Package 'DEoptimR'

Bound-Constrained and Nonlinear Constrained Single-Objective Optimization via Differential Evolution

Description

A bespoke implementation of the ‘jDE’ variant by Brest et al. (2006) doi:10.1109/TEVC.2006.872133.

Usage

JDEoptim(lower, upper, fn,
         constr = NULL, meq = 0, eps = 1e-05,
         NP = 10*length(lower), Fl = 0.1, Fu = 1,
         tau_F = 0.1, tau_CR = 0.1, tau_pF = 0.1,
         jitter_factor = 0.001,
         tol = 1e-15, maxiter = 200*length(lower), fnscale = 1,
         compare_to = c("median", "max"),
         add_to_init_pop = NULL,
         trace = FALSE, triter = 1,
         details = FALSE, ...)

Arguments

lower, upper	numeric vectors of lower and upper bounds for the parameters to be optimized over. Must be finite (is.finite) as they bound the hyper-rectangle of the initial random population.
fn	(nonlinear) objective function to be minimized. It takes as first argument the vector of parameters over which minimization is to take place. It must return the value of the function at that point.
constr	an optional function for specifying the left-hand side of nonlinear constraints under which we want to minimize fn. Nonlinear equalities should be given first and defined to equal zero ($h_j(X) = 0$), followed by nonlinear inequalities defined as lesser than zero ($g_i(X) \le 0$). This function takes the vector of parameters as its first argument and returns a real vector with the length of the total number of constraints. It defaults to NULL, meaning that bound-constrained minimization is used.
meq	an optional positive integer specifying that the first meq constraints are treated as equality constraints, and all the remaining as inequality constraints. Defaults to 0 (inequality constraints only).
eps	maximal admissible constraint violation for equality constraints. An optional real vector of small positive tolerance values with length meq used in the transformation of equalities into inequalities of the form $|h_j(X)| - \epsilon \le 0$. A scalar value is expanded to apply to all equality constraints. Default is 1e-5.
NP	an optional positive integer giving the number of candidate solutions in the randomly distributed initial population. Defaults to 10*length(lower).
Fl	an optional scalar which represents the minimum value that the scaling factor F could take. Default is 0.1, which is almost always satisfactory.
Fu	an optional scalar which represents the maximum value that the scaling factor F could take. Default is 1, which is almost always satisfactory.
tau_F	an optional scalar which represents the probability that the scaling factor F is updated. Defaults to 0.1, which is almost always satisfactory.
tau_CR	an optional constant value which represents the probability that the crossover probability CR is updated. Defaults to 0.1, which is almost always satisfactory.
tau_pF	an optional scalar which represents the probability that the mutation probability $p_F$ in the mutation strategy DE/rand/1/either-or is updated. Defaults to 0.1.
jitter_factor	an optional tuning constant for jitter. If NULL only dither is used. Defaults to 0.001.
tol	an optional positive scalar giving the tolerance for the stopping criterion. Default is 1e-15.
maxiter	an optional positive integer specifying the maximum number of iterations that may be performed before the algorithm is halted. Defaults to 200*length(lower).
fnscale	an optional positive scalar specifying the typical magnitude of fn. It is used only in the stopping criterion. Defaults to 1. See ‘Details’.
compare_to	an optional character string controlling which function should be applied to the fn values of the candidate solutions in a generation to be compared with the so-far best one when evaluating the stopping criterion. If “median” the median function is used; else, if “max” the max function is used. It defaults to “median”. See ‘Details’.
add_to_init_pop	an optional real vector of length length(lower) or matrix with length(lower) rows specifying initial values of the parameters to be optimized which are appended to the randomly generated initial population. It defaults to NULL.
trace	an optional logical value indicating if a trace of the iteration progress should be printed. Default is FALSE.
triter	an optional positive integer giving the frequency of tracing (every triter iterations) when trace = TRUE. Default is triter = 1, in which case iteration : < value of stopping test > ( value of best solution ) best solution { index of violated constraints } is printed at each iteration.
details	an optional logical value. If TRUE the output will contain the parameters in the final population and their respective fn values. Defaults to FALSE.
...	optional additional arguments passed to fn and constr.

