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Method of Moving Asymptotes

Source: R/mma.R
mma.Rd

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.

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

“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.

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

Author

Hans W. Borchers

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)
#> 
#> Call:
#> nloptr(x0 = x0, eval_f = fn, eval_grad_f = gr, lb = lower, ub = upper, 
#>     eval_g_ineq = hin, eval_jac_g_ineq = hinjac, opts = opts)
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because 
#> xtol_rel or xtol_abs (above) was reached. )
#> 
#> Number of Iterations....: 229 
#> Termination conditions:  stopval: -Inf	xtol_rel: 1e-08	maxeval: 1000	ftol_rel: 0	ftol_abs: 0 
#> Number of inequality constraints:  4 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  680.630044400453 
#> Optimal value of controls: 2.3305 1.951372 -0.477538 4.365726 -0.6244892 1.038134 1.594228
#> 
#> 

# Using computed Jacobian
S <- mma(x0.hs100, fn.hs100, hin = hin.hs100,
      nl.info = TRUE, control = list(xtol_rel = 1e-8),
      deprecatedBehavior = FALSE)
#> 
#> Call:
#> nloptr(x0 = x0, eval_f = fn, eval_grad_f = gr, lb = lower, ub = upper, 
#>     eval_g_ineq = hin, eval_jac_g_ineq = hinjac, opts = opts)
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because 
#> xtol_rel or xtol_abs (above) was reached. )
#> 
#> Number of Iterations....: 277 
#> Termination conditions:  stopval: -Inf	xtol_rel: 1e-08	maxeval: 1000	ftol_rel: 0	ftol_abs: 0 
#> Number of inequality constraints:  4 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  680.630053840435 
#> Optimal value of controls: 2.330504 1.951372 -0.477535 4.365726 -0.6244867 1.038129 1.59423
#> 
#>