knitr::opts_chunk$set(echo = TRUE) ## Added 20190717 to get vignette to build pkgbuild::compile_dll()

In 2011 the authors of the L-BFGSB program published a correction
and update to their 1995 code. The latter is the basis of the L-BFGS-B
method of the `optim()`

function in base-R. The package `lbfgsb3`

wrapped
the updated code using a `.Fortran`

call after removing a very large number
of Fortran output statements. Matthew Fidler used this Fortran code and
an `Rcpp`

interface to produce package `lbfgsb3c`

where the function
`lbfgsb3c()`

returns an object similar to that of base-R `optim()`

and
that of `optimx::optimr()`

. Subsequently, in a fine example of the
collaborations that have made **R** so useful, we have merged the
functionality of package `lbfgsb3`

into `lbfgsb3c`

, as explained
in this vignette. Note that this document is intended primarily to
document our efforts to check the differences in variants of the
code rather than be expository.

The base-R code lbfgsb.c (at writing in R-3.5.2/src/appl/) is commented:

/* l-bfgs-b.f -- translated by f2c (version 19991025). From ?optim: The code for method ‘"L-BFGS-B"’ is based on Fortran code by Zhu, Byrd, Lu-Chen and Nocedal obtained from Netlib (file 'opt/lbfgs_bcm.shar') The Fortran files contained no copyright information. Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995) A limited memory algorithm for bound constrained optimization. \emph{SIAM J. Scientific Computing}, \bold{16}, 1190--1208. */

The paper @Byrd95 builds on @Lu94limitedmemory. There have been a number
of other workers who have followed-up on this work, but **R** code and
packages seem to have largely stayed with codes derived from these original
papers. Though the date of the paper is 1995, the ideas it embodies were
around for a decade and a half at least, in particular in Nocedal80 and
LiuN89. The definitive Fortran code was published as @Zhu1997LBFGS. This
is available as `toms/778.zip`

on www.netlib.org. A side-by-side comparison of the
main subroutines in the two downloads from Netlib unfortunately shows a lot of
differences. I have not tried to determine if these affect performance or are
simply cosmetic.

More seriously perhaps, there were some deficiencies in the code(s), and in 2011
Nocedal's team published a Fortran code with some corrections (@Morales2011).
Since the **R** code predates this, I prepared package `lbfgsb3`

(@lbfgsb3JN) to wrap
the Fortran code. However, I did not discover any test cases where the
`optim::L-BFGS-B`

and `lbfgsb3`

were different, though I confess to only
running some limited tests. There are, in fact, more in this vignette.

In 2016, I was at a Fields Institute optimization conference in Toronto
for the 70th birthday of Andy Conn. By sheer serendipity, Nocedal did not attend
the conference, but
sat down next to me at the conference dinner. When I asked him about the key changes,
he said that the most important one was to fix the computation of the machine
precision, which was not always correct in the 1995 code. Since **R** gets this
number as `.Machine$double.eps`

, the offending code is irrelevant.

Within @Morales2011, there is also reported an improvement in the subspace
minimization that is applied in cases of bounds constraints. Since few of the
tests I have applied impose such constraints, it is reasonable that I will
not have observed performance differences between the base-R `optim`

code
and my `lbfsgb3`

package. More appropriate tests are welcome, and on my agenda.

Besides the ACM TOMS code, there are two related codes from the Northwestern team on NETLIB: http://netlib.org/opt/lbfgs_um.shar is for unconstrained minimization, while http://netlib.org/opt/lbfgs_bcm.shar handles bounds constrained problems. To these are attached references @LiuN89 and @Byrd1995 respectively, most likely reflecting the effort required to implement the constraints.

The unconstrained code has been converted to **C** under the leadership of
Naoaki Okazaki (see http://www.chokkan.org/software/liblbfgs/, or the fork at https://github.com/MIRTK/LBFGS).
This has been wrapped for **R** as @Coppola2014 as the
`lbfgs`

package. This can be called from `optimx::optimr()`

.

