DEoptim {RcppDE}

Differential Evolution Optimization

Description

Performs evolutionary global optimization via the Differential Evolution algorithm.

Usage

DEoptim(fn, lower, upper, control = DEoptim.control(), ...)

Arguments

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. 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.

lower, upper

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.

control

a list of control parameters; see DEoptim.control.

...

further arguments to be passed to fn.

Details

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.

Value

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.

Note

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 \mathit{NP} denote the number of parameter vectors (members) x \in R^d in the population. In order to create the initial generation, \mathit{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, \ldots, \mathit{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 \doteq x_{r_0} + \mathit{F} \cdot (x_{r_1} - x_{r_2})

where \mathit{F} is a positive scale factor, effective values for which are typically less than one. After the first mutation operation, mutation is continued until d mutations have been made, with a crossover probability \mathit{CR} \in [0,1]. The crossover probability \mathit{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 \mathit{NP}, \mathit{F} and \mathit{CR}, Storn et al. (2006) state the following: Set the number of parents \mathit{NP} to 10 times the number of parameters, select weighting factor \mathit{F} = 0.8 and crossover constant \mathit{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 \mathit{NP}, but often you only have to adjust \mathit{F} to be a little lower or higher than 0.8. If you increase \mathit{NP} and simultaneously lower \mathit{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 \mathit{F} than it is to the choice of \mathit{CR}. \mathit{CR} is more like a fine tuning element. High values of \mathit{CR} like \mathit{CR} = 1 give faster convergence if convergence occurs. Sometimes, however, you have to go down as much as \mathit{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).

Author(s)

For RcppDE: Dirk Eddelbuettel.

For DEoptim: David Ardia, Katharine Mullen [e-mail], Brian Peterson and Joshua Ulrich.

References

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

See Also

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.

Examples

  ## 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))