nloptr: R Interface to NLopt

# Copyright (C) 2010 Jelmer Ypma. All Rights Reserved.
# SPDX-License-Identifier: LGPL-3.0-or-later
#
# File:   test-hs071.R
# Author: Jelmer Ypma
# Date:   10 June 2010
#
# Maintenance assumed by Avraham Adler (AA) on 2023-02-10
#
# Example problem, number 71 from the Hock-Schittkowsky test suite.
#
# \min_{x} x1*x4*(x1 + x2 + x3) + x3
# s.t.
#    x1*x2*x3*x4 >= 25
#    x1^2 + x2^2 + x3^2 + x4^2 = 40
#    1 <= x1,x2,x3,x4 <= 5
#
# we re-write the inequality as
#   25 - x1*x2*x3*x4 <= 0
#
# and the equality as
#   x1^2 + x2^2 + x3^2 + x4^2 - 40 = 0
#
# x0 = (1,5,5,1)
#
# Optimal solution = (1.00000000, 4.74299963, 3.82114998, 1.37940829)
#
# CHANGELOG:
#   2014-05-05: 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)

# f(x) = x1*x4*(x1 + x2 + x3) + x3
eval_f <- function(x) {
  list("objective" = x[1] * x[4] * (x[1] + x[2] + x[3]) + x[3],
       "gradient" = c(x[1] * x[4] + x[4] * (x[1] + x[2] + x[3]),
                      x[1] * x[4],
                      x[1] * x[4] + 1,
                      x[1] * (x[1] + x[2] + x[3])))
}

# Inequality constraints.
eval_g_ineq <- function(x) {
  constr <- c(25 - x[1] * x[2] * x[3] * x[4])
  grad <- c(-x[2] * x[3] * x[4],
            -x[1] * x[3] * x[4],
            -x[1] * x[2] * x[4],
            -x[1] * x[2] * x[3])
  list("constraints" = constr, "jacobian" = grad)
}

# Equality constraints.
eval_g_eq <- function(x) {
  constr <- c(x[1] ^ 2 + x[2] ^ 2 + x[3] ^ 2 + x[4] ^ 2 - 40)
  grad <- c(2.0 * x[1],
            2.0 * x[2],
            2.0 * x[3],
            2.0 * x[4])
  list("constraints" = constr, "jacobian" = grad)
}

# Initial values.
x0 <- c(1, 5, 5, 1)

# Lower and upper bounds of control.
lb <- c(1, 1, 1, 1)
ub <- c(5, 5, 5, 5)

# Optimal solution.
solution.opt <- c(1.00000000, 4.74299963, 3.82114998, 1.37940829)

# Set optimization options.
local_opts <- list("algorithm" = "NLOPT_LD_MMA", "xtol_rel"  = 1.0e-7)
opts <- list("algorithm"   = "NLOPT_LD_AUGLAG",
             "xtol_rel"    = 1.0e-7,
             "maxeval"     = 1000,
             "local_opts"  = local_opts,
             "print_level" = 0)

# Do optimization.
res <- nloptr(x0          = x0,
              eval_f      = eval_f,
              lb          = lb,
              ub          = ub,
              eval_g_ineq = eval_g_ineq,
              eval_g_eq   = eval_g_eq,
              opts        = opts)

# Run some checks on the optimal solution.
expect_equal(res$solution, solution.opt, tolerance = 1e-5)
expect_true(all(res$solution >= lb))
expect_true(all(res$solution <= ub))

# Check whether constraints are violated (up to specified tolerance).

expect_true(
  eval_g_ineq(res$solution)$constr <= res$options$tol_constraints_ineq
)

expect_equal(eval_g_eq(res$solution)$constr, 0,
             tolerance = res$options$tol_constraints_eq)