Details

Overview:

The setting of the control parameters of canonical Differential Evolution (DE) is crucial for the algorithm's performance. Unfortunately, when the generally recommended values for these parameters (see, e.g., Storn and Price, 1997) are unsuitable for use, their determination is often difficult and time consuming. The jDE algorithm proposed in Brest et al. (2006) employs a simple self-adaptive scheme to perform the automatic setting of control parameters scale factor F and crossover rate CR.

This implementation differs from the original description, most notably in the use of the DE/rand/1/either-or mutation strategy (Price et al., 2005), combination of jitter with dither (Storn, 2008), and the random initialization of F and CR. The mutation operator brings an additional control parameter, the mutation probability $p_F$, which is self-adapted in the same manner as CR.

As done by jDE and its variants (Brest et al., 2021) each worse parent in the current population is immediately replaced (asynchronous update) by its newly generated better or equal offspring (Babu and Angira, 2006) instead of updating the current population with all the new solutions at the same time as in classical DE (synchronous update).

As the algorithm subsamples via sample() which from R version 3.6.0 depends on RNGkind(*, sample.kind), exact reproducibility of results from R versions 3.5.3 and earlier requires setting RNGversion("3.5.0"). In any case, do use set.seed() additionally for reproducibility!

Constraint Handling:

Constraint handling is done using the approach described in Zhang and Rangaiah (2012), but with a different reduction updating scheme for the constraint relaxation value ($\mu$). Instead of doing it once for every generation or iteration, the reduction is triggered for two cases when the constraints only contain inequalities. Firstly, every time a feasible solution is selected for replacement in the next generation by a new feasible trial candidate solution with a better objective function value. Secondly, whenever a current infeasible solution gets replaced by a feasible one. If the constraints include equalities, then the reduction is not triggered in this last case. This constitutes an original feature of the implementation.

The performance of any constraint handling technique for metaheuristics is severely impaired by a small feasible region. Therefore, equality constraints are particularly difficult to handle due to the tiny feasible region they define. So, instead of explicitly including all equality constraints in the formulation of the optimization problem, it might prove advantageous to eliminate some of them. This is done by expressing one variable $x_k$ in terms of the remaining others for an equality constraint $h_j(X) = 0$ where $X = [x_1,\ldots,x_k,\ldots,x_d]$ is the vector of solutions, thereby obtaining a relationship as $x_k = R_{k,j}([x_1,\ldots,x_{k-1},x_{k+1},\ldots,x_d])$. In this way both the variable $x_k$ and the equality constraint $h_j(X) = 0$ can be removed altogether from the original optimization formulation, since the value of $x_k$ can be calculated during the search process by the relationship $R_{k,j}$. Notice, however, that two additional inequalities

$l_k \le R_{k,j}([x_1,\ldots,x_{k-1},x_{k+1},\ldots,x_d]) \le u_k,$

where the values $l_k$ and $u_k$ are the lower and upper bounds of $x_k$, respectively, must be provided in order to obtain an equivalent formulation of the problem. For guidance and examples on applying this approach see Wu et al. (2015).

Bound constraints are enforced by the midpoint base approach (see, e.g., Biedrzycki et al., 2019).

Discrete and Integer Variables:

Any DE variant is easily extended to deal with mixed integer nonlinear programming problems using a small variation of the technique presented by Lampinen and Zelinka (1999). Integer values are obtained by means of the floor() function only in the evaluation of the objective function and constraints, whereas DE itself still uses continuous variables. Additionally, each upper bound of the integer variables should be added by 1.

Notice that the final solution needs to be converted with floor() to obtain its integer elements.

Stopping Criterion:

The algorithm is stopped if

$\frac{\mathrm{compare\_to}\{[\mathrm{fn}(X_1),\ldots,\mathrm{fn}(X_\mathrm{npop})]\} - \mathrm{fn}(X_\mathrm{best})}{\mathrm{fnscale}} \le \mathrm{tol},$

where the “best” individual $X_\mathrm{best}$ is the feasible solution with the lowest objective function value in the population and the total number of elements in the population, npop, is NP+NCOL(add_to_init_pop). For compare_to = "max" this is the Diff criterion studied by Zielinski and Laur (2008) among several other alternatives, which was found to yield the best results.

Value

A list with the following components:

par	The best set of parameters found.
value	The value of fn corresponding to par.
iter	Number of iterations taken by the algorithm.
convergence	An integer code. 0 indicates successful completion. 1 indicates that the iteration limit maxiter has been reached.

and if details = TRUE:

poppar	Matrix of dimension (length(lower), npop), with columns corresponding to the parameter vectors remaining in the population.
popcost	The values of fn associated with poppar, vector of length npop.

