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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.

Note

The original code, written in Fortran by Powell, was converted in C for the Scipy project.

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.

See also

Author

Hans W. Borchers

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