mma: 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(), ...)

Arguments

starting point for searching the optimum.

objective function that is to be minimized.

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.

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:

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

Run this code

# NOT RUN {
##  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 <- mma(x0.hs100, fn.hs100, gr = gr.hs100,
            hin = hin.hs100, hinjac = hinjac.hs100,
            nl.info = TRUE, control = list(xtol_rel = 1e-8))

# }
# NOT RUN {
# This example is put in donttest because it runs for more than
# 40 seconds under 32-bit Windows. The difference in time needed
# to execute the code between 32-bit Windows and 64-bit Windows
# can probably be explained by differences in rounding/truncation
# on the different systems. On Windows 32-bit more iterations
# are needed resulting in a longer runtime.
# correct result with inexact jacobian
S <- mma(x0.hs100, fn.hs100, hin = hin.hs100,
            nl.info = TRUE, control = list(xtol_rel = 1e-8))
# }
# NOT RUN {
# }

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