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)

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).

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))
#> 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, eval_grad_f = gr, lb = lower, ub = upper, 
#>     eval_g_ineq = hin, eval_jac_g_ineq = hinjac, 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....: 95 
#> Termination conditions:  stopval: -Inf	xtol_rel: 1e-08	maxeval: 1000	ftol_rel: 0	ftol_abs: 0 
#> Number of inequality constraints:  4 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  700.832432136289 
#> Optimal value of controls: 1.016144 2.110428 0.0002795978 4.044003 0.001397989 1.000001 1.006676
#> 
#> 

# \donttest{
S <- ccsaq(x0.hs100, fn.hs100, hin = hin.hs100,
      nl.info = TRUE, control = list(xtol_rel = 1e-8))
#> 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, eval_grad_f = gr, lb = lower, ub = upper, 
#>     eval_g_ineq = hin, eval_jac_g_ineq = hinjac, 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....: 53 
#> Termination conditions:  stopval: -Inf	xtol_rel: 1e-08	maxeval: 1000	ftol_rel: 0	ftol_abs: 0 
#> Number of inequality constraints:  4 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  680.630056418528 
#> Optimal value of controls: 2.330494 1.951373 -0.477532 4.365726 -0.6244891 1.038135 1.59422
#> 
#> 
# }