IPOPT 求解优化问题示例

求解模型

状态变量 $ x = [x_1,x_2,x_3,x_4]^\mathrm{T}$
目标函数 $ f(x) = x_1 * x_4 * (x_1 + x_2+ x_3) + x_3$
约束条件：$ g_1(x) = x_1 * x_2 * x_3 * x_4 \ge 25 $; $ g_2(x) = x^2_1 + x^2_2 + x^2_3 + x^2_4 = 40 $

求解代码

example.cpp

#include 
#include 
//#include 

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] = 1.0;
    xi[2] = 1.0;
    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     100\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
        );
        std::cout << check_x[i] << ", " << solution.x[i] << std::endl;
        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;
}

int main(int argc, const char** argv) {
    //MPI_Init();
    get_started();
    return 0;
}

CMakeLists.txt

project(ipopt_example)
cmake_minimum_required(VERSION 2.8)

message(status "IPOPT_CFLAGS: " ${IPOPT_CFLAGS})
message(status "IPOPT_CFLAGS_OTHER: " ${IPOPT_CFLAGS_OTHER})
set(CMAKE_CXX_FLAGS "-DHAVE_CSTDDEF -DHAVE_MPI_INITIALIZED")

include_directories(/usr/local/include)
link_directories(/usr/local/lib)

add_executable(solver example.cpp)

target_link_libraries(solver ipopt)

编译求解结果：

$ mkdir build
$ cd build
$ cmake ..
$ make -j8
$ ./solver
1, 1
4.743, 4.743
3.82115, 3.82115
1.37941, 1.37941

参考： https://github.com/qianqian121/ipopt_example