COBYLA is an algorithm for derivative-free optimization with nonlinear inequality and equality constraints (but see below).
Usage
cobyla(
x0,
fn,
lower = NULL,
upper = NULL,
hin = NULL,
nl.info = FALSE,
control = list(),
deprecatedBehavior = TRUE,
...
)Arguments
- x0
starting point for searching the optimum.
- fn
objective function that is to be minimized.
- lower, upper
lower and upper bound constraints.
- hin
function defining the inequality constraints, that is
hin>=0for all components.- nl.info
logical; shall the original NLopt info be shown.
- control
list of options, see
nl.optsfor help.- deprecatedBehavior
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.
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
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.
References
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.
Examples
## 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."
#>