| optim {stats} | R Documentation |
General-purpose optimization based on Nelder–Mead, quasi-Newton and conjugate-gradient algorithms. It includes an option for box-constrained optimization and simulated annealing.
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())
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 For the |
... |
Further arguments to be passed to |
method |
The method to be used. See ‘Details’. |
lower, upper |
Bounds on the variables for the |
control |
A list of control parameters. See ‘Details’. |
hessian |
Logical. Should a numerically differentiated Hessian matrix be returned? |
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. 890);
specifically, the temperature is set to
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 stats4.
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:
traceNon-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. (To understand exactly what
these do see the source code: higher levels give more detail.)
fnscaleAn 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.
parscaleA 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. Not used (nor needed)
for method = "Brent".
ndepsA vector of step sizes for the finite-difference
approximation to the gradient, on par/parscale
scale. Defaults to 1e-3.
maxitThe maximum number of iterations. Defaults to
100 for the derivative-based methods, and
500 for "Nelder-Mead".
For "SANN" maxit gives the total number of function
evaluations: there is no other stopping criterion. Defaults to
10000.
abstolThe absolute convergence tolerance. Only useful for non-negative functions, as a tolerance for reaching zero.
reltolRelative 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.
alpha, beta, gammaScaling 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).
REPORTThe frequency of reports for the "BFGS",
"L-BFGS-B" and "SANN" methods if control$trace
is positive. Defaults to every 10 iterations for "BFGS" and
"L-BFGS-B", or every 100 temperatures for "SANN".
typefor the conjugate-gradients method. Takes value
1 for the Fletcher–Reeves update, 2 for
Polak–Ribiere and 3 for Beale–Sorenson.
lmmis an integer giving the number of BFGS updates
retained in the "L-BFGS-B" method, It defaults to 5.
factrcontrols 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.
pgtolhelps 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.
tempcontrols the "SANN" method. It is the
starting temperature for the cooling schedule. Defaults to
10.
tmaxis the number of function evaluations at each
temperature for the "SANN" method. Defaults to 10.
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.
For optim, a list with components:
par |
The best set of parameters found. |
value |
The value of |
counts |
A two-element integer vector giving the number of calls
to |
convergence |
An integer code.
|
message |
A character string giving any additional information
returned by the optimizer, or |
hessian |
Only if argument |
For optimHess, the description of the hessian component
applies.
optim will work with one-dimensional pars, 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.
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
‘opt/lbfgs_bcm.shar’: another version is in ‘toms/778’).
The code for method "SANN" was contributed by A. Trapletti.
Belisle, C. J. P. (1992) Convergence theorems for a class of simulated annealing algorithms on Rd. 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.
optimize for one-dimensional minimization and
constrOptim for constrained optimization.
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)