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).
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.optsfor 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.
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
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).
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))
#>
#> Call:
#> nloptr(x0 = x0, eval_f = fn, 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: -3.32236800687327
#> Current value of controls: 0.2016884 0.1500025 0.4768667 0.2753391 0.311648 0.6572931
#>
#>
## 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