nloptr (version 0.8.4)

nloptr: R interface to NLopt

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 http://ab-initio.mit.edu/nlopt{the NLopt website}, where more details are available.

NLopt addresses general nonlinear optimization problems of the form:

min f(x) x in R^n

s.t. g(x)

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

x0

vector with starting values for the optimization.

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).

eval_grad_f

function that returns the value of the gradient of the objective function. Not all of the algorithms require a gradient.

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.

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.

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 "objective" and "jacobian" (see below for an example).

eval_jac_g_ineq

function to evaluate the jacobian of the (non-)linear inequality constraints that should hold in the solution.

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 "objective" and "jacobian" (see below for an example).

eval_jac_g_eq

function to evaluate the jacobian of the (non-)linear equality constraints that should hold in the solution.

opts

list with options. The option "algorithm" is required. Check the 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_

...

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
• call
  the call that was made to solve
• status
  integer value with the status of the optimization (0 is success)
• message
  more informative message with the status of the optimization
• iterations
  number of iterations that were executed
• objective
  value if the objective function in the solution
• solution
  optimal value of the controls
• version
  version op NLopt that was used

References

Steven G. Johnson, The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt

See Also

nloptr.print.options check.derivatives optim nlm nlminb Rsolnp ssolnp

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 )