The Controlled Random Search (CRS) algorithm (and in particular, the CRS2 variant) with the `local mutation' modification.
crs2lm(
x0,
fn,
lower,
upper,
maxeval = 10000,
pop.size = 10 * (length(x0) + 1),
ranseed = NULL,
xtol_rel = 1e-06,
nl.info = FALSE,
...
)initial point for searching the optimum.
objective function that is to be minimized.
lower and upper bound constraints.
maximum number of function evaluations.
population size.
prescribe seed for random number generator.
stopping criterion for relative change reached.
logical; shall the original NLopt info be shown.
additional arguments passed to the function.
List with components:
the optimal solution found so far.
the function value corresponding to par.
number of (outer) iterations, see maxeval.
integer code indicating successful completion (> 0) or a possible error number (< 0).
character string produced by NLopt and giving additional information.
The CRS algorithms are sometimes compared to genetic algorithms, in that they start with a random population of points, and randomly evolve these points by heuristic rules. In this case, the evolution somewhat resembles a randomized Nelder-Mead algorithm.
The published results for CRS seem to be largely empirical.
The initial population size for CRS defaults to \(10x(n+1)\) in \(n\) dimensions, but this can be changed. The initial population must be at least \(n+1\).
W. L. Price, “Global optimization by controlled random search,” J. Optim. Theory Appl. 40 (3), p. 333-348 (1983).
P. Kaelo and M. M. Ali, “Some variants of the controlled random search algorithm for global optimization,” J. Optim. Theory Appl. 130 (2), 253-264 (2006).
## Minimize the Hartmann 6-Dimensional function
## See https://www.sfu.ca/~ssurjano/hart6.html
a <- c(1.0, 1.2, 3.0, 3.2)
A <- matrix(c(10, 0.05, 3, 17,
3, 10, 3.5, 8,
17, 17, 1.7, 0.05,
3.5, 0.1, 10, 10,
1.7, 8, 17, 0.1,
8, 14, 8, 14), nrow = 4)
B <- matrix(c(.1312, .2329, .2348, .4047,
.1696, .4135, .1451, .8828,
.5569, .8307, .3522, .8732,
.0124, .3736, .2883, .5743,
.8283, .1004, .3047, .1091,
.5886, .9991, .6650, .0381), nrow = 4)
hartmann6 <- function(x, a, A, B) {
fun <- 0
for (i in 1:4) {
fun <- fun - a[i] * exp(-sum(A[i, ] * (x - B[i, ]) ^ 2))
}
fun
}
## The function has a global minimum of -3.32237 at
## (0.20169, 0.150011, 0.476874, 0.275332, 0.311652, 0.6573)
S <- crs2lm(x0 = rep(0, 6), hartmann6, lower = rep(0, 6), upper = rep(1, 6),
ranseed = 10L, nl.info = TRUE, xtol_rel=1e-8, maxeval = 10000,
a = a, A = A, B = B)
#>
#> Call:
#> nloptr(x0 = x0, eval_f = fn, lb = lower, ub = upper, opts = opts)
#>
#>
#> Minimization using NLopt version 2.7.1
#>
#> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because
#> xtol_rel or xtol_abs (above) was reached. )
#>
#> Number of Iterations....: 5690
#> Termination conditions: maxeval: 10000 xtol_rel: 1e-08
#> Number of inequality constraints: 0
#> Number of equality constraints: 0
#> Optimal value of objective function: -3.32236801141551
#> Optimal value of controls: 0.2016895 0.1500107 0.476874 0.2753324 0.3116516 0.6573005
#>
#>
S
#> $par
#> [1] 0.2016895 0.1500107 0.4768740 0.2753324 0.3116516 0.6573005
#>
#> $value
#> [1] -3.322368
#>
#> $iter
#> [1] 5690
#>
#> $convergence
#> [1] 4
#>
#> $message
#> [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
#>