mma: Method of Moving Asymptotes

In nloptr: R Interface to NLopt

Method of Moving Asymptotes

Description

Globally-convergent method-of-moving-asymptotes (MMA) algorithm for gradient-based local optimization, including nonlinear inequality constraints (but not equality constraints).

Usage

mma(
  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.

Details

This is an improved CCSA ("conservative convex separable approximation") variant of the original MMA algorithm published by Svanberg in 1987, which has become popular for topology optimization.

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; rather, it means that the algorithm is guaranteed to converge to some local minimum from any feasible starting point.

Author(s)

Hans W. Borchers

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

slsqp

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)

# Using analytic Jacobian
S <- mma(x0.hs100, fn.hs100, gr = gr.hs100,
      hin = hin.hs100, hinjac = hinjac.hs100,
      nl.info = TRUE, control = list(xtol_rel = 1e-8),
      deprecatedBehavior = FALSE)

# Using computed Jacobian
S <- mma(x0.hs100, fn.hs100, hin = hin.hs100,
      nl.info = TRUE, control = list(xtol_rel = 1e-8),
      deprecatedBehavior = FALSE)

nloptr documentation built on July 4, 2024, 1:08 a.m.