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 been 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 L-BFGS 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 Hartmann6 function
hartmann6 <- function(x) {
n <- length(x)
a <- c(1.0, 1.2, 3.0, 3.2)
A <- matrix(c(10.0, 0.05, 3.0, 17.0,
3.0, 10.0, 3.5, 8.0,
17.0, 17.0, 1.7, 0.05,
3.5, 0.1, 10.0, 10.0,
1.7, 8.0, 17.0, 0.1,
8.0, 14.0, 8.0, 14.0), nrow=4, ncol=6)
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, ncol=6)
fun <- 0.0
for (i in 1:4) {
fun <- fun - a[i] * exp(-sum(A[i,]*(x-B[i,])^2))
}
return(fun)
}
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))
#>
#> 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
#>
#>
## 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: -3.32236801141552
## Current value of controls:
## 0.2016895 0.1500107 0.476874 0.2753324 0.3116516 0.6573005