Using Rcpp (see @RCppDERF2011) and the Fortran code in package
`lbfgs3`

, Matthew Fidler developed package `lbfgsb3c`

(@lbfgsb3cMF). As this provides a more standard call and return than
`lbfgsb3`

Fidler and I are unified the two packages and released them
both under the same name `lbfgsb3c`

.

`lbfgsb3c`

There is really only one optimizer function in the package, but it may be called by four (4) names:

`lbfgsb3c()`

uses Rcpp (@RCppDE2013, @RCppDERF2011, @RCppDEJJB2017) to streamline the call to the underlying Fortran. This is the base function used.`lbfgsb3x()`

is an alias of`lbfgsb3c()`

. We were using this name for a while, and have kept the alias to avoid having to edit test scripts.`lbfgsb3`

, which imitates a`.Fortran`

call of the compiled 2011 Fortran code. The object returned by this routine is NOT equivalent to the object returned by base-R`optim()`

or by`optimx::optimr()`

. Instead, it includes a structure`info`

which contains the detailed diagnostic information of the Fortran code. For most users, this is not of interest, and I only recommend use of this function for those needing to examine how the optimization has been carried out.`lbfgsb3f()`

is an alias of`lbfsgb3()`

.

We recommend using the `lbfsgb3c()`

call for most uses.

# candlestick function # J C Nash 2011-2-3 cstick.f<-function(x,alpha=100){ x<-as.vector(x) r2<-crossprod(x) f<-as.double(r2+alpha/r2) return(f) } cstick.g<-function(x,alpha=100){ x<-as.vector(x) r2<-as.numeric(crossprod(x)) g1<-2*x g2 <- (-alpha)*2*x/(r2*r2) g<-as.double(g1+g2) return(g) } library(lbfgsb3c) nn <- 2 x0 <- c(10,10) lo <- c(1, 1) up <- c(10,10) print(x0) ## c2o <- opm(x0, cstick.f, cstick.g, lower=lo, upper=up, method=meths, control=list(trace=0)) ## print(summary(c2o, order=value)) c2l1 <- lbfgsb3c(x0, cstick.f, cstick.g, lower=lo, upper=up) c2l1 ## meths <- c("L-BFGS-B", "lbfgsb3c", "Rvmmin", "Rcgmin", "Rtnmin") ## require(optimx) ## cstick2a <- opm(x0, cstick.f, cstick.g, method=meths, upper=up, lower=lo, control=list(kkt=FALSE)) ## print(summary(cstick2a, par.select=1:2, order=value)) lo <- c(4, 4) ## c2ob <- opm(x0, cstick.f, cstick.g, lower=lo, upper=up, method=meths, control=list(trace=0)) ## print(summary(c2ob, order=value)) c2l2 <- lbfgsb3c(x0, cstick.f, cstick.g, lower=lo, upper=up) c2l2 ## cstick2b <- opm(x0, cstick.f, cstick.g, method=meths, upper=up, lower=lo, control=list(kkt=FALSE)) ## print(summary(cstick2b, par.select=1:2, order=value)) ## nn <- 100 ## x0 <- rep(10, nn) ## up <- rep(10, nn) ## lo <- rep(1e-4, nn) ## cco <- opm(x0, cstick.f, cstick.g, lower=lo, upper=up, method=meths, control=list(trace=0, kkt=FALSE)) ## print(summary(cco, par.select=1:4, order=value))

