QPSolverExample.cpp
Code presents how to solve quadratic programming problemThis example shows how to solve a quadratic programming problem formulted by equation:
![\[ \arg \min_{x y} 4 x^2 + 2 xy + 5 y^2 + 1.5 x - 2 y\]](form_1.png)
![\[\begin{array}{rcr} \text{subject to } & \\ & 2 \leq 2x + y \leq \infty \\ & -\infty \leq -x + 2y \leq 6 \\ & x \in [0, 20] \text{ and } y \in [0, \infty] \end{array}\]](form_2.png)
Above problem can be rewritten:
![\[ \arg \min_{x y} \frac{1}{2} (8 x^2 + 4 xy + 10 y^2) + 1.5 x - 2 y = \arg \min_{x y} \frac{1}{2} \mathbf{x^T} Q \mathbf{x} + C \mathbf{x^T} \]](form_3.png)
![\[\begin{array}{rcr} \text{subject to } & \\ & \textbf{a} \leq A\textbf{x} \leq \textbf{b} \\ & \textbf{c} \leq \textbf{x} \leq \textbf{d} \end{array}\]](form_4.png)
![\[\begin{array}{rcr} \text{where } & \\ Q = & \begin{bmatrix} 8 & 2 \\ 2 & 10 \end{bmatrix} \\ C = & \begin{bmatrix} 1.5 \\ -2 \end{bmatrix} \\ A = & \begin{bmatrix} 2 & 1 \\ -1 & 2 \end{bmatrix} \\ \textbf{a} = & \begin{bmatrix} 2 \\ -\infty \end{bmatrix} \\ \textbf{b} = & \begin{bmatrix} \infty \\ 6 \end{bmatrix} \\ \textbf{c} = & \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ \textbf{d} = & \begin{bmatrix} 20 \\ \infty \end{bmatrix} \end{array}\]](form_5.png)
Matrix form of QP problem allows us to use QuadraticProgrammingSolver to find the minimum.
#include <armadillo>
#include <juliant.hpp>
using namespace julian;
int main() {
arma::mat Q, C, A;
arma::mat a, b, c, d;
Q << 8.0 << 2.0 << arma::endr
<< 2.0 << 10.0 << arma::endr;
C << 1.5 << arma::endr
<< -2.0 << arma::endr;
A << 2.0 << 1.0 << arma::endr
<<-1.0 << 2.0 << arma::endr;
a << 2.0 << arma::endr
<< arma::datum::inf << arma::endr;
b << arma::datum::inf << arma::endr
<< 6.0 << arma::endr;
c << 0.0 << arma::endr
<< 0.0 << arma::endr;
d << 20.0 << arma::endr
<< arma::datum::inf << arma::endr;
/******** setting solver ************/
QuadraticProgrammingSolver qp_solver;
qp_solver.setQuadraticTerm(Q);
qp_solver.setLinearTerm(C);
qp_solver.setNonEqualityConstrainsMatrix(A);
qp_solver.setNonEqualityConstrainsLowerVector(a);
qp_solver.setNonEqualityConstrainsUpperVector(b);
qp_solver.setArgumentLowerConstrains(c);
qp_solver.setArgumentUpperConstrains(d);
/******** results ************/
SHOW(result);
return 0;
}
