xref: /dports/math/R-cran-nloptr/nloptr/man/nloptr.Rd

% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/nloptr.R
\name{nloptr}
\alias{nloptr}
\title{R interface to NLopt}
\usage{
nloptr(x0, eval_f, eval_grad_f = NULL, lb = NULL, ub = NULL,
  eval_g_ineq = NULL, eval_jac_g_ineq = NULL, eval_g_eq = NULL,
  eval_jac_g_eq = NULL, opts = list(), ...)
}
\arguments{
\item{x0}{vector with starting values for the optimization.}

\item{eval_f}{function that returns the value of the objective function. It
can also return gradient information at the same time in a list with
elements "objective" and "gradient" (see below for an example).}

\item{eval_grad_f}{function that returns the value of the gradient of the
objective function. Not all of the algorithms require a gradient.}

\item{lb}{vector with lower bounds of the controls (use -Inf for controls
without lower bound), by default there are no lower bounds for any of the
controls.}

\item{ub}{vector with upper bounds of the controls (use Inf for controls
without upper bound), by default there are no upper bounds for any of the
controls.}

\item{eval_g_ineq}{function to evaluate (non-)linear inequality constraints
that should hold in the solution.  It can also return gradient information
at the same time in a list with elements "constraints" and "jacobian" (see
below for an example).}

\item{eval_jac_g_ineq}{function to evaluate the jacobian of the (non-)linear
inequality constraints that should hold in the solution.}

\item{eval_g_eq}{function to evaluate (non-)linear equality constraints that
should hold in the solution.  It can also return gradient information at the
same time in a list with elements "constraints" and "jacobian" (see below for
an example).}

\item{eval_jac_g_eq}{function to evaluate the jacobian of the (non-)linear
equality constraints that should hold in the solution.}

\item{opts}{list with options. The option "algorithm" is required. Check the
\href{http://ab-initio.mit.edu/wiki/index.php/NLopt_Algorithms}{NLopt website}
for a full list of available algorithms. Other options control the
termination conditions (minf_max, ftol_rel, ftol_abs, xtol_rel, xtol_abs,
maxeval, maxtime). Default is xtol_rel = 1e-4. More information
\href{http://ab-initio.mit.edu/wiki/index.php/NLopt_Introduction\#Termination_conditions}{here}.
A full description of all options is shown by the function
\code{nloptr.print.options()}.

Some algorithms with equality constraints require the option local_opts,
which contains a list with an algorithm and a termination condition for the
local algorithm. See ?`nloptr-package` for an example.

The option print_level controls how much output is shown during the
optimization process. Possible values: \tabular{ll}{ 0 (default) \tab no
output \cr 1 \tab show iteration number and value of objective function \cr
2 \tab 1 + show value of (in)equalities \cr 3 \tab 2 + show value of
controls }

The option check_derivatives (default = FALSE) can be used to run to compare
the analytic gradients with finite difference approximations.  The option
check_derivatives_print ('all' (default), 'errors', 'none') controls the
output of the derivative checker, if it is run, showing all comparisons,
only those that resulted in an error, or none.  The option
check_derivatives_tol (default = 1e-04), determines when a difference
between an analytic gradient and its finite difference approximation is
flagged as an error.}

\item{...}{arguments that will be passed to the user-defined objective and
constraints functions.}
}
\value{
The return value contains a list with the inputs, and additional
elements
  \item{call}{the call that was made to solve}
  \item{status}{integer value with the status of the optimization (0 is
    success)}
\item{message}{more informative message with the status of the optimization}
\item{iterations}{number of iterations that were executed}
\item{objective}{value if the objective function in the solution}
\item{solution}{optimal value of the controls}
\item{version}{version of NLopt that was used}
}
\description{
nloptr is an R interface to NLopt, a free/open-source library for nonlinear
optimization started by Steven G. Johnson, providing a common interface for
a number of different free optimization routines available online as well as
original implementations of various other algorithms. The NLopt library is
available under the GNU Lesser General Public License (LGPL), and the
copyrights are owned by a variety of authors. Most of the information here
has been taken from \href{http://ab-initio.mit.edu/nlopt}{the NLopt website},
where more details are available.
}
\details{
NLopt addresses general nonlinear optimization problems of the form:

min f(x) x in R^n

s.t.  g(x) <= 0 h(x) = 0 lb <= x <= ub

where f is the objective function to be minimized and x represents the n
optimization parameters. This problem may optionally be subject to the bound
constraints (also called box constraints), lb and ub. For partially or
totally unconstrained problems the bounds can take -Inf or Inf. One may also
optionally have m nonlinear inequality constraints (sometimes called a
nonlinear programming problem), which can be specified in g(x), and equality
constraints that can be specified in h(x). Note that not all of the
algorithms in NLopt can handle constraints.
}
\note{
See ?`nloptr-package` for an extended example.
}
\examples{

library('nloptr')

