sq, 2)
sum(eurodistmat[cbind(sq2[,2], sq2[,1])])
}
genseq <- function(sq) { # Generate new candidate sequence
idx <- seq(2, NROW(eurodistmat)-1)
changepoints <- sample(idx, size = 2, replace = FALSE)
tmp <- sq[changepoints[1]]
sq[changepoints[1]] <- sq[changepoints[2]]
sq[changepoints[2]] <- tmp
sq
}
sq <- c(1:nrow(eurodistmat), 1) # Initial sequence: alphabetic
distance(sq)
# rotate for conventional orientation
loc <- -cmdscale(eurodist, add = TRUE)$points
x <- loc[,1]; y <- loc[,2]
s <- seq_len(nrow(eurodistmat))
tspinit <- loc[sq,]
plot(x, y, type = "n", asp = 1, xlab = "", ylab = "",
main = "initial solution of traveling salesman problem", axes = FALSE)
arrows(tspinit[s,1], tspinit[s,2], tspinit[s+1,1], tspinit[s+1,2],
angle = 10, col = "green")
text(x, y, labels(eurodist), cex = 0.8)
set.seed(123) # chosen to get a good soln relatively quickly
res <- optim(sq, distance, genseq, method = "SANN",
control = list(maxit = 30000, temp = 2000, trace = TRUE,
REPORT = 500))
res # Near optimum distance around 12842
tspres <- loc[res$par,]
plot(x, y, type = "n", asp = 1, xlab = "", ylab = "",
main = "optim() 'solving' traveling salesman problem", axes = FALSE)
arrows(tspres[s,1], tspres[s,2], tspres[s+1,1], tspres[s+1,2],
angle = 10, col = "red")
text(x, y, labels(eurodist), cex = 0.8)
```

*Documentation reproduced from package stats, version 3.3, License: Part of R 3.3*