paper is availabe at here

introduction

Before that, we should understand the Quaratic Programming (QP). The QP has a quaratic