auglag: Augmented Lagrangian Algorithm

In nloptr: R Interface to NLopt

Augmented Lagrangian Algorithm

Description

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 NULL and 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.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.

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.)

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.

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.

Author(s)

Hans W. Borchers

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).

See Also

alabama::auglag, Rsolnp::solnp

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:   0.8228761 0.9114382
# $value:   1.393464
# $iter:  1001

auglag(x0, fn, gr = NULL, hin = hin, heq = heq, localsolver = "SLSQP",
       deprecatedBehavior = FALSE)

# $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:   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: -1.7173645  1.5959655  1.8268352 -0.7636185 -0.7636185
# $value:   0.05394987
# $iter:  916

nloptr documentation built on July 4, 2024, 1:08 a.m.