mlsl function

Multi-level Single-linkage

The Multi-Level Single-Linkage (MLSL ) algorithm for global optimization searches by a sequence of local optimizations from random starting points. A modification of MLSL is included using a low-discrepancy sequence (LDS ) instead of pseudorandom numbers.

mlsl(
  x0,
  fn,
  gr = NULL,
  lower,
  upper,
  local.method = "LBFGS",
  low.discrepancy = TRUE,
  nl.info = FALSE,
  control = list(),
  ...
)

Arguments

• x0: initial point for searching the optimum.
• fn: objective function that is to be minimized.
• gr: gradient of function fn; will be calculated numerically if not specified.
• lower, upper: lower and upper bound constraints.
• local.method: only BFGS for the moment.
• low.discrepancy: logical; shall a low discrepancy variation be used.
• nl.info: logical; shall the original NLopt info be shown.
• control: list of options, see nl.opts for help.
• ...: 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

MLSL is a multistart algorithm: it works by doing a sequence of local optimizations---using some other local optimization algorithm---from random or low-discrepancy starting points. MLSL is distinguished, however, by a `clustering' heuristic that helps it to avoid repeated searches of the same local optima and also has some theoretical guarantees of finding all local optima in a finite number of local minimizations.

The local-search portion of MLSL can use any of the other algorithms in NLopt , and, in particular, can use either gradient-based or derivative-free algorithms. For this wrapper only gradient-based LBFGS is available as local method.

Note

If you don't set a stopping tolerance for your local-optimization algorithm, MLSL defaults to ftol_rel = 1e-15 and xtol_rel = 1e-7 for the local searches.

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 <- mlsl(x0 = rep(0, 6), hartmann6, lower = rep(0, 6), upper = rep(1, 6),
      nl.info = TRUE, control = list(xtol_rel = 1e-8, maxeval = 1000),
      a = a, A = A, B = B)

References

A. H. G. Rinnooy Kan and G. T. Timmer, Stochastic global optimization methods Mathematical Programming, vol. 39, p. 27-78 (1987).

Sergei Kucherenko and Yury Sytsko, Application of deterministic low-discrepancy sequences in globaloptimization , Computational Optimization and Applications, vol. 30, p. 297-318 (2005).

See Also

direct

Author(s)

Hans W. Borchers