# mma

##### 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).

##### Usage

```
mma(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.

##### 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:

##### Value

List with components:

the optimal solution found so far.

the function value corresponding to `par`

.

number of (outer) iterations, see `maxeval`

.

integer code indicating successful completion (> 1) or a possible error number (< 0).

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).

##### See Also

##### Examples

```
# 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 {
# }
```

*Documentation reproduced from package nloptr, version 1.2.1, License: LGPL-3*