# require(funconstrain) ## not in CRAN, so explicit inclusion of this function # exrosen <- ex_rosen() # exrosenf <- exrosen$fn exrosenf <- function (par) { n <- length(par) if (n%%2 != 0) { stop("Extended Rosenbrock: n must be even") } fsum <- 0 for (i in 1:(n/2)) { p2 <- 2 * i p1 <- p2 - 1 f_p1 <- 10 * (par[p2] - par[p1]^2) f_p2 <- 1 - par[p1] fsum <- fsum + f_p1 * f_p1 + f_p2 * f_p2 } fsum } # exroseng <- exrosen$gr exroseng <- function (par) { n <- length(par) if (n%%2 != 0) { stop("Extended Rosenbrock: n must be even") } grad <- rep(0, n) for (i in 1:(n/2)) { p2 <- 2 * i p1 <- p2 - 1 xx <- par[p1] * par[p1] yx <- par[p2] - xx f_p1 <- 10 * yx f_p2 <- 1 - par[p1] grad[p1] <- grad[p1] - 400 * par[p1] * yx - 2 * f_p2 grad[p2] <- grad[p2] + 200 * yx } grad } exrosenx0 <- function (n = 20) { if (n%%2 != 0) { stop("Extended Rosenbrock: n must be even") } rep(c(-1.2, 1), n/2) } require(lbfgsb3c) ## require(optimx) ## require(optimx) for (n in seq(2,12, by=2)) { cat("ex_rosen try for n=",n,"\n") x0 <- exrosenx0(n) lo <- rep(-1.5, n) up <- rep(3, n) print(x0) cat("optim L-BFGS-B\n") eo <- optim(x0, exrosenf, exroseng, lower=lo, upper=up, method="L-BFGS-B", control=list(trace=0)) print(eo) cat("lbfgsb3c\n") el <- lbfgsb3c(x0, exrosenf, exroseng, lower=lo, upper=up, control=list(trace=0)) print(el) ## erfg <- opm(x0, exrosenf, exroseng, method=meths, lower=lo, upper=up) ## print(summary(erfg, par.select=1:2, order=value)) }

While you may use the same interface as described in the writing R extensions to interface compiled code with this function, see L-BFGS-B, it is sometimes more convenient to use your own compiled code.

The following example shows how this is done using the file `jrosen.f`

. We have unfortunately
found that compilation is not always portable across systems, so this example is
presented without execution.

subroutine rosen(n, x, fval) double precision x(n), fval, dx integer n, i fval = 0.0D0 do 10 i=1,(n-1) dx = x(i + 1) - x(i) * x(i) fval = fval + 100.0 * dx * dx dx = 1.0 - x(i) fval = fval + dx * dx 10 continue return end

Here is the example script. Note that we must have the file `jrosen.f`

available.
Because the executable files on different systems use different conventions and
structures, we have turned evaluation off here so this vignette can be built
on multiple platforms. However, we wished to provide examples of how compiled code
could be used.

system("R CMD SHLIB jrosen.f") dyn.load("jrosen.so") is.loaded("rosen") x0 <- as.double(c(-1.2,1)) fv <- as.double(-999) n <- as.double(2) testf <- .Fortran("rosen", n=as.integer(n), x=as.double(x0), fval=as.double(fv)) testf rrosen <- function(x) { fval <- 0.0 for (i in 1:(n-1)) { dx <- x[i + 1] - x[i] * x[i] fval <- fval + 100.0 * dx * dx dx <- 1.0 - x[i] fval <- fval + dx * dx } fval } (rrosen(x0)) frosen <- function(x){ nn <- length(x) if (nn > 100) { stop("max number of parameters is 100")} fv <- -999.0 val <- .Fortran("rosen", n=as.integer(nn), x=as.double(x), fval=as.double(fv)) val$fval # NOTE--need ONLY function value returned } # Test the funcion tval <- frosen(x0) str(tval) cat("Run with Nelder-Mead using R function\n") mynm <- optim(x0, rrosen, control=list(trace=0)) print(mynm) cat("\n\n Run with Nelder-Mead using Fortran function") mynmf <- optim(x0, frosen, control=list(trace=0)) print(mynmf) library(lbfgsb3c) library(microbenchmark) cat("try lbfgsb3c, no Gradient \n") cat("R function\n") tlR<-microbenchmark(myopR <- lbfgsb3c(x0, rrosen, gr=NULL, control=list(trace=0))) print(tlR) print(myopR) cat("Fortran function\n") tlF<-microbenchmark(myop <- lbfgsb3c(x0, frosen, gr=NULL, control=list(trace=0))) print(tlF) print(myop)

In this example, Fortran execution was actually SLOWER than plain R on the system where it was run.

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