Note

It is possible to perform a warm start, i.e., starting from the previous run and resume optimization, using NP = 0 and the component poppar for the add_to_init_pop argument.

Author(s)

Eduardo L. T. Conceicao [email protected]

References

Babu, B. V. and Angira, R. (2006) Modified differential evolution (MDE) for optimization of non-linear chemical processes. Computers and Chemical Engineering 30, 989–1002. doi:10.1016/j.compchemeng.2005.12.020.

Biedrzycki, R., Arabas, J. and Jagodzinski, D. (2019) Bound constraints handling in differential evolution: An experimental study. Swarm and Evolutionary Computation 50, 100453. doi:10.1016/j.swevo.2018.10.004.

Brest, J., Greiner, S., Boskovic, B., Mernik, M. and Zumer, V. (2006) Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems. IEEE Transactions on Evolutionary Computation 10, 646–657. doi:10.1109/TEVC.2006.872133.

Brest, J., Maucec, M. S. and Boskovic, B. (2021) Self-adaptive differential evolution algorithm with population size reduction for single objective bound-constrained optimization: Algorithm j21; in 2021 IEEE Congress on Evolutionary Computation (CEC). IEEE, pp. 817–824. doi:10.1109/CEC45853.2021.9504782.

Lampinen, J. and Zelinka, I. (1999). Mechanical engineering design optimization by differential evolution; in Corne, D., Dorigo, M. and Glover, F., Eds., New Ideas in Optimization. McGraw-Hill, pp. 127–146.

Price, K. V., Storn, R. M. and Lampinen, J. A. (2005) Differential evolution: A practical approach to global optimization. Springer, Berlin, Heidelberg, pp. 117–118. doi:10.1007/3-540-31306-0_2.

Storn, R. (2008) Differential evolution research — Trends and open questions; in Chakraborty, U. K., Ed., Advances in differential evolution. SCI 143, Springer, Berlin, Heidelberg, pp. 11–12. doi:10.1007/978-3-540-68830-3_1.

Storn, R. and Price, K. (1997) Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11, 341–359. doi:10.1023/A:1008202821328.

Wu, G., Pedrycz, W., Suganthan, P. N. and Mallipeddi, R. (2015) A variable reduction strategy for evolutionary algorithms handling equality constraints. Applied Soft Computing 37, 774–786. doi:10.1016/j.asoc.2015.09.007.

Zhang, H. and Rangaiah, G. P. (2012) An efficient constraint handling method with integrated differential evolution for numerical and engineering optimization. Computers and Chemical Engineering 37, 74–88. doi:10.1016/j.compchemeng.2011.09.018.

Zielinski, K. and Laur, R. (2008) Stopping criteria for differential evolution in constrained single-objective optimization; in Chakraborty, U. K., Ed., Advances in differential evolution. SCI 143, Springer, Berlin, Heidelberg, pp. 111–138. doi:10.1007/978-3-540-68830-3_4.

See Also

Function DEoptim() in the DEoptim package has many more options than JDEoptim(), but does not allow constraints in the same flexible manner.

Examples

# NOTE: Examples were excluded from testing
#       to reduce package check time.

# Use a preset seed so test values are reproducible.
set.seed(1234)

# Bound-constrained optimization

#   Griewank function
#
#   -600 <= xi <= 600, i = {1, 2, ..., n}
#   The function has a global minimum located at
#   x* = (0, 0, ..., 0) with f(x*) = 0. Number of local minima
#   for arbitrary n is unknown, but in the two dimensional case
#   there are some 500 local minima.
#
#   Source:
#     Ali, M. Montaz, Khompatraporn, Charoenchai, and
#     Zabinsky, Zelda B. (2005).
#     A numerical evaluation of several stochastic algorithms
#     on selected continuous global optimization test problems.
#     Journal of Global Optimization 31, 635-672.
#     https://doi.org/10.1007/s10898-004-9972-2
griewank <- function(x) {
    1 + crossprod(x)/4000 - prod( cos(x/sqrt(seq_along(x))) )
}

JDEoptim(rep(-600, 10), rep(600, 10), griewank,
         tol = 1e-7, trace = TRUE, triter = 50)

