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IPOPT的使用
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IPOPT是一种常用的非线性优化求解器，使用内点法进行求解。对于复杂问题，需要借助自动微分工具，帮助求解梯度、雅各比矩阵、Hessian矩阵，如ADOL-C，CppAD

安装

参考非线性优化求解器IPOPT 和 ubuntu 环境下 IPOPT 安装与使用

我安装完后，部分文件和官方说明的不同，目录coin在我的电脑上是coin-or，sudo vim /usr/include/cppad/ipopt/solve_callback.hpp，修改如下

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# include <coin-or/IpIpoptApplication.hpp>
# include <coin-or/IpTNLP.hpp>

配置

CMake配置

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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)

举例

状态变量 $ x = [x_1,x_2,x_3,x_4]^\mathrm{T}$

目标函数

约束条件:

起始点

优化结果：

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#include <cppad/ipopt/solve.hpp>
#include <iostream>
//#include <mpi.h>
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;
    // x[i] 设置第i个变量的初始迭代值
    Dvector xi(nx);
    xi[0] = 1.0;
    xi[1] = 1.0;
    xi[2] = 1.0;
    xi[3] = 1.0;

    // x_l[i]设置xi的下界值, x_u[i]设置xi的上界值
    Dvector xl(nx), xu(nx);
    for(i = 0; i < nx; i++)
    {     
        xl[i] = 1.0;
        xu[i] = 5.0;
    }
    // g_l[i]设置约束i的下界值, g_u[i]设置约束i的上界值
    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 rel_tol = 1e-6;  // relative tolerance
    double abs_tol = 1e-6;  // absolute tolerance
    std::cout << "solution x: "<< std::endl;
    for(i = 0; i < nx; i++)
    {     
        std::cout << solution.x[i] << std::endl;
    }
    return ok;
}

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