Quadratic Programming
Mathwrist Code Example Series

(October 5, 2023)

This document gives some code examples on how to use the quadratic programming (QP) features in Mathwrist’s C++ Numerical Programming Library (NPL).

Indefinite QP

This example is taken from chapter 16.5 of [1] to solve the following indefinite QP problem.

\arg\min_{x_{1},x_{2}}[x_{1},x_{2}]\left[\begin{array}[]{cc}1&0\cr 0&-1\end{% array}\right]\left[\begin{array}[]{c}x_{1}\cr x_{2}\end{array}\right]\mbox{ s.% t. }

1 // 2 general linear constraints, 2 unknowns

2 // x1 + 2 x2 >= 2

3 // -5 x1 + 4 x2 <= 10

4 LinearProg::Constraints constr(2, 2);

5 constr.A(0, 0) = 1;

6 constr.A(0, 1) = 2;

7 constr.A(1, 0) = -5;

8 constr.A(1, 1) = 4;

9 constr.general_bounds[0] = Bound::lower(2);

10 constr.general_bounds[1] = Bound::upper(10);

12 // simple bounds x1 <= 3, x2 >= 0

13 constr.simple_bounds[0] = Bound::upper(3);

14 constr.simple_bounds[1] = Bound::lower(0);

16 // Indefinite Hessian

17 Matrix G(2,2);

18 G(0, 0) = 1;

19 G(1, 1) = -1;

21 Matrix d;

22 QP_ActiveSet qp(G, d, constr);

24 qp.set_crash_start_radius(0.1);

26 Matrix x(2, 1);

27 x(0, 0) = 2;

28 x(1, 0) = 0;

30 qp(x);

32 // Compare with known solution.

33 assert(std::abs(x(0, 0) - 3) < 1.e-10);

34 assert(std::abs(x(1, 0) - 25. / 4) < 1.e-10);