## Rosenbrock Banana function and gradient in separate functions
eval_f <- function(x) {
    return( 100 * (x[2] - x[1] * x[1])^2 + (1 - x[1])^2 )
}

eval_grad_f <- function(x) {
    return( c( -400 * x[1] * (x[2] - x[1] * x[1]) - 2 * (1 - x[1]),
                200 * (x[2] - x[1] * x[1])) )
}


# initial values
x0 <- c( -1.2, 1 )

opts <- list("algorithm"="NLOPT_LD_LBFGS",
             "xtol_rel"=1.0e-8)

# solve Rosenbrock Banana function
res <- nloptr( x0=x0,
               eval_f=eval_f,
               eval_grad_f=eval_grad_f,
               opts=opts)
print( res )


## Rosenbrock Banana function and gradient in one function
# this can be used to economize on calculations
eval_f_list <- function(x) {
    return( list( "objective" = 100 * (x[2] - x[1] * x[1])^2 + (1 - x[1])^2,
                  "gradient"  = c( -400 * x[1] * (x[2] - x[1] * x[1]) - 2 * (1 - x[1]),
                                    200 * (x[2] - x[1] * x[1])) ) )
}

# solve Rosenbrock Banana function using an objective function that
# returns a list with the objective value and its gradient
res <- nloptr( x0=x0,
               eval_f=eval_f_list,
               opts=opts)
print( res )



# Example showing how to solve the problem from the NLopt tutorial.
#
# min sqrt( x2 )
# s.t. x2 >= 0
#      x2 >= ( a1*x1 + b1 )^3
#      x2 >= ( a2*x1 + b2 )^3
# where
# a1 = 2, b1 = 0, a2 = -1, b2 = 1
#
# re-formulate constraints to be of form g(x) <= 0
#      ( a1*x1 + b1 )^3 - x2 <= 0
#      ( a2*x1 + b2 )^3 - x2 <= 0

library('nloptr')


# objective function
eval_f0 <- function( x, a, b ){
    return( sqrt(x[2]) )
}

# constraint function
eval_g0 <- function( x, a, b ) {
    return( (a*x[1] + b)^3 - x[2] )
}

# gradient of objective function
eval_grad_f0 <- function( x, a, b ){
    return( c( 0, .5/sqrt(x[2]) ) )
}

# jacobian of constraint
eval_jac_g0 <- function( x, a, b ) {
    return( rbind( c( 3*a[1]*(a[1]*x[1] + b[1])^2, -1.0 ),
                   c( 3*a[2]*(a[2]*x[1] + b[2])^2, -1.0 ) ) )
}


# functions with gradients in objective and constraint function
# this can be useful if the same calculations are needed for
# the function value and the gradient
eval_f1 <- function( x, a, b ){
    return( list("objective"=sqrt(x[2]),
                 "gradient"=c(0,.5/sqrt(x[2])) ) )
}

eval_g1 <- function( x, a, b ) {
    return( list( "constraints"=(a*x[1] + b)^3 - x[2],
                  "jacobian"=rbind( c( 3*a[1]*(a[1]*x[1] + b[1])^2, -1.0 ),
                                    c( 3*a[2]*(a[2]*x[1] + b[2])^2, -1.0 ) ) ) )
}


# define parameters
a <- c(2,-1)
b <- c(0, 1)

# Solve using NLOPT_LD_MMA with gradient information supplied in separate function
res0 <- nloptr( x0=c(1.234,5.678),
                eval_f=eval_f0,
                eval_grad_f=eval_grad_f0,
                lb = c(-Inf,0),
                ub = c(Inf,Inf),
                eval_g_ineq = eval_g0,
                eval_jac_g_ineq = eval_jac_g0,
                opts = list("algorithm"="NLOPT_LD_MMA"),
                a = a,
                b = b )
print( res0 )

# Solve using NLOPT_LN_COBYLA without gradient information
res1 <- nloptr( x0=c(1.234,5.678),
                eval_f=eval_f0,
                lb = c(-Inf,0),
                ub = c(Inf,Inf),
                eval_g_ineq = eval_g0,
                opts = list("algorithm"="NLOPT_LN_COBYLA"),
                a = a,
                b = b )
print( res1 )


# Solve using NLOPT_LD_MMA with gradient information in objective function
res2 <- nloptr( x0=c(1.234,5.678),
                eval_f=eval_f1,
                lb = c(-Inf,0),
                ub = c(Inf,Inf),
                eval_g_ineq = eval_g1,
                opts = list("algorithm"="NLOPT_LD_MMA", "check_derivatives"=TRUE),
                a = a,
                b = b )
print( res2 )

}
\references{
Steven G. Johnson, The NLopt nonlinear-optimization package,
\url{http://ab-initio.mit.edu/nlopt}
}
\seealso{
\code{\link[nloptr:nloptr.print.options]{nloptr.print.options}}
  \code{\link[nloptr:check.derivatives]{check.derivatives}}
  \code{\link{optim}}
  \code{\link{nlm}}
  \code{\link{nlminb}}
  \code{Rsolnp::Rsolnp}
  \code{Rsolnp::solnp}
}
\author{
Steven G. Johnson and others (C code) \cr Jelmer Ypma (R interface)
}
\keyword{interface}
\keyword{optimize}