To illustrate the different APIs we consider the simple QP
\[\begin{aligned} &\underset{x_1, x_2}{\text{minimize}}&& \frac{1}{2} (x_1^2 + x_2^2) + x_1 + x_2\\ &\text{subject to} &&-1\leq x_1\leq 1 \\ & &&-2\leq x_2\leq 2 \\ & &&-3\leq x_1 + 2 x_2\leq 3 \\ & &&-4\leq x_1 - x_2\leq 4, \\ \end{aligned}\]which can be put in the form
\[\begin{aligned} &\underset{x}{\text{minimize}}&& \frac{1}{2} x^T H x + f^T x\\ &\text{subject to} && l\:\: \leq \:x \:\:\leq u, \\ & && b_l \leq A x \leq b_u, \\ \end{aligned}\]with
\[H = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad f = \begin{pmatrix} 1 \\ 1 \end{pmatrix},\quad\] \[u =\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \quad l =-\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \quad A = \begin{pmatrix} 1 & 2 \\ 1 & -1 \end{pmatrix},\quad b_u =\begin{pmatrix} 3 \\ 4 \end{pmatrix},\quad b_l =-\begin{pmatrix} 3 \\ 4 \end{pmatrix}\]