mma function

Method of Moving Asymptotes

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

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.

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.

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.

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.

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)

References

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

See Also

slsqp

Author(s)

Hans W. Borchers