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(),
...
)
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>=0
for all components.- 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
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. As this does not give full satisfaction with the implementation in NLOPT, it has not been made available here.
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 Hock-Schittkowski no. 100
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) {
h <- numeric(4)
h[1] <- 127 - 2*x[1]^2 - 3*x[2]^4 - x[3] - 4*x[4]^2 - 5*x[5]
h[2] <- 282 - 7*x[1] - 3*x[2] - 10*x[3]^2 - x[4] + x[5]
h[3] <- 196 - 23*x[1] - x[2]^2 - 6*x[6]^2 + 8*x[7]
h[4] <- -4*x[1]^2 - x[2]^2 + 3*x[1]*x[2] -2*x[3]^2 - 5*x[6] +11*x[7]
return(h)
}
S <- cobyla(x0.hs100, fn.hs100, hin = hin.hs100,
nl.info = TRUE, control = list(xtol_rel = 1e-8, maxeval = 2000))
#> For consistency with the rest of the package the inequality sign may be switched from >= to <= in a future nloptr version.
#>
#> 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....: 1912
#> 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: 680.630057374431
#> Optimal value of controls: 2.330499 1.951372 -0.477545 4.365726 -0.6244869 1.038131 1.594227
#>
#>
## Optimal value of objective function: 680.630057374431