nloptr: R Interface to NLopt

# Copyright (C) 2010-14 Jelmer Ypma. All Rights Reserved.
# SPDX-License-Identifier: LGPL-3.0-or-later
#
# File:   test-example.R
# Author: Jelmer Ypma
# Date:   10 June 2010
#
# Maintenance assumed by Avraham Adler (AA) on 2023-02-10
#
# 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
#
# Optimal solution: (1/3, 8/27)
#
# CHANGELOG:
#   2014-05-03: Changed example to use unit testing framework testthat.
#   2019-12-12: Corrected warnings and using updated testtthat framework (AA)
#   2023-02-07: Remove wrapping tests in "test_that" to reduce duplication. (AA)

library(nloptr)

tol <- sqrt(.Machine$double.eps)

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

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

# gradient of objective function
eval_grad_f0 <- function(x, a, b) c(0, 0.5 / sqrt(x[2]))

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

# 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) {
  list("objective" = sqrt(x[2]), "gradient" = c(0, 0.5 / sqrt(x[2])))
}

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

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

# Define optimal solution.
solution.opt <- c(1 / 3, 8 / 27)

# Test NLopt tutorial example with NLOPT_LD_MMA with gradient information. 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("xtol_rel" = 1e-4, "algorithm" = "NLOPT_LD_MMA"),
  a               = a,
  b               = b
)

expect_equal(res0$solution, solution.opt, tolerance = tol)

# Test NLopt tutorial example with NLOPT_LN_COBYLA with gradient information.

# Solve using NLOPT_LN_COBYLA without gradient information A tighter convergence
# tolerance is used here (1e-6), to ensure that the final solution is equal to
# the optimal solution (within some tolerance).
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("xtol_rel" = 1e-6, "algorithm" = "NLOPT_LN_COBYLA"),
  a           = a,
  b           = b
)

expect_equal(res1$solution, solution.opt, tolerance = tol)

# Test NLopt tutorial example with NLOPT_LN_COBYLA with gradient information
# using combined function.

# 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("xtol_rel" = 1e-4, "algorithm" = "NLOPT_LD_MMA"),
  a           = a,
  b           = b
)

expect_equal(res2$solution, solution.opt, tolerance = tol)