optim | R Documentation |
Description
General-purpose optimization based on Nelder–Mead, quasi-Newton andconjugate-gradient algorithms. It includes an option forbox-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 firstargument the vector of parameters over which minimization is to takeplace. 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’. Can be abbreviated. |
lower, upper | Bounds on the variables for the |
control | a |
hessian | Logical. Should a numerically differentiated Hessianmatrix be returned? |
Details
Note that arguments after ...
must be matched exactly.
By default optim
performs minimization, but it will maximizeif control$fnscale
is negative. optimHess
is anauxiliary function to compute the Hessian at a later stage ifhessian = 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 relativelyslow. It will work reasonably well for non-differentiable functions.
Method "BFGS"
is a quasi-Newton method (also known as a variablemetric algorithm), specifically that published simultaneously in 1970by Broyden, Fletcher, Goldfarb and Shanno. This uses function valuesand gradients to build up a picture of the surface to be optimized.
Method "CG"
is a conjugate gradients method based on that byFletcher and Reeves (1964) (but with the option of Polak–Ribiere orBeale–Sorenson updates). Conjugate gradient methods will generallybe more fragile than the BFGS method, but as they do not store amatrix they may be successful in much larger optimization problems.
Method "L-BFGS-B"
is that of Byrd et. al. (1995) whichallows box constraints, that is each variable can be given a lowerand/or upper bound. The initial value must satisfy the constraints.This uses a limited-memory modification of the BFGS quasi-Newtonmethod. If non-trivial bounds are supplied, this method will beselected, with a warning.
Nocedal and Wright (1999) is a comprehensive reference for theprevious three methods.
Method "SANN"
is by default a variant of simulated annealinggiven in Belisle (1992). Simulated-annealing belongs to the class ofstochastic global optimization methods. It uses only function valuesbut is relatively slow. It will also work for non-differentiablefunctions. This implementation uses the Metropolis function for theacceptance probability. By default the next candidate point isgenerated from a Gaussian Markov kernel with scale proportional to theactual temperature. If a function to generate a new candidate point isgiven, method "SANN"
can also be used to solve combinatorialoptimization problems. Temperatures are decreased according to thelogarithmic cooling schedule as given in Belisle (1992, p. 890);specifically, the temperature is set totemp / log(((t-1) %/% tmax)*tmax + exp(1))
, where t
isthe current iteration step and temp
and tmax
arespecifiable via control
, see below. Note that the"SANN"
method depends critically on the settings of the controlparameters. It is not a general-purpose method but can be very usefulin getting to a good value on a very rough surface.
Method "Brent"
is for one-dimensional problems only, usingoptimize()
. It can be useful in cases whereoptim()
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 functioncannot be evaluated at the supplied value, but the initial value musthave a computable finite value of fn
.(Except for method "L-BFGS-B"
where the values should always befinite.)
optim
can be used recursively, and for a single parameteras well as many. It also accepts a zero-length par
, and justevaluates the function with that argument.
The control
argument is a list that can supply any of thefollowing components:
trace
Non-negative integer. If positive,tracing information on theprogress of the optimization is produced. Higher values mayproduce more tracing information: for method
"L-BFGS-B"
there are six levels of tracing. (To understand exactly whatthese do see the source code: higher levels give more detail.)fnscale
An overall scaling to be applied to the valueof
fn
andgr
during optimization. If negative,turns the problem into a maximization problem. Optimization isperformed onfn(par)/fnscale
.parscale
A vector of scaling values for the parameters.Optimization is performed on
par/parscale
and these should becomparable in the sense that a unit change in any element producesabout a unit change in the scaled value. Not used (nor needed)formethod = "Brent"
.ndeps
A vector of step sizes for the finite-differenceapproximation to the gradient, on
par/parscale
scale. Defaults to1e-3
.maxit
The maximum number of iterations. Defaults to
100
for the derivative-based methods, and500
for"Nelder-Mead"
.For
"SANN"
maxit
gives the total number of functionevaluations: there is no other stopping criterion. Defaults to10000
.abstol
The absolute convergence tolerance. Onlyuseful for non-negative functions, as a tolerance for reaching zero.
