# mlsl

##### 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.

##### 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

`BFGS`

for 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.opts`

for help.- ...
additional arguments passed to the function.

##### 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 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.

##### Value

List with components:

the optimal solution found so far.

the function value corresponding to `par`

.

number of (outer) iterations, see `maxeval`

.

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

character string produced by NLopt and giving additional information.

##### 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).

##### See Also

##### Examples

```
# NOT RUN {
### 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 0.6573005
# }
```

*Documentation reproduced from package nloptr, version 1.2.1, License: LGPL-3*