COBYLA is an algorithm for derivative-free optimization with nonlinear inequality and equality constraints (but see below).
cobyla(
x0,
fn,
lower = NULL,
upper = NULL,
hin = NULL,
nl.info = FALSE,
control = list(),
deprecatedBehavior = TRUE,
...
)starting point for searching the optimum.
objective function that is to be minimized.
lower and upper bound constraints.
function defining the inequality constraints, that is
hin>=0 for all components.
logical; shall the original NLopt info be shown.
list of options, see nl.opts for help.
logical; if TRUE (default for now), the old
behavior of the Jacobian function is used, where the equality is \(\ge 0\)
instead of \(\le 0\). This will be reversed in a future release and
eventually removed.
additional arguments passed to the function.
List with components:
the optimal solution found so far.
the function value corresponding to par.
number of (outer) iterations, see maxeval.
integer code indicating successful completion (> 0) or a possible error number (< 0).
character string produced by NLopt and giving additional information.
It constructs successive linear approximations of the objective function and constraints via a simplex of \(n+1\) points (in \(n\) dimensions), and optimizes these approximations in a trust region at each step.
COBYLA supports equality constraints by transforming them into two
inequality constraints. This functionality has not been added to the wrapper.
To use COBYLA with equality constraints, please use the full
nloptr invocation.
The original code, written in Fortran by Powell, was converted in C for the SciPy project.
M. J. D. Powell, “A direct search optimization method that models the objective and constraint functions by linear interpolation,” in Advances in Optimization and Numerical Analysis, eds. S. Gomez and J.-P. Hennart (Kluwer Academic: Dordrecht, 1994), p. 51-67.
## Solve the Hock-Schittkowski problem no. 100 with analytic gradients
## See https://apmonitor.com/wiki/uploads/Apps/hs100.apm
x0.hs100 <- c(1, 2, 0, 4, 0, 1, 1)
fn.hs100 <- function(x) {(x[1] - 10) ^ 2 + 5 * (x[2] - 12) ^ 2 + x[3] ^ 4 +
3 * (x[4] - 11) ^ 2 + 10 * x[5] ^ 6 + 7 * x[6] ^ 2 +
x[7] ^ 4 - 4 * x[6] * x[7] - 10 * x[6] - 8 * x[7]}
hin.hs100 <- function(x) {c(
2 * x[1] ^ 2 + 3 * x[2] ^ 4 + x[3] + 4 * x[4] ^ 2 + 5 * x[5] - 127,
7 * x[1] + 3 * x[2] + 10 * x[3] ^ 2 + x[4] - x[5] - 282,
23 * x[1] + x[2] ^ 2 + 6 * x[6] ^ 2 - 8 * x[7] - 196,
4 * x[1] ^ 2 + x[2] ^ 2 - 3 * x[1] * x[2] + 2 * x[3] ^ 2 + 5 * x[6] -
11 * x[7])
}
S <- cobyla(x0.hs100, fn.hs100, hin = hin.hs100,
nl.info = TRUE, control = list(xtol_rel = 1e-8, maxeval = 2000),
deprecatedBehavior = FALSE)
#>
#> Call:
#> nloptr(x0 = x0, eval_f = fn, lb = lower, ub = upper, eval_g_ineq = hin,
#> opts = opts)
#>
#>
#> Minimization using NLopt version 2.7.1
#>
#> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because
#> xtol_rel or xtol_abs (above) was reached. )
#>
#> Number of Iterations....: 1623
#> Termination conditions: stopval: -Inf xtol_rel: 1e-08 maxeval: 2000 ftol_rel: 0 ftol_abs: 0
#> Number of inequality constraints: 4
#> Number of equality constraints: 0
#> Optimal value of objective function: 680.630057374426
#> Optimal value of controls: 2.330499 1.951372 -0.4775447 4.365726 -0.624487 1.038131 1.594227
#>
#>
## The optimum value of the objective function should be 680.6300573
## A suitable parameter vector is roughly
## (2.330, 1.9514, -0.4775, 4.3657, -0.6245, 1.0381, 1.5942)
S
#> $par
#> [1] 2.3304990 1.9513724 -0.4775447 4.3657263 -0.6244870 1.0381308 1.5942267
#>
#> $value
#> [1] 680.6301
#>
#> $iter
#> [1] 1623
#>
#> $convergence
#> [1] 4
#>
#> $message
#> [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
#>