reltol
Relative convergence tolerance. The algorithmstops if it is unable to reduce the value by a factor of
reltol * (abs(val) + reltol)
at a step. Defaults tosqrt(.Machine$double.eps)
, typically about1e-8
.alpha
,beta
,gamma
Scaling parametersfor the
"Nelder-Mead"
method.alpha
is the reflectionfactor (default 1.0),beta
the contraction factor (0.5) andgamma
the expansion factor (2.0).REPORT
The frequency of reports for the
"BFGS"
,"L-BFGS-B"
and"SANN"
methods ifcontrol$trace
is positive. Defaults to every 10 iterations for"BFGS"
and"L-BFGS-B"
, or every 100 temperatures for"SANN"
.warn.1d.NelderMead
a
logical
indicatingif the (default)"Nelder-Mean"
method should signal awarning when used for one-dimensional minimization. As thewarning is sometimes inappropriate, you can suppress it by settingthis option to false.type
for the conjugate-gradients method. Takes value
1
for the Fletcher–Reeves update,2
forPolak–Ribiere and3
for Beale–Sorenson.lmm
is an integer giving the number of BFGS updatesretained in the
"L-BFGS-B"
method, It defaults to5
.factr
controls the convergence of the
"L-BFGS-B"
method. Convergence occurs when the reduction in the objective iswithin this factor of the machine tolerance. Default is1e7
,that is a tolerance of about1e-8
.pgtol
helps control the convergence of the
"L-BFGS-B"
method. It is a tolerance on the projected gradient in the currentsearch direction. This defaults to zero, when the check issuppressed.temp
controls the
"SANN"
method. It is thestarting temperature for the cooling schedule. Defaults to10
.tmax
is the number of function evaluations at eachtemperature for the
"SANN"
method. Defaults to10
.
Any names given to par
will be copied to the vectors passed tofn
and gr
. Note that no other attributes of par
are copied over.
The parameter vector passed to fn
has special semantics and maybe 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 |
counts | A two-element integer vector giving the number of callsto |
convergence | An integer code.
|
message | A character string giving any additional informationreturned by the optimizer, or |
hessian | Only if argument |
For optimHess
, the description of the hessian
componentapplies.
Note
optim
will work with one-dimensional par
s, but thedefault 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.
Source
The code for methods "Nelder-Mead"
, "BFGS"
and"CG"
was based originally on Pascal code in Nash (1990) that wastranslated by p2c
and then hand-optimized. Dr Nash has agreedthat 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.
References
Belisle, C. J. P. (1992).Convergence theorems for a class of simulated annealing algorithms onRd.Journal of Applied Probability, 29, 885–895.\Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.2307/3214721")}.
Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995).A limited memory algorithm for bound constrained optimization.SIAM Journal on Scientific Computing, 16, 1190–1208.\Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1137/0916069")}.
Fletcher, R. and Reeves, C. M. (1964).Function minimization by conjugate gradients.Computer Journal 7, 148–154.\Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1093/comjnl/7.2.149")}.
Nash, J. C. (1990).Compact Numerical Methods for Computers. Linear Algebra andFunction Minimisation.Adam Hilger.
Nelder, J. A. and Mead, R. (1965).A simplex algorithm for function minimization.Computer Journal, 7, 308–313.\Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1093/comjnl/7.4.308")}.
Nocedal, J. and Wright, S. J. (1999).Numerical Optimization.Springer.
See Also
nlm
, nlminb
.
optimize
for one-dimensional minimization andconstrOptim
for constrained optimization.
Examples
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 stepsoptim(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 constrainedoptim(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 -15.81515fw <- function (x) 10*sin(0.3*x)*sin(1.3*x^2) + 0.00001*x^4 + 0.2*x+80plot(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$value, pch = 8, col = "red", cex = 2)## Combinatorial optimization: Traveling salesman problemlibrary(stats) # normally loadedeurodistmat <- as.matrix(eurodist)distance <- function(sq) { # Target function sq2 <- embed(sq, 2) sum(eurodistmat[cbind(sq2[,2], sq2[,1])])}genseq <- function(sq) { # Generate new candidate sequence idx <- seq(2, NROW(eurodistmat)-1) changepoints <- sample(idx, size = 2, replace = FALSE) tmp <- sq[changepoints[1]] sq[changepoints[1]] <- sq[changepoints[2]] sq[changepoints[2]] <- tmp sq}sq <- c(1:nrow(eurodistmat), 1) # Initial sequence: alphabeticdistance(sq)# rotate for conventional orientationloc <- -cmdscale(eurodist, add = TRUE)$pointsx <- 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 quicklyres <- optim(sq, distance, genseq, method = "SANN", control = list(maxit = 30000, temp = 2000, trace = TRUE, REPORT = 500))res # Near optimum distance around 12842tspres <- loc[res$par,]plot(x, y, type = "n", asp = 1, xlab = "", ylab = "", main = "optim() 'solving' traveling salesman problem", axes = FALSE)arrows(tspres[s,1], tspres[s,2], tspres[s+1,1], tspres[s+1,2], angle = 10, col = "red")text(x, y, labels(eurodist), cex = 0.8)## 1-D minimization: "Brent" or optimize() being preferred.. but NM may be ok and "unavoidable",## ---------------- so we can suppress the check+warning :system.time(rO <- optimize(function(x) (x-pi)^2, c(0, 10)))system.time(ro <- optim(1, function(x) (x-pi)^2, control=list(warn.1d.NelderMead = FALSE)))rO$minimum - pi # 0 (perfect), on one platformro$par - pi # ~= 1.9e-4 on one platformutils::str(ro)