Differential Evolution Optimization
Performs evolutionary global optimization via the Differential Evolution algorithm.
DEoptim(fn, lower, upper, control = DEoptim.control(), ..., fnMap=NULL)
fn |
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. |
lower, upper |
two vectors specifying scalar real lower and upper
bounds on each parameter to be optimized, so that the i-th element
of |
control |
a list of control parameters; see |
fnMap |
an optional function that will be run after each population is created, but before the population is passed to the objective function. This allows the user to impose integer/cardinality constriants. |
... |
further arguments to be passed to |
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 version 2.2-0, the package allows for parallel
operation, so that the evaluations of the objective function may be
performed using all available cores. This is accomplished using either
the built-in parallel package or the foreach package. If
parallel operation is desired, the user should set parallelType
and make sure that the arguments and packages needed by the objective
function are available; see DEoptim.control, the example
below and examples in the sandbox directory for details.
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. (2011) 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 (other 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 may be obtained 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
consider using the package in parallel mode (so that all CPU's
available are used), and also
ensure that evaluation of the objective function is as efficient as
possible (e.g. by using vectorization in pure R code, or
writing parts of the objective function in a
lower-level language like C or Fortran).
Further details and examples of the R package DEoptim can be found
in Mullen et al. (2011) and Ardia et al. (2011a, 2011b) or look at the
package's vignette by typing vignette("DEoptim"). Also, an illustration of
the package usage for a high-dimensional non-linear portfolio optimization problem
is available by typing vignette("DEoptimPortfolioOptimization").
Please cite the package in publications. Use citation("DEoptim").
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.
Differential Evolution (DE) is a search heuristic introduced by Storn and Price (1997). Its remarkable performance as a global optimization algorithm on continuous numerical minimization problems has been extensively explored; see Price et al. (2006). DE belongs to the class of genetic algorithms which use biology-inspired operations of crossover, mutation, and selection on a population in order to minimize an objective function over the course of successive generations (see Mitchell, 1998). As with other evolutionary algorithms, DE solves optimization problems by evolving a population of candidate solutions using alteration and selection operators. DE uses floating-point instead of bit-string encoding of population members, and arithmetic operations instead of logical operations in mutation. DE is particularly well-suited to find the global optimum of a real-valued function of real-valued parameters, and does not require that the function be either continuous or differentiable.
Let NP denote the number of parameter vectors (members) x in R^d in the population. In order to create the initial generation, NP guesses for the optimal value of the parameter vector are made, either using random values between lower and upper bounds (defined by the user) or using values given by the user. Each generation involves creation of a new population from the current population members {x_i | i=1,...,NP}, where i indexes the vectors that make up the population. This is accomplished using differential mutation of the population members. An initial mutant parameter vector v_i is created by choosing three members of the population, x_{r_0}, x_{r_1} and x_{r_2}, at random. Then v_i is generated as
v_i := x_{r_0} + F * (x_{r_1} - x_{r_2})
where F is the differential weighting factor, effective values for which are
typically between 0 and 1. After the first mutation operation, mutation is
continued until d mutations have been made, with a crossover probability
CR in [0,1].
The crossover probability CR controls the fraction of the parameter
values that are copied from the mutant. If an element of the trial parameter vector is found to violate the
bounds after mutation and crossover, it is reset in such a way that the bounds are respected (with the
specific protocol depending on the implementation).
Then, the objective function values associated with the children are determined. If a trial
vector has equal or lower objective function value than the previous
vector it replaces the previous vector in the population;
otherwise the previous vector remains. Variations of this scheme have also
been proposed; see Price et al. (2006) and DEoptim.control.
Intuitively, the effect of the scheme is that the shape of the distribution of the population in the search space is converging with respect to size and direction towards areas with high fitness. The closer the population gets to the global optimum, the more the distribution will shrink and therefore reinforce the generation of smaller difference vectors.
As a general advice regarding the choice of NP, F and CR, Storn et al. (2006) state the following: Set the number of parents NP to 10 times the number of parameters, select differential weighting factor F = 0.8 and crossover constant CR = 0.9. Make sure that you initialize your parameter vectors by exploiting their full numerical range, i.e., if a parameter is allowed to exhibit values in the range [-100, 100] it is a good idea to pick the initial values from this range instead of unnecessarily restricting diversity. If you experience misconvergence in the optimization process you usually have to increase the value for NP, but often you only have to adjust F to be a little lower or higher than 0.8. If you increase NP and simultaneously lower F a little, convergence is more likely to occur but generally takes longer, i.e., DE is getting more robust (there is always a convergence speed/robustness trade-off).
DE is much more sensitive to the choice of F than it is to the choice of CR. CR is more like a fine tuning element. High values of CR like CR = 1 give faster convergence if convergence occurs. Sometimes, however, you have to go down as much as CR = 0 to make DE robust enough for a particular problem. For more details on the DE strategy, we refer the reader to Storn and Price (1997) and Price et al. (2006).
David Ardia, Katharine Mullen mullenkate@gmail.com, Brian Peterson and Joshua Ulrich.
Ardia, D., Boudt, K., Carl, P., Mullen, K.M., Peterson, B.G. (2011) Differential Evolution with DEoptim. An Application to Non-Convex Portfolio Optimization. R Journal, 3(1), 27-34. doi: 10.32614/RJ-2011-005
Ardia, D., Ospina Arango, J.D., Giraldo Gomez, N.D. (2011) Jump-Diffusion Calibration using Differential Evolution. Wilmott Magazine, 55 (September), 76-79. doi: 10.1002/wilm.10034
Mitchell, M. (1998) An Introduction to Genetic Algorithms. The MIT Press. ISBN 0262631857.
Mullen, K.M, Ardia, D., Gil, D., Windover, D., Cline,J. (2011). DEoptim: An R Package for Global Optimization by Differential Evolution. Journal of Statistical Software, 40(6), 1-26. doi: 10.18637/jss.v040.i06
Price, K.V., Storn, R.M., Lampinen J.A. (2006) Differential Evolution - A Practical Approach to Global Optimization. Berlin Heidelberg: Springer-Verlag. ISBN 3540209506.
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.
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 (1,1).
## 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))
## The below examples shows how the call to DEoptim can be
## parallelized.
## Note that if your objective function requires packages to be
## loaded or has arguments supplied via \code{...}, these should be
## specified using the \code{packages} and \code{parVar} arguments
## in control.
## Not run:
Genrose <- function(x) {
## One generalization of the Rosenbrock banana valley function (n parameters)
n <- length(x)
## make it take some time ...
Sys.sleep(.001)
1.0 + sum (100 * (x[-n]^2 - x[-1])^2 + (x[-1] - 1)^2)
}
# get some run-time on simple problems
maxIt <- 250
n <- 5
oneCore <- system.time( DEoptim(fn=Genrose, lower=rep(-25, n), upper=rep(25, n),
control=list(NP=10*n, itermax=maxIt)))
withParallel <- system.time( DEoptim(fn=Genrose, lower=rep(-25, n), upper=rep(25, n),
control=list(NP=10*n, itermax=maxIt, parallelType=1)))
## Compare timings
(oneCore)
(withParallel)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.