ccsaq() R function from [nloptr]

Conservative Convex Separable Approximation with Affine Approximation plus Quadratic Penalty

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)


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

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.
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.

Returns

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.

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])
}

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),
           nrow = 4, byrow = TRUE)
}

##  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)

# Results 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),
      deprecatedBehavior = FALSE)

# Results without Jacobian
S <- ccsaq(x0.hs100, fn.hs100, hin = hin.hs100,
      nl.info = TRUE, control = list(xtol_rel = 1e-8),
      deprecatedBehavior = FALSE)

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).

ccsaq function