R interface to NLopt

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 the NLopt website, where more details are available.

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 "constraints" 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 "constraints" 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 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 here. #nolint A full description of all options is shown by the function 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:

0 (default)	no output
1	show iteration number and value of objective function
2	1 + show value of (in)equalities
3	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.

...

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 of NLopt that was used

Details

NLopt addresses general nonlinear optimization problems of the form:

$$\min f(x)\quad x\in R^n$$

$$\textrm{s.t. }\\ g(x) \leq 0\\ h(x) = 0\\ lb \leq x \leq ub$$

where $f(x)$ 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.

References

Steven G. Johnson, The NLopt nonlinear-optimization package, https://github.com/stevengj/nlopt

Author

Steven G. Johnson and others (C code)
Jelmer Ypma (R interface)

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)
#> 
#> Call:
#> nloptr(x0 = x0, eval_f = eval_f, eval_grad_f = eval_grad_f, opts = opts)
#> 
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 1 ( NLOPT_SUCCESS: Generic success return value. )
#> 
#> Number of Iterations....: 56 
#> Termination conditions:  xtol_rel: 1e-08 
#> Number of inequality constraints:  0 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  7.35727226897802e-23 
#> Optimal value of controls: 1 1
#> 
#> 


## 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)
#> 
#> Call:
#> nloptr(x0 = x0, eval_f = eval_f_list, opts = opts)
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 1 ( NLOPT_SUCCESS: Generic success return value. )
#> 
#> Number of Iterations....: 56 
#> Termination conditions:  xtol_rel: 1e-08 
#> Number of inequality constraints:  0 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  7.35727226897802e-23 
#> Optimal value of controls: 1 1
#> 
#> 



# 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)
#> Warning: No termination criterion specified, using default(relative x-tolerance = 1e-04)
print(res0)
#> 
#> Call:
#> 
#> 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)
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because 
#> xtol_rel or xtol_abs (above) was reached. )
#> 
#> Number of Iterations....: 11 
#> Termination conditions:  relative x-tolerance = 1e-04 (DEFAULT) 
#> Number of inequality constraints:  2 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  0.54433104762009 
#> Optimal value of controls: 0.3333333 0.2962963
#> 
#> 

# 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)
#> Warning: No termination criterion specified, using default(relative x-tolerance = 1e-04)
print(res1)
#> 
#> Call:
#> 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)
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because 
#> xtol_rel or xtol_abs (above) was reached. )
#> 
#> Number of Iterations....: 31 
#> Termination conditions:  relative x-tolerance = 1e-04 (DEFAULT) 
#> Number of inequality constraints:  2 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  0.544242301658176 
#> Optimal value of controls: 0.3333292 0.2961997
#> 
#> 


# 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)
#> Warning: No termination criterion specified, using default(relative x-tolerance = 1e-04)
#> Checking gradients of objective function.
#> Derivative checker results: 0 error(s) detected.
#> 
#>   eval_grad_f[1] = 0.000000e+00 ~ 0.000000e+00   [0.000000e+00]
#>   eval_grad_f[2] = 2.098323e-01 ~ 2.098323e-01   [1.422937e-09]
#> 
#> Checking gradients of inequality constraints.
#> Derivative checker results: 0 error(s) detected.
#> 
#>   eval_jac_g_ineq[1, 1] =  3.654614e+01 ~  3.654614e+01   [1.667794e-08]
#>   eval_jac_g_ineq[2, 1] = -1.642680e-01 ~ -1.642680e-01   [2.103453e-07]
#>   eval_jac_g_ineq[1, 2] = -1.000000e+00 ~ -1.000000e+00   [0.000000e+00]
#>   eval_jac_g_ineq[2, 2] = -1.000000e+00 ~ -1.000000e+00   [0.000000e+00]
#> 
print(res2)
#> 
#> Call:
#> 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)
#> 
#> 
#> Minimization using NLopt version 2.7.1 
#> 
#> NLopt solver status: 4 ( NLOPT_XTOL_REACHED: Optimization stopped because 
#> xtol_rel or xtol_abs (above) was reached. )
#> 
#> Number of Iterations....: 11 
#> Termination conditions:  relative x-tolerance = 1e-04 (DEFAULT) 
#> Number of inequality constraints:  2 
#> Number of equality constraints:    0 
#> Optimal value of objective function:  0.54433104762009 
#> Optimal value of controls: 0.3333333 0.2962963
#> 
#>