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())"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.
fn and gr."L-BFGS-B"
method, or bounds in which to search for method "Brent".optim, a list with components:
fn corresponding to par.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.0 indicates successful
completion (which is always the case for "SANN" and
"Brent"). Possible error codes are
1maxit
had been reached.1051"L-BFGS-B"
method; see component message for further details.52"L-BFGS-B"
method; see component message for further details.NULL.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.optimHess, the description of the hessian component
applies.
"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.... 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.\ifelse{latex}{\out{~}}{ } 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"L-BFGS-B"
there are six levels of tracing. (To understand exactly what
these do see the source code: higher levels give more detail.)
fnscalefn and gr during optimization. If negative,
turns the problem into a maximization problem. Optimization is
performed on fn(par)/fnscale.
parscalepar/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".
ndepspar/parscale
scale. Defaults to 1e-3.
maxit100 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.
abstol
reltolreltol * (abs(val) + reltol) at a step. Defaults to
sqrt(.Machine$double.eps), typically about 1e-8.
alpha, beta, gamma"Nelder-Mead" method. alpha is the reflection
factor (default 1.0), beta the contraction factor (0.5) and
gamma the expansion factor (2.0).
REPORT"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".
type1 for the Fletcher--Reeves update, 2 for
Polak--Ribiere and 3 for Beale--Sorenson.
lmm"L-BFGS-B" method, It defaults to 5.
factr"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.
pgtol"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"SANN" method. It is the
starting temperature for the cooling schedule. Defaults to
10.
tmax"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.
The parameter vector passed to fn has special semantics and may
be shared between calls: the function should not change or copy it.
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.
nlm, nlminb. optimize for one-dimensional minimization and
constrOptim for constrained optimization.