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Performs evolutionary global optimization via the Differential Evolution algorithm.
DEoptim(fn, lower, upper, control = DEoptim.control(), ...)The output of the function DEoptim is a member of the S3 class DEoptim. More precisely,
this is a list (of length 2) containing the following elements:
optim, a list containing the following elements:
bestmem: the best set of parameters found.
bestval: the value of fn corresponding to bestmem.
nfeval: number of function evaluations.
iter: number of procedure iterations.
member, a list containing the following elements:
lower: the lower boundary.
upper: the upper boundary.
bestvalit: the best value of fn at each iteration.
bestmemit: the best member at each iteration.
pop: the population generated at the last iteration.
storepop: a list containing the intermediate populations.
Members of the class DEoptim have a plot method that
accepts the argument plot.type. plot.type = "bestmemit" results
in a plot of the parameter values that represent the lowest value of the objective function
each generation. plot.type = "bestvalit" plots the best value of
the objective function each generation. Finally, plot.type = "storepop" results in a plot of
stored populations (which are only available if these have been saved by
setting the control argument of DEoptim appropriately). Storing intermediate populations
allows us to examine the progress of the optimization in detail.
A summary method also exists and returns the best parameter vector, the best value of the objective function,
the number of generations optimization ran, and the number of times the
objective function was evaluated.
the function to be optimized (minimized). The function should have as its first
argument the vector of real-valued parameters to optimize, and return a scalar real result. NA
and NaN values are not allowed. Note that fn can also
be an external pointer object encapsulating a C/C++-level function pointer
to a compiled functions which may offer considerable speed improvements.
two vectors specifying scalar real lower and upper bounds on each parameter to be optimized, so that the i-th element
of lower and upper applied to the i-th parameter. The implementation searches
between lower and upper for the global optimum (minimum) of fn.
a list of control parameters; see DEoptim.control.
further arguments to be passed to fn.
For RcppDE: Dirk Eddelbuettel.
For DEoptim: David Ardia, Katharine Mullen katharine.mullen@nist.gov, Brian Peterson and Joshua Ulrich.
DEoptim performs optimization (minimization) of fn.
The control argument is a list; see the help file for
DEoptim.control for details.
The R implementation of Differential Evolution (DE), DEoptim, was first published on the Comprehensive R Archive Network (CRAN) in 2005 by David Ardia. Early versions were written in pure R. Since version 2.0-0 (published to CRAN in 2009) the package has relied on an interface to a C implementation of DE, which is significantly faster on most problems as compared to the implementation in pure R. The C interface is in many respects similar to the MS Visual C++ v5.0 implementation of the Differential Evolution algorithm distributed with the book Differential Evolution -- A Practical Approach to Global Optimization by Price, K.V., Storn, R.M., Lampinen J.A, Springer-Verlag, 2006. Since version 2.0-3 the C implementation dynamically allocates the memory required to store the population, removing limitations on the number of members in the population and length of the parameter vectors that may be optimized. Since becoming publicly available, the package DEoptim has been used by several authors to solve optimization problems arising in diverse domains; see Mullen et al. (2009) for a review.
To perform a maximization (instead of minimization) of a given
function, simply define a new function which is the opposite of the
function to maximize and apply DEoptim to it.
To integrate additional constraints (than box constraints) on the parameters x of
fn(x), for instance x[1] + x[2]^2 < 2, integrate the
constraint within the function to optimize, for instance:
fn <- function(x){
if (x[1] + x[2]^2 < 2){
r <- Inf
else{
...
}
return(r)
}
This simplistic strategy usually does not work all that well for gradient-based or Newton-type methods. It is likely to be alright when the solution is in the interior of the feasible region, but when the solution is on the boundary, optimization algorithm would have a difficult time converging. Furthermore, when the solution is on the boundary, this strategy would make the algorithm converge to an inferior solution in the interior. However, for methods such as DE which are not gradient based, this strategy might not be that bad.
Note that DEoptim stops if any NA or NaN value is
obtained. You have to redefine your function to handle these values
(for instance, set NA to Inf in your objective function).
It is important to emphasize that the result of DEoptim is a random variable,
i.e., different results will obtain when the algorithm is run repeatedly with the same
settings. Hence, the user should set the random seed if they want to reproduce the results, e.g., by
setting set.seed(1234) before the call of DEoptim.
DEoptim relies on repeated evaluation of the objective function
in order to move the population toward a global minimum. Users
interested in making DEoptim run as fast as possible should
ensure that evaluation of the objective function is as efficient as
possible. Using pure R code, this may often be accomplished
using vectorization. Writing parts of the objective function in a
lower-level language like C or Fortran may also increase speed.
Further details and examples of the R package DEoptim can be found in Mullen et al. (2009) and Ardia et al. (2010).
Please cite the package in publications.
Storn, R. and Price, K. (1997) Differential Evolution -- A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces, Journal of Global Optimization, 11:4, 341--359.
Price, K.V., Storn, R.M., Lampinen J.A. (2006) Differential Evolution - A Practical Approach to Global Optimization. Berlin Heidelberg: Springer-Verlag. ISBN 3540209506.
Mitchell, M. (1998) An Introduction to Genetic Algorithms. The MIT Press. ISBN 0262631857.
Mullen, K.M., Ardia, D., Gil, D.L, Windover, D., Cline, J. (2009) DEoptim: An R Package for Global Optimization by Differential Evolution. URL https://www.ssrn.com/abstract=1526466
Ardia, D., Boudt, K., Carl, P., Mullen, K.M., Peterson, B.G. (2010) Differential Evolution (DEoptim) for Non-Convex Portfolio Optimization. URL https://www.ssrn.com/abstract=1584905
DEoptim.control for control arguments,
DEoptim-methods for methods on DEoptim objects,
including some examples in plotting the results;
optim or constrOptim
for alternative optimization algorithms.
## Rosenbrock Banana function
## The function has a global minimum f(x) = 0 at the point (0,0).
## Note that the vector of parameters to be optimized must be the first
## argument of the objective function passed to DEoptim.
Rosenbrock <- function(x){
x1 <- x[1]
x2 <- x[2]
100 * (x2 - x1 * x1)^2 + (1 - x1)^2
}
## DEoptim searches for minima of the objective function between
## lower and upper bounds on each parameter to be optimized. Therefore
## in the call to DEoptim we specify vectors that comprise the
## lower and upper bounds; these vectors are the same length as the
## parameter vector.
lower <- c(-10,-10)
upper <- -lower
## run DEoptim and set a seed first for replicability
set.seed(1234)
DEoptim(Rosenbrock, lower, upper)
## increase the population size
DEoptim(Rosenbrock, lower, upper, DEoptim.control(NP = 100))
## change other settings and store the output
outDEoptim <- DEoptim(Rosenbrock, lower, upper, DEoptim.control(NP = 80,
itermax = 400, F = 1.2, CR = 0.7))
## plot the output
plot(outDEoptim)
## 'Wild' function, global minimum at about -15.81515
Wild <- function(x)
10 * sin(0.3 * x) * sin(1.3 * x^2) +
0.00001 * x^4 + 0.2 * x + 80
plot(Wild, -50, 50, n = 1000, main = "'Wild function'")
outDEoptim <- DEoptim(Wild, lower = -50, upper = 50,
control = DEoptim.control(trace = FALSE))
plot(outDEoptim)
DEoptim(Wild, lower = -50, upper = 50,
control = DEoptim.control(NP = 50))
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