# Nonlinear constrained optimization

#   0 <= x1 <= 34, 0 <= x2 <= 17, 100 <= x3 <= 300
#   The global optimum is
#   (x1, x2, x3; f) = (0, 16.666667, 100; 189.311627).
#
#   Source:
#     Westerberg, Arthur W., and Shah, Jigar V. (1978).
#     Assuring a global optimum by the use of an upper bound
#     on the lower (dual) bound.
#     Computers and Chemical Engineering 2, 83-92.
#     https://doi.org/10.1016/0098-1354(78)80012-X
fcn <-
    list(obj = function(x) {
             35*x[1]^0.6 + 35*x[2]^0.6
         },
         eq = 2,
         con = function(x) {
             x1 <- x[1]; x3 <- x[3]
             c(600*x1 - 50*x3 - x1*x3 + 5000,
               600*x[2] + 50*x3 - 15000)
         })

JDEoptim(c(0, 0, 100), c(34, 17, 300),
         fn = fcn$obj, constr = fcn$con, meq = fcn$eq,
         tol = 1e-7, trace = TRUE, triter = 50)

#   Designing a pressure vessel
#   Case A: all variables are treated as continuous
#
#   1.1 <= x1 <= 12.5*, 0.6 <= x2 <= 12.5*,
#   0.0 <= x3 <= 240.0*, 0.0 <= x4 <= 240.0
#   Roughly guessed*
#   The global optimum is (x1, x2, x3, x4; f) =
#   (1.100000, 0.600000, 56.99482, 51.00125; 7019.031).
#
#   Source:
#     Lampinen, Jouni, and Zelinka, Ivan (1999).
#     Mechanical engineering design optimization
#     by differential evolution.
#     In: David Corne, Marco Dorigo and Fred Glover (Editors),
#     New Ideas in Optimization, McGraw-Hill, pp 127-146
pressure_vessel_A <-
    list(obj = function(x) {
             x1 <- x[1]; x2 <- x[2]; x3 <- x[3]; x4 <- x[4]
             0.6224*x1*x3*x4 + 1.7781*x2*x3^2 +
             3.1611*x1^2*x4 + 19.84*x1^2*x3
         },
         con = function(x) {
             x1 <- x[1]; x2 <- x[2]; x3 <- x[3]; x4 <- x[4]
             c(0.0193*x3 - x1,
               0.00954*x3 - x2,
               750.0*1728.0 - pi*x3^2*x4 - 4/3*pi*x3^3)
         })

JDEoptim(c( 1.1,  0.6,   0.0,   0.0),
         c(12.5, 12.5, 240.0, 240.0),
         fn = pressure_vessel_A$obj,
         constr = pressure_vessel_A$con,
         tol = 1e-7, trace = TRUE, triter = 50)

# Mixed integer nonlinear programming

#   Designing a pressure vessel
#   Case B: solved according to the original problem statements
#           steel plate available in thicknesses multiple
#           of 0.0625 inch
#
#   wall thickness of the
#   shell 1.1 [18*0.0625] <= x1 <= 12.5 [200*0.0625]
#   heads 0.6 [10*0.0625] <= x2 <= 12.5 [200*0.0625]
#         0.0 <= x3 <= 240.0, 0.0 <= x4 <= 240.0
#   The global optimum is (x1, x2, x3, x4; f) =
#   (1.125 [18*0.0625], 0.625 [10*0.0625],
#    58.29016, 43.69266; 7197.729).
pressure_vessel_B <-
    list(obj = function(x) {
             x1 <- floor(x[1])*0.0625
             x2 <- floor(x[2])*0.0625
             x3 <- x[3]; x4 <- x[4]
             0.6224*x1*x3*x4 + 1.7781*x2*x3^2 +
             3.1611*x1^2*x4 + 19.84*x1^2*x3
         },
         con = function(x) {
             x1 <- floor(x[1])*0.0625
             x2 <- floor(x[2])*0.0625
             x3 <- x[3]; x4 <- x[4]
             c(0.0193*x3 - x1,
               0.00954*x3 - x2,
               750.0*1728.0 - pi*x3^2*x4 - 4/3*pi*x3^3)
         })

res <- JDEoptim(c( 18,    10,     0.0,   0.0),
                c(200+1, 200+1, 240.0, 240.0),
                fn = pressure_vessel_B$obj,
                constr = pressure_vessel_B$con,
                tol = 1e-7, trace = TRUE, triter = 50)
res
# Now convert to integer x1 and x2
c(floor(res$par[1:2]), res$par[3:4])

---

Bound-Constrained and Nonlinear Constrained Multimodal Optimization via Differential Evolution

Description

A bespoke implementation of the ‘NCDE’ (neighborhood based crowding DE) algorithm by Qu et al. (2012) doi:10.1109/TEVC.2011.2161873, assisted with the dynamic archive mechanism of Epitropakis et al. (2013) doi:10.1109/CEC.2013.6557556.

