Nonlinear Constrained and Unconstrained Optimization via Differential Evolution
An bespoke implementation of the ‘jDE’ variant by Brest et al. (2006) doi: 10.1109/TEVC.2006.872133.
JDEoptim(lower, upper, fn,
constr = NULL, meq = 0, eps = 1e-05,
NP = 10*d, 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*d, fnscale = 1,
compare_to = c("median", "max"),
add_to_init_pop = NULL,
trace = FALSE, triter = 1,
details = FALSE, ...)lower, upper |
numeric vectors of lower or upper
bounds, respectively, for the parameters to be optimized over. Must
be finite ( |
fn |
(nonlinear) objective |
constr |
an optional |
meq |
an optional positive integer specifying that the first
|
eps |
maximal admissible constraint violation for equality constraints.
An optional real vector of small positive tolerance values with length
|
NP |
an optional positive integer giving the number of candidate
solutions in the randomly distributed initial population. Defaults to
|
Fl |
an optional scalar which represents the minimum value that the
scaling factor |
Fu |
an optional scalar which represents the maximum value that
the scaling factor |
tau_F |
an optional scalar which represents the probability that
the scaling factor |
tau_CR |
an optional constant value which represents the probability
that the crossover probability |
tau_pF |
an optional scalar which represents the probability that
the mutation probability pF in the mutation strategy
DE/rand/1/either-or is updated. Defaults to |
jitter_factor |
an optional tuning constant for jitter.
If |
tol |
an optional positive scalar giving the tolerance for the
stopping criterion. Default is |
maxiter |
an optional positive integer specifying the maximum
number of iterations that may be performed before the algorithm is halted.
Defaults to |
fnscale |
an optional positive scalar specifying the typical
magnitude of |
compare_to |
an optional character string controlling which function
should be applied to the |
add_to_init_pop |
an optional real vector of length |
trace |
an optional logical value indicating if a trace of the
iteration progress should be printed. Default is |
triter |
an optional positive integer that controls the frequency
of tracing when |
details |
an optional logical value. If |
... |
optional additional arguments passed to |
The setting of the control parameters of standard
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 for immediately replacing each worse parent in the current population 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.
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 is done using the approach described in Zhang and Rangaiah (2012), but with a different reduction updating scheme for the constraint relaxation value (μ). 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 the constraint handling technique 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,…,x_k,…,x_d] is the vector of solutions, thereby obtaining a relationship as x_k = R_{k,j}([x_1,…,x_{k-1},x_{k+1},…,x_d]). But this means that 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 ≤ R_{k,j}([x_1,…,x_{k-1},x_{k+1},…,x_d]) ≤ 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).
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 for the evaluation
of the objective function. This is because DE itself works with
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.
The algorithm is stopped if
( compare_to{ [fn(X_1),…,fn(X_npop)] } - fn(X_best) )/fnscale <= tol,
where the “best” individual X_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). This is a variant of
the Diff criterion studied by Zielinski and Laur (2008), which
was found to yield the best results.
A list with the following components:
par |
The best set of parameters found. |
value |
The value of |
iter |
Number of iterations taken by the algorithm. |
convergence |
An integer code. |
and if details = TRUE:
poppar |
Matrix of dimension |
popcost |
The values of |
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.
Eduardo L. T. Conceicao mail@eduardoconceicao.org
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.
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.
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, pp. 117–118.
Storn, R. (2008) Differential evolution research — trends and open questions; in Chakraborty, U. K., Ed., Advances in differential evolution. SCI 143, Springer-Verlag, Berlin, pp. 11–12.
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.
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.
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.
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-Verlag, Berlin, pp. 111–138.
# 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.
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.
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])Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.