R: Conservative Convex Separable Approximation with Affine...

ccsaq {nloptr} | R Documentation

## Conservative Convex Separable Approximation with Affine Approximation plus Quadratic Penalty

### Description

This is a variant of CCSA ("conservative convex separable approximation") which, instead of constructing local MMA approximations, constructs simple quadratic approximations (or rather, affine approximations plus a quadratic penalty term to stay conservative)

### Usage

    ccsaq(x0, fn, gr = NULL, lower = NULL, upper = NULL, hin = NULL,
      hinjac = NULL, nl.info = FALSE, control = list(), ...)

### Arguments

`x0` | starting point for searching the optimum.
`fn` | objective function that is to be minimized.
`gr` | gradient of function `fn`; will be calculated numerically if not specified.
`lower, upper` | lower and upper bound constraints.
`hin` | function defining the inequality constraints, that is `hin>=0` for all components.
`hinjac` | Jacobian of function `hin`; will be calculated numerically if not specified.
`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 (> 1) or a possible error number (< 0).
`message` | character string produced by NLopt and giving additional information.

### Note

“Globally convergent” does not mean that this algorithm converges to the global optimum; it means that it is guaranteed to converge to some local minimum from any feasible starting point.

### References

Krister Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12 (2), p. 555-573 (2002).

### See Also

`mma`

### Examples

    ##  Solve the Hock-Schittkowski problem no. 100 with analytic gradients
    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)
    }
    gr.hs100 <- function(x) {
       c(  2 * x[1] -  20,
          10 * x[2] - 120,
           4 * x[3]^3,
           6 * x[4] - 66,
          60 * x[5]^5,
          14 * x[6]   - 4 * x[7] - 10,
           4 * x[7]^3 - 4 * x[6] -  8 )}
    hinjac.hs100 <- function(x) {
        matrix(c(4*x[1], 12*x[2]^3, 1, 8*x[4], 5, 0, 0,
            7, 3, 20*x[3], 1, -1, 0, 0,
            23, 2*x[2], 0, 0, 0, 12*x[6], -8,
            8*x[1]-3*x[2], 2*x[2]-3*x[1], 4*x[3], 0, 0, 5, -11), 4, 7, byrow=TRUE)
    }

    # incorrect result with exact jacobian
    S <- ccsaq(x0.hs100, fn.hs100, gr = gr.hs100,
                hin = hin.hs100, hinjac = hinjac.hs100,
                nl.info = TRUE, control = list(xtol_rel = 1e-8))

    S <- ccsaq(x0.hs100, fn.hs100, hin = hin.hs100,
                nl.info = TRUE, control = list(xtol_rel = 1e-8))

[Package nloptr version 1.2.2.2 Index]