Usage

NCDEoptim(lower, upper, fn,
          constr = NULL, meq = 0, eps = 1e-5,
          crit = 1e-5, niche_radius = NULL, archive_size = 100,
          reinit_if_solu_in_arch = TRUE,
          NP = 100, Fl = 0.1, Fu = 1, CRl = 0, CRu = 1.1,
          nbngbrsl = NP/20, nbngbrsu = NP/5,
          tau_F = 0.1, tau_CR = 0.1, tau_pF = 0.1,
          tau_nbngbrs = 0.1,
          jitter_factor = 0.001,
          maxiter = 2000,
          add_to_init_pop = NULL,
          trace = FALSE, triter = 1,
          ...)

Arguments

lower, upper	numeric vectors, the lower and upper bounds of the search space (box constraints); must be finite (is.finite).
fn	a function to be minimized that takes a numeric vector $X_i$ as first argument and returns the value of the objective.
constr	a vector function specifying the left-hand side of equality constraints defined to equal zero ($h_j(X_i) = 0,\; j = 1,\ldots,\mathrm{meq}$), followed by inequality constraints defined as lesser than zero ($g_j(X_i) \le 0,\; j = \mathrm{meq}+1,\ldots$). This function takes $X_i$ as its first argument and returns a numeric vector with the same length of the total number of constraints. It defaults to NULL, which means that bound-constrained minimization is used.
meq	an integer, the first meq constraints are equality constraints whereas the remaining ones are inequality constraints. Defaults to 0 (inequality constraints only).
eps	the maximal admissible constraint violation for equality constraints. A numeric vector of small positive tolerance values with length meq used in the transformation of equalities into inequalities of the form $|h_j(X_i)| - \epsilon \le 0$. A scalar value is expanded to apply to all equality constraints. Default is 1e-5.
crit	a numeric, the acceptance threshold on the archive strategy. If isTRUE(all.equal(fn(X_best_so_far_in_archive), fn(X_i), tolerance = crit)), a solution $X_i$ is checked for possible insertion into the dynamic archive. Defaults to 1e-5.
niche_radius	a numeric, the absolute tolerance used to decide whether the solution $X_i$ is identical to an already existing local or global solution in the archive. It defaults to NULL, meaning that the niche radius is adaptively chosen during the search. Results are much better if one is able to provide a reasonable value.
archive_size	an integer, the maximum number of solutions that can be kept in the archive; entries above this limit are discarded. Default is 100.
reinit_if_solu_in_arch	a logical, if TRUE, any solution $X_i$ already in the archive reinitializes its nearest neighbor in the population within the range $[\mathrm{lower}, \mathrm{upper}]$. Default is TRUE.
NP	an integer, the population size. Defaults to 100.
Fl	a numeric, the minimum value that the scaling factor F could take. It defaults to 0.1.
Fu	a numeric, the maximum value that the scaling factor F could take. It defaults to 1.
CRl	a numeric, the minimum value to be used for the crossover constant CR. It defaults to 0.
CRu	a numeric, the maximum value to be used for the crossover constant CR. It defaults to 1.1.
nbngbrsl	an integer, the lower limit for the neighborhood size nbngbrs. It defaults to 1/20 of the population size.
nbngbrsu	an integer, the upper limit for the neighborhood size nbngbrs. It defaults to 1/5 of the population size.
tau_F	a numeric, the probability that the scaling factor F is updated. Defaults to 0.1.
tau_CR	a numeric, the probability that the crossover constant CR is updated. Defaults to 0.1.
tau_pF	a numeric, the probability that the mutation probability $p_F$ in the mutation strategy DE/rand/1/either-or is updated. Defaults to 0.1.
tau_nbngbrs	a numeric, the probability that the neighborhood size nbngbrs is updated. Defaults to 0.1.
jitter_factor	a numeric, the tuning constant for jitter. If NULL only dither is used. Default is 0.001.
maxiter	an integer, the maximum number of iterations allowed which is the stopping condition. Default is 2000.
add_to_init_pop	numeric vector of length length(lower) or column-wise matrix with length(lower) rows specifying initial candidate solutions which are appended to the randomly generated initial population. Default is NULL.
trace	a logical, determines whether or not to monitor the iteration process. Default is FALSE.
triter	an integer, trace output is printed at every triter iterations. Default is 1.
...	additional arguments passed to fn and constr.

