Subplex Algorithm

Subplex is a variant of Nelder-Mead that uses Nelder-Mead on a sequence of subspaces.

sbplx(
  x0,
  fn,
  lower = NULL,
  upper = NULL,
  nl.info = FALSE,
  control = list(),
  ...
)

Arguments

x0

starting point for searching the optimum.

fn

objective function that is to be minimized.

lower, upper

lower and upper bound constraints.

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

SUBPLEX is claimed to be much more efficient and robust than the original Nelder-Mead while retaining the latter's facility with discontinuous objectives.

This implementation has explicit support for bound constraints via the method in the Box paper as described on the neldermead help page.

Note

It is the request of Tom Rowan that reimplementations of his algorithm shall not use the name `subplex'.

References

T. Rowan, “Functional Stability Analysis of Numerical Algorithms”, Ph.D. thesis, Department of Computer Sciences, University of Texas at Austin, 1990.

See also

subplex::subplex

Examples

# Fletcher and Powell's helic valley
fphv <- function(x)
  100*(x[3] - 10*atan2(x[2], x[1])/(2*pi))^2 +
    (sqrt(x[1]^2 + x[2]^2) - 1)^2 +x[3]^2
x0 <- c(-1, 0, 0)
sbplx(x0, fphv)  #  1 0 0
#> $par
#> [1] 1.000000e+00 3.706887e-12 5.858708e-12
#> 
#> $value
#> [1] 3.449246e-23
#> 
#> $iter
#> [1] 994
#> 
#> $convergence
#> [1] 4
#> 
#> $message
#> [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
#> 

# Powell's Singular Function (PSF)
psf <- function(x)  (x[1] + 10*x[2])^2 + 5*(x[3] - x[4])^2 +
          (x[2] - 2*x[3])^4 + 10*(x[1] - x[4])^4
x0 <- c(3, -1, 0, 1)
sbplx(x0, psf, control = list(maxeval = Inf, ftol_rel = 1e-6)) #  0 0 0 0 (?)
#> Warning: NAs introduced by coercion to integer range
#> $par
#> [1]  0.012385093 -0.001238441  0.007823193  0.007827134
#> 
#> $value
#> [1] 8.567466e-08
#> 
#> $iter
#> [1] 78796
#> 
#> $convergence
#> [1] 3
#> 
#> $message
#> [1] "NLOPT_FTOL_REACHED: Optimization stopped because ftol_rel or ftol_abs (above) was reached."
#>