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(), ...)
x0 |
initial point for searching the optimum. |
fn |
objective function that is to be minimized. |
gr |
gradient of function |
lower, upper |
lower and upper bound constraints. |
local.method |
only |
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 |
... |
additional arguments passed to the function. |
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 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.
List with components:
par |
the optimal solution found so far. |
value |
the function value corresponding to |
iter |
number of (outer) iterations, see |
convergence |
integer code indicating successful completion (> 0) or a possible error number (< 0). |
message |
character string produced by NLopt and giving additional information. |
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.
Hans W. Borchers
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).
### 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))
## 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 0.6573005Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.