nloptr (version 2.2.0)

direct: DIviding RECTangles Algorithm for Global Optimization

Description

DIRECT is a deterministic search algorithm based on systematic division of the search domain into smaller and smaller hyperrectangles. The DIRECT_L makes the algorithm more biased towards local search (more efficient for functions without too many minima).

Usage

direct(
  fn,
  lower,
  upper,
  scaled = TRUE,
  original = FALSE,
  nl.info = FALSE,
  control = list(),
  ...
)
directL(
  fn,
  lower,
  upper,
  randomized = FALSE,
  original = FALSE,
  nl.info = FALSE,
  control = list(),
  ...
)

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.

Arguments

fn

objective function that is to be minimized.

lower, upper

lower and upper bound constraints.

scaled

logical; shall the hypercube be scaled before starting.

original

logical; whether to use the original implementation by Gablonsky -- the performance is mostly similar.

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.

randomized

logical; shall some randomization be used to decide which dimension to halve next in the case of near-ties.

Author

Hans W. Borchers

Details

The DIRECT and DIRECT-L algorithms start by rescaling the bound constraints to a hypercube, which gives all dimensions equal weight in the search procedure. If your dimensions do not have equal weight, e.g. if you have a ``long and skinny'' search space and your function varies at about the same speed in all directions, it may be better to use unscaled variant of the DIRECT algorithm.

The algorithms only handle finite bound constraints which must be provided. The original versions may include some support for arbitrary nonlinear inequality, but this has not been tested.

The original versions do not have randomized or unscaled variants, so these options will be disregarded for these versions.

References

D. R. Jones, C. D. Perttunen, and B. E. Stuckmann, ``Lipschitzian optimization without the Lipschitz constant,'' J. Optimization Theory and Applications, vol. 79, p. 157 (1993).

J. M. Gablonsky and C. T. Kelley, ``A locally-biased form of the DIRECT algorithm," J. Global Optimization, vol. 21 (1), p. 27-37 (2001).

See Also

The dfoptim package will provide a pure R version of this algorithm.

Examples

### Minimize the Hartmann6 function
hartmann6 <- function(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
  for (i in 1:4) {
    fun <- fun - a[i] * exp(-sum(A[i,] * (x - B[i,]) ^ 2))
  }
  fun
}
S <- directL(hartmann6, rep(0, 6), 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:  -3.32236800687327
## Current value of controls:
##   0.2016884 0.1500025 0.4768667 0.2753391 0.311648 0.6572931