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.
Usage
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
BFGSfor 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.optsfor help.- ...
additional arguments passed to the function.
Value
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.
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 global optimization”, Computational Optimization and Applications, vol. 30, p. 297-318 (2005).
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)
#>
#> Call:
#> nloptr(x0 = x0, eval_f = fn, eval_grad_f = gr, lb = lower, ub = upper,
#> opts = opts)
#>
#>
#> Minimization using NLopt version 2.7.1
#>
#> NLopt solver status: 5 ( NLOPT_MAXEVAL_REACHED: Optimization stopped because
#> maxeval (above) was reached. )
#>
#> Number of Iterations....: 1000
#> Termination conditions: stopval: -Inf xtol_rel: 1e-08 maxeval: 1000 ftol_rel: 0 ftol_abs: 0
#> Number of inequality constraints: 0
#> Number of equality constraints: 0
#> Current value of objective function: -3.32236801141544
#> Current value of controls: 0.2016895 0.1500107 0.476874 0.2753324 0.3116516 0.6573005
#>
#>