R: Constrained Optimization by Linear Approximations

cobyla {nloptr} | R Documentation

## Constrained Optimization by Linear Approximations

### Description

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.

### 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.

### 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.

### Note

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

### Author(s)

Hans W. Borchers

### 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

`bobyqa`, `newuoa`

### 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))
    ## Optimal value of objective function:  680.630057374431

[Package nloptr version 1.2.2.2 Index]