The Augmented Lagrangian method adds additional terms to the unconstrained objective function, designed to emulate a Lagrangian multiplier.
Usage
auglag(
x0,
fn,
gr = NULL,
lower = NULL,
upper = NULL,
hin = NULL,
hinjac = NULL,
heq = NULL,
heqjac = NULL,
localsolver = "COBYLA",
localtol = 1e-06,
ineq2local = FALSE,
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 the objective function; will be provided provided is
NULLand the solver requires derivatives.- lower, upper
lower and upper bound constraints.
- hin, hinjac
defines the inequality constraints,
hin(x) >= 0- heq, heqjac
defines the equality constraints,
heq(x) = 0.- localsolver
available local solvers: COBYLA, LBFGS, MMA, or SLSQP.
- localtol
tolerance applied in the selected local solver.
- ineq2local
logical; shall the inequality constraints be treated by the local solver?; not possible at the moment.
- nl.info
logical; shall the original NLopt info been shown.
- control
list of options, see
nl.optsfor 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.- global_solver
the global NLOPT solver used.
- local_solver
the local NLOPT solver used, LBFGS or COBYLA.
- convergence
integer code indicating successful completion (> 0) or a possible error number (< 0).
- message
character string produced by NLopt and giving additional information.
Details
This method combines the objective function and the nonlinear inequality/equality constraints (if any) in to a single function: essentially, the objective plus a `penalty' for any violated constraints.
This modified objective function is then passed to another optimization algorithm with no nonlinear constraints. If the constraints are violated by the solution of this sub-problem, then the size of the penalties is increased and the process is repeated; eventually, the process must converge to the desired solution (if it exists).
Since all of the actual optimization is performed in this subsidiary optimizer, the subsidiary algorithm that you specify determines whether the optimization is gradient-based or derivative-free.
The local solvers available at the moment are COBYLA'' (for the derivative-free approach) and LBFGS'', MMA'', or SLSQP'' (for smooth
functions). The tolerance for the local solver has to be provided.
There is a variant that only uses penalty functions for equality constraints while inequality constraints are passed through to the subsidiary algorithm to be handled directly; in this case, the subsidiary algorithm must handle inequality constraints. (At the moment, this variant has been turned off because of problems with the NLOPT library.)
Note
Birgin and Martinez provide their own free implementation of the method as part of the TANGO project; other implementations can be found in semi-free packages like LANCELOT.
References
Andrew R. Conn, Nicholas I. M. Gould, and Philippe L. Toint, ``A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds,'' SIAM J. Numer. Anal. vol. 28, no. 2, p. 545-572 (1991).
E. G. Birgin and J. M. Martinez, ``Improving ultimate convergence of an augmented Lagrangian method," Optimization Methods and Software vol. 23, no. 2, p. 177-195 (2008).
Examples
x0 <- c(1, 1)
fn <- function(x) (x[1] - 2) ^ 2 + (x[2] - 1) ^ 2
hin <- function(x) 0.25 * x[1]^2 + x[2] ^ 2 - 1 # hin <= 0
heq <- function(x) x[1] - 2 * x[2] + 1 # heq = 0
gr <- function(x) nl.grad(x, fn)
hinjac <- function(x) nl.jacobian(x, hin)
heqjac <- function(x) nl.jacobian(x, heq)
# with COBYLA
auglag(x0, fn, gr = NULL, hin = hin, heq = heq, deprecatedBehavior = FALSE)
#> $par
#> [1] 0.8228755 0.9114379
#>
#> $value
#> [1] 1.393465
#>
#> $iter
#> [1] 1001
#>
#> $global_solver
#> [1] "NLOPT_LN_AUGLAG"
#>
#> $local_solver
#> [1] "NLOPT_LN_COBYLA"
#>
#> $convergence
#> [1] 5
#>
#> $message
#> [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."
#>
# $par: 0.8228761 0.9114382
# $value: 1.393464
# $iter: 1001
auglag(x0, fn, gr = NULL, hin = hin, heq = heq, localsolver = "SLSQP",
deprecatedBehavior = FALSE)
#> $par
#> [1] 0.8228757 0.9114378
#>
#> $value
#> [1] 1.393465
#>
#> $iter
#> [1] 184
#>
#> $global_solver
#> [1] "NLOPT_LD_AUGLAG"
#>
#> $local_solver
#> [1] "NLOPT_LD_SLSQP"
#>
#> $convergence
#> [1] 4
#>
#> $message
#> [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
#>
# $par: 0.8228757 0.9114378
# $value: 1.393465
# $iter 184
## Example from the alabama::auglag help page
## Parameters should be roughly (0, 0, 1) with an objective value of 1.
fn <- function(x) (x[1] + 3 * x[2] + x[3]) ^ 2 + 4 * (x[1] - x[2]) ^ 2
heq <- function(x) x[1] + x[2] + x[3] - 1
# hin restated from alabama example to be <= 0.
hin <- function(x) c(-6 * x[2] - 4 * x[3] + x[1] ^ 3 + 3, -x[1], -x[2], -x[3])
set.seed(12)
auglag(runif(3), fn, hin = hin, heq = heq, localsolver= "lbfgs",
deprecatedBehavior = FALSE)
#> $par
#> [1] 4.861756e-08 4.732373e-08 9.999999e-01
#>
#> $value
#> [1] 1
#>
#> $iter
#> [1] 145
#>
#> $global_solver
#> [1] "NLOPT_LD_AUGLAG"
#>
#> $local_solver
#> [1] "NLOPT_LD_LBFGS"
#>
#> $convergence
#> [1] 4
#>
#> $message
#> [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
#>
# $par: 4.861756e-08 4.732373e-08 9.999999e-01
# $value: 1
# $iter: 145
## Powell problem from the Rsolnp::solnp help page
## Parameters should be roughly (-1.7171, 1.5957, 1.8272, -0.7636, -0.7636)
## with an objective value of 0.0539498478.
x0 <- c(-2, 2, 2, -1, -1)
fn1 <- function(x) exp(x[1] * x[2] * x[3] * x[4] * x[5])
eqn1 <-function(x)
c(x[1] * x[1] + x[2] * x[2] + x[3] * x[3] + x[4] * x[4] + x[5] * x[5] - 10,
x[2] * x[3] - 5 * x[4] * x[5],
x[1] * x[1] * x[1] + x[2] * x[2] * x[2] + 1)
auglag(x0, fn1, heq = eqn1, localsolver = "mma", deprecatedBehavior = FALSE)
#> $par
#> [1] -1.7172821 1.5958701 1.8269884 -0.7636277 -0.7636277
#>
#> $value
#> [1] 0.05394986
#>
#> $iter
#> [1] 932
#>
#> $global_solver
#> [1] "NLOPT_LD_AUGLAG"
#>
#> $local_solver
#> [1] "NLOPT_LD_MMA"
#>
#> $convergence
#> [1] 4
#>
#> $message
#> [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
#>
# $par: -1.7173645 1.5959655 1.8268352 -0.7636185 -0.7636185
# $value: 0.05394987
# $iter: 916