R: DIviding RECTangles Algorithm for Global Optimization

direct {nloptr} | R Documentation

## 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(), ...)

### 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.

### 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.

### 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.

### Note

The DIRECT_L algorithm should be tried first.

### Author(s)

Hans W. Borchers

### 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) {
        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 <- directL(hartmann6, rep(0,6), rep(1,6),
                 nl.info = TRUE, control=list(xtol_rel=1e-8, maxeval=1000))
    ## Number of Iterations....: 500
    ## Termination conditions:  stopval: -Inf
    ##     xtol_rel: 1e-08,  maxeval: 500,  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

[Package nloptr version 1.2.2.2 Index]