mma: Method of Moving Asymptotes
In nloptwrap: Wraps solvers in Package nloptr

Description Usage Arguments Details Value Note References See Also Examples

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

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

`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.

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:

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.

“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.

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

slsqp

##  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))

# correct result with inexact jacobian
S <- mma(x0.hs100, fn.hs100, hin = hin.hs100,
            nl.info = TRUE, control = list(xtol_rel = 1e-8))