nloptr (version 2.0.3)

mlsl: Multi-level Single-linkage

Description

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.

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 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.

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

direct

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

# }