Low-storage BFGS

Low-storage version of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.

lbfgs(
  x0,
  fn,
  gr = NULL,
  lower = NULL,
  upper = NULL,
  nl.info = FALSE,
  control = list(),
  ...
)

Arguments

x0: initial point for searching the optimum.
fn: objective function to be minimized.
gr: gradient of function fn; will be calculated numerically if not specified.
lower, upper: lower and upper bound constraints.
nl.info: logical; shall the original NLopt info been shown.
control: list of control parameters, see nl.opts for help.
...: further arguments to be 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 (> 0) or a possible error number (< 0).
message: character string produced by NLopt and giving additional information.

Details

The low-storage (or limited-memory) algorithm is a member of the class of quasi-Newton optimization methods. It is well suited for optimization problems with a large number of variables.

One parameter of this algorithm is the number m of gradients to remember from previous optimization steps. NLopt sets m to a heuristic value by default. It can be changed by the NLopt function set_vector_storage.

Note

Based on a Fortran implementation of the low-storage BFGS algorithm written by L. Luksan, and posted under the GNU LGPL license.

References

J. Nocedal, “Updating quasi-Newton matrices with limited storage,” Math. Comput. 35, 773-782 (1980).

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Programming 45, p. 503-528 (1989).

Author

Hans W. Borchers

Examples


flb <- function(x) {
  p <- length(x)
  sum(c(1, rep(4, p-1)) * (x - c(1, x[-p])^2)^2)
}
# 25-dimensional box constrained: par[24] is *not* at the boundary
S <- lbfgs(rep(3, 25), flb, lower=rep(2, 25), upper=rep(4, 25),
     nl.info = TRUE, control = list(xtol_rel=1e-8))
#> 
#> Call:
#> nloptr(x0 = x0, eval_f = fn, eval_grad_f = gr, lb = lower, ub = upper, 
#>     opts = opts)
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 1 ( NLOPT_SUCCESS: Generic success return value. )
#> 
#> Number of Iterations....: 19 
#> Termination conditions:  stopval: -Inf	xtol_rel: 1e-08	maxeval: 1000	ftol_rel: 0	ftol_abs: 0 
#> Number of inequality constraints:  0 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  368.105912874334 
#> Optimal value of controls: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2.109093 4
#> 
#> 
## Optimal value of objective function:  368.105912874334
## Optimal value of controls: 2  ...  2  2.109093  4