Details

This implementation differs mainly from the original ‘NCDE’ algorithm of Qu et al. (2012) by employing the archiving procedure proposed in Epitropakis et al. (2013) and the adaptive ‘jDE’ strategy instead of canonical Diferential Evolution. The key reason for archiving good solutions during the search process is to prevent them from being lost during evolution. Constraints are tackled through the $\varepsilon$-constrained method as proposed in Poole and Allen (2019). The ‘jDE’ and $\varepsilon$-constrained mechanisms are applied in the same way as in JDEoptim, but with synchronous mode of population update. In contrast, the reinitialization in the current population triggered by already found solutions is done asynchronously.

Each line of trace output follows the format of:

iteration : < value of niche radius > population>> ( value of best solution ) best solution { index of violated constraints } archive>> [ number of solutions found ] ( value of best solution ) best solution

Value

A list with the following components:

solution_arch	a matrix whose columns are the local and global minima stored in the archive of feasible solutions in ascending order of the objective function values.
objective_arch	the values of $\mathrm{fn}(X_i)$ for the corresponding columns of solution_arch.
solution_pop	a matrix whose columns are the local and global minima stored in the final population in ascending order of the objective function values; feasible solutions come first followed by the infeasible ones.
objective_pop	the values of $\mathrm{fn}(X_i)$ for the corresponding columns of solution_pop.
iter	the number of iterations used.

and if there are general constraints present:

constr_value_arch	a matrix whose columns contain the values of the constraints for solution_arch.
constr_value_pop	a matrix whose columns contain the values of the constraints for solution_pop.

Note

This function is in an experimental stage.

Author(s)

Eduardo L. T. Conceicao [email protected]

References

Epitropakis, M. G., Li, X. and Burke, E. K. (2013) A dynamic archive niching differential evolution algorithm for multimodal optimization; in 2013 IEEE Congress on Evolutionary Computation (CEC). IEEE, pp. 79–86. doi:10.1109/CEC.2013.6557556.

Poole, D. J. and Allen, C. B. (2019) Constrained niching using differential evolution. Swarm and Evolutionary Computation 44, 74–100. doi:10.1016/j.swevo.2018.11.004.

Qu, B. Y., Suganthan, P. N. and Liang, J. J. (2012) Differential evolution with neighborhood mutation for multimodal optimization. IEEE Transactions on Evolutionary Computation 16, 601–614. doi:10.1109/TEVC.2011.2161873.

Examples

# NOTE: Examples were excluded from testing
#       to reduce package check time.

# Use a preset seed so test values are reproducible.
set.seed(1234)

# Warning: the examples are run using a very small number of
# iterations to decrease execution time.

# Bound-constrained optimization

#   Vincent function
#
#   f(x) = -mean(sin(10*log(x)))
#
#   0.25 <= xi <= 10, i = {1, 2, ..., n}
#   The function has 6^n global minima without local minima.

NCDEoptim(c(0.25, 0.25), c(10, 10),
          function(x) -mean(sin(10*log(x))),
          niche_radius = 0.2,
          maxiter = 200, trace = TRUE, triter = 20)

# Nonlinear constrained optimization

#   Function F10 of Poole and Allen (2019)
#
#   f(x) = -sin(5*pi*x)^6 + 1
#   subject to:
#   g(x) = -cos(10*pi*x) <= 0
#
#   0 <= x <= 1
#   The 10 global optima are
#   (x1*, ..., x10*; f*) = ((2*(1:10) - 1)/20; 0.875).

NCDEoptim(0, 1,
          function(x) -sin(5*pi*x)^6 + 1,
          function(x) -cos(10*pi*x),
          niche_radius = 0.05,
          maxiter = 200, trace = TRUE, triter = 20)

---