Nonlinear Programming Using CppAD and Ipopt: Example and Test

ipopt_solve_get_started.cpp

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation . Nonlinear Programming Using CppAD and Ipopt: Example and Test
Purpose
This example program demonstrates how to use ipopt_solve to solve the example problem in the Ipopt documentation; i.e., the problem @[@ \begin{array}{lc} {\rm minimize \; } & x_1 * x_4 * (x_1 + x_2 + x_3) + x_3 \\ {\rm subject \; to \; } & x_1 * x_2 * x_3 * x_4 \geq 25 \\ & x_1^2 + x_2^2 + x_3^2 + x_4^2 = 40 \\ & 1 \leq x_1, x_2, x_3, x_4 \leq 5 \end{array} @]@

Configuration Requirement
This example will be compiled and tested provided include_ipopt is on the cmake command line.

# include <cppad/ipopt/solve.hpp>

namespace {
    using CppAD::AD;

    class FG_eval {
    public:
        typedef CPPAD_TESTVECTOR( AD<double> ) ADvector;
        void operator()(ADvector& fg, const ADvector& x)
        {   assert( fg.size() == 3 );
            assert( x.size()  == 4 );

            // Fortran style indexing
            AD<double> x1 = x[0];
            AD<double> x2 = x[1];
            AD<double> x3 = x[2];
            AD<double> x4 = x[3];
            // f(x)
            fg[0] = x1 * x4 * (x1 + x2 + x3) + x3;
            // g_1 (x)
            fg[1] = x1 * x2 * x3 * x4;
            // g_2 (x)
            fg[2] = x1 * x1 + x2 * x2 + x3 * x3 + x4 * x4;
            //
            return;
        }
    };
}

bool get_started(void)
{   bool ok = true;
    size_t i;
    typedef CPPAD_TESTVECTOR( double ) Dvector;

    // number of independent variables (domain dimension for f and g)
    size_t nx = 4;
    // number of constraints (range dimension for g)
    size_t ng = 2;
    // initial value of the independent variables
    Dvector xi(nx);
    xi[0] = 1.0;
    xi[1] = 5.0;
    xi[2] = 5.0;
    xi[3] = 1.0;
    // lower and upper limits for x
    Dvector xl(nx), xu(nx);
    for(i = 0; i < nx; i++)
    {   xl[i] = 1.0;
        xu[i] = 5.0;
    }
    // lower and upper limits for g
    Dvector gl(ng), gu(ng);
    gl[0] = 25.0;     gu[0] = 1.0e19;
    gl[1] = 40.0;     gu[1] = 40.0;

    // object that computes objective and constraints
    FG_eval fg_eval;

    // options
    std::string options;
    // turn off any printing
    options += "Integer print_level  0\n";
    options += "String  sb           yes\n";
    // maximum number of iterations
    options += "Integer max_iter     10\n";
    // approximate accuracy in first order necessary conditions;
    // see Mathematical Programming, Volume 106, Number 1,
    // Pages 25-57, Equation (6)
    options += "Numeric tol          1e-6\n";
    // derivative testing
    options += "String  derivative_test            second-order\n";
    // maximum amount of random pertubation; e.g.,
    // when evaluation finite diff
    options += "Numeric point_perturbation_radius  0.\n";

    // place to return solution
    CppAD::ipopt::solve_result<Dvector> solution;

    // solve the problem
    CppAD::ipopt::solve<Dvector, FG_eval>(
        options, xi, xl, xu, gl, gu, fg_eval, solution
    );
    //
    // Check some of the solution values
    //
    ok &= solution.status == CppAD::ipopt::solve_result<Dvector>::success;
    //
    double check_x[]  = { 1.000000, 4.743000, 3.82115, 1.379408 };
    double check_zl[] = { 1.087871, 0.,       0.,      0.       };
    double check_zu[] = { 0.,       0.,       0.,      0.       };
    double rel_tol    = 1e-6;  // relative tolerance
    double abs_tol    = 1e-6;  // absolute tolerance
    for(i = 0; i < nx; i++)
    {   ok &= CppAD::NearEqual(
            check_x[i],  solution.x[i],   rel_tol, abs_tol
        );
        ok &= CppAD::NearEqual(
            check_zl[i], solution.zl[i], rel_tol, abs_tol
        );
        ok &= CppAD::NearEqual(
            check_zu[i], solution.zu[i], rel_tol, abs_tol
        );
    }

    return ok;
}

Input File: example/ipopt_solve/get_started.cpp