Generalpurpose Optimization
Description
Generalpurpose optimization based on Nelder–Mead, quasiNewton and conjugategradient algorithms. It includes an option for boxconstrained optimization and simulated annealing.
Usage
1 2 3 4 5 6 7 
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 
... 
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 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 nondifferentiable functions.
Method "BFGS"
is a quasiNewton 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 "LBFGSB"
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 limitedmemory modification of the BFGS quasiNewton
method. If nontrivial 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). Simulatedannealing belongs to the class of
stochastic global optimization methods. It uses only function values
but is relatively slow. It will also work for nondifferentiable
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.\sspace890);
specifically, the temperature is set to
temp / log(((t1) %/% 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 generalpurpose method but can be very useful
in getting to a good value on a very rough surface.
Method "Brent"
is for onedimensional 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 "LBFGSB"
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 zerolength par
, and just
evaluates the function with that argument.
The control
argument is a list that can supply any of the
following components:
trace
Nonnegative integer. If positive, tracing information on the progress of the optimization is produced. Higher values may produce more tracing information: for method
"LBFGSB"
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 finitedifference approximation to the gradient, on
par/parscale
scale. Defaults to1e3
.maxit
The maximum number of iterations. Defaults to
100
for the derivativebased methods, and500
for"NelderMead"
.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 nonnegative 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 about1e8
.alpha
,beta
,gamma
Scaling parameters for the
"NelderMead"
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"
,"LBFGSB"
and"SANN"
methods ifcontrol$trace
is positive. Defaults to every 10 iterations for"BFGS"
and"LBFGSB"
, or every 100 temperatures for"SANN"
.type
for the conjugategradients 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
"LBFGSB"
method, It defaults to5
.factr
controls the convergence of the
"LBFGSB"
method. Convergence occurs when the reduction in the objective is within this factor of the machine tolerance. Default is1e7
, that is a tolerance of about1e8
.pgtol
helps control the convergence of the
"LBFGSB"
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.
Value
For optim
, a list with components:
par 
The best set of parameters found. 
value 
The value of 
counts 
A twoelement 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.
Note
optim
will work with onedimensional 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.
Source
The code for methods "NelderMead"
, "BFGS"
and
"CG"
was based originally on Pascal code in Nash (1990) that was
translated by p2c
and then handoptimized. Dr Nash has agreed
that the code can be made freely available.
The code for method "LBFGSB"
is based on Fortran code by Zhu,
Byrd, LuChen 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 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.
See Also
nlm
, nlminb
.
optimize
for onedimensional minimization and
constrOptim
for constrained optimization.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85  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$par, fr, grr)
optim(c(1.2,1), fr, NULL, method = "BFGS", hessian = TRUE)
## These do not converge in the default number of steps
optim(c(1.2,1), fr, grr, method = "CG")
optim(c(1.2,1), fr, grr, method = "CG", control = list(type = 2))
optim(c(1.2,1), fr, grr, method = "LBFGSB")
flb < function(x)
{ p < length(x); sum(c(1, rep(4, p1)) * (x  c(1, x[p])^2)^2) }
## 25dimensional box constrained
optim(rep(3, 25), flb, NULL, method = "LBFGSB",
lower = rep(2, 25), upper = rep(4, 25)) # par[24] is *not* at boundary
## "wild" function , global minimum at about 15.81515
fw < function (x)
10*sin(0.3*x)*sin(1.3*x^2) + 0.00001*x^4 + 0.2*x+80
plot(fw, 50, 50, n = 1000, main = "optim() minimising 'wild function'")
res < optim(50, fw, method = "SANN",
control = list(maxit = 20000, temp = 20, parscale = 20))
res
## Now improve locally {typically only by a small bit}:
(r2 < optim(res$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)

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