Description
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 For the "BFGS"
, "CG"
and "L-BFGS-B"
methods. If it is NULL
, a finite-difference approximation will be used."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?
Value
For optim
, a list with components:
The best set of parameters found.
The value of fn
corresponding to par
.
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.
An integer code. 0
indicates successful completion (which is always the case for "SANN"
and "Brent"
). Possible error codes are
1
indicates that the iteration limit
maxit
had been reached.10
indicates degeneracy of the Nelder--Mead simplex.
51
indicates a warning from the
"L-BFGS-B"
method; see componentmessage
for further details.52
indicates an error from the
"L-BFGS-B"
method; see componentmessage
for further details.See Alsooptimr function - RDocumentation
A character string giving any additional information returned by the optimizer, or NULL
.
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.
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.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:
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. (To understand exactly what these do see the source code: higher levels give more detail.)fnscale
An overall scaling to be applied to the value of
fn
andgr
during optimization. If negative, turns the problem into a maximization problem. Optimization is performed onfn(par)/fnscale
.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. Not used (nor needed)formethod = "Brent"
.ndeps
A vector of step sizes for the finite-difference approximation 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 function evaluations: there is no other stopping criterion. Defaults to10000
.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 tosqrt(.Machine$double.eps)
, typically about1e-8
.alpha
,beta
,gamma
Scaling parameters for the
"Nelder-Mead"
method.alpha
is the reflection factor (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
indicating if the (default)"Nelder-Mean"
method should signal a warning when used for one-dimensional minimization. As the warning is sometimes inappropriate, you can suppress it by setting this option to false.type
for the conjugate-gradients method. Takes value
1
for the Fletcher--Reeves update,2
for Polak--Ribiere and3
for Beale--Sorenson.lmm
is an integer giving the number of BFGS updates retained 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 is within 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 current search direction. This defaults to zero, when the check is suppressed.temp
controls the
"SANN"
method. It is the starting temperature for the cooling schedule. Defaults to10
.tmax
is the number of function evaluations at each temperature for the
"SANN"
method. Defaults to10
.
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.
References
Belisle, C. J. P. (1992). Convergence theorems for a class of simulated annealing algorithms on \(R^d\). Journal of Applied Probability, 29, 885--895. 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. 10.1137/0916069.
Fletcher, R. and Reeves, C. M. (1964). Function minimization by conjugate gradients. Computer Journal 7, 148--154. 10.1093/comjnl/7.2.149.
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. 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 and constrOptim
for constrained optimization.
Examples
# NOT RUN {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)# }
Run the code above in your browser using DataLab