crs2lm function

Controlled Random Search

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,
  ...
)

Arguments

• x0: initial point for searching the optimum.
• fn: objective function that is to be minimized.
• lower, upper: lower and upper bound constraints.
• maxeval: maximum number of function evaluations.
• pop.size: population size.
• ranseed: prescribe seed for random number generator.
• xtol_rel: stopping criterion for relative change reached.
• nl.info: logical; shall the original NLopt info be shown.
• ...: additional arguments passed to the function.

Returns

List with components: - par: the optimal solution found so far.

• value: the function value corresponding to par.

• iter: number of (outer) iterations, see maxeval.

• convergence: integer code indicating successful completion (> 0) or a possible error number (< 0).

• message: character string produced by NLopt and giving additional information.

Details

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.

Note

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 $n+1$.

Examples

## 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)

S

References

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).