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,
...
)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 and upper bound constraints.
function defining the inequality constraints, that is
hin <= 0 for all components.
Jacobian of function hin; will be calculated
numerically if not specified.
logical; shall the original NLopt info been shown.
list of options, see nl.opts for help.
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.
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.
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.
“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.
Krister Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations”, SIAM J. Optim. 12 (2), p. 555-573 (2002).
# 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
#>
#>