C++中求解二次规划问题（OSQP）的一个使用实例

参考

库的安装（eigen/OSQP/OSQP-eigen）及其他实例可以参考这个链接
https://blog.csdn.net/nudt_zrs/article/details/124258999?spm=1001.2014.3001.5501

实例

以下面的一个优化问题求解为例
$$\begin{array}{c}\begin{array}{rl}& min(x_1-1{)}^2+(x_2-1{)}^2\\ & s.t.0\le x_1\le 1.5\\ & 0\le x_2\le 1.5\end{array}\end{array}$$

首先对目标方程进行简化，写成矩阵的形式：
$$\begin{array}{c}\begin{array}{rl}obj& =(x_1^2-2x_1+1)+(x_2^2-2x_2+1)\\ & =x_1^2+x_2^2-2x_1-2x_2+2\\ & =\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]-2\left[\begin{array}{cc}1& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+2\end{array}\end{array}$$

由于二次规划问题求解器默认的求解形式为$obj=\frac{1}{2}{x}^{T}Hx+f^{T}x$，其中常数项对二次型求解没有影响，可以直接忽略。因此将上式写为
$$\begin{array}{c}\begin{array}{rl}obj& =\frac{1}{2}{x}^T\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]x+{\left[\begin{array}{c}-2\\ -2\end{array}\right]}^{T}x\\ & =\frac{1}{2}{x}^{T}Hx+f^{T}x\end{array}\end{array}$$

而其不等式约束则可改写为如下形式
$$\begin{array}{c}\begin{array}{rl}\left[\begin{array}{c}0\\ 0\end{array}\right]\le & \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\le \left[\begin{array}{c}1.5\\ 1.5\end{array}\right]\\ \Rightarrow & lb\le Ax\le ub\end{array}\end{array}$$

// osqp-eigen
#include "OsqpEigen/OsqpEigen.h"
 
// eigen
#include <Eigen/Dense>
#include <iostream>
 
int main()
{
    // allocate QP problem matrices and vectores
    Eigen::SparseMatrix<double> hessian(2, 2);      //P: n*n正定矩阵,必须为稀疏矩阵SparseMatrix
    Eigen::VectorXd gradient(2);                    //Q: n*1向量
    Eigen::SparseMatrix<double> linearMatrix(2, 2); //A: m*n矩阵,必须为稀疏矩阵SparseMatrix
    Eigen::VectorXd lowerBound(2);                  //L: m*1下限向量
    Eigen::VectorXd upperBound(2);                  //U: m*1上限向量
 
    hessian.insert(0, 0) = 2.0; //注意稀疏矩阵的初始化方式,无法使用<<初始化
    hessian.insert(1, 1) = 2.0;
    // std::cout << "hessian:" << std::endl
    //           << hessian << std::endl;
    gradient << -2, -2;
    linearMatrix.insert(0, 0) = 1.0; //注意稀疏矩阵的初始化方式,无法使用<<初始化
    linearMatrix.insert(1, 1) = 1.0;
    // std::cout << "linearMatrix:" << std::endl
    //           << linearMatrix << std::endl;
    lowerBound << 1, 1;
    upperBound << 1.5, 1.5;
 
    // instantiate the solver
    OsqpEigen::Solver solver;
 
    // settings
    solver.settings()->setVerbosity(false);
    solver.settings()->setWarmStart(true);
 
    // set the initial data of the QP solver
    solver.data()->setNumberOfVariables(2);   //变量数n
    solver.data()->setNumberOfConstraints(2); //约束数m
    if (!solver.data()->setHessianMatrix(hessian))
        return 1;
    if (!solver.data()->setGradient(gradient))
        return 1;
    if (!solver.data()->setLinearConstraintsMatrix(linearMatrix))
        return 1;
    if (!solver.data()->setLowerBound(lowerBound))
        return 1;
    if (!solver.data()->setUpperBound(upperBound))
        return 1;
 
    // instantiate the solver
    if (!solver.initSolver())
        return 1;
 
    Eigen::VectorXd QPSolution;
 
    // solve the QP problem
    if (!solver.solve())
    {
        return 1;
    }
 
    QPSolution = solver.getSolution();
    std::cout << "QPSolution" << std::endl
              << QPSolution << std::endl; //输出为m*1的向量
    return 0;
}

Cmakelist文件如下

cmake_minimum_required(VERSION 3.1)
 
set(CMAKE_CXX_STANDARD 11)
set(CMAKE_CXX_STANDARD_REQUIRED ON)
 
project(OSQP_test LANGUAGES CXX)
 
find_package(OsqpEigen)
find_package(Eigen3)
 
include_directories(SYSTEM ${EIGEN3_INCLUDE_DIR})
 
#MPCExample
add_executable(OSQP_test main.cpp)
 
target_link_libraries(OSQP_test OsqpEigen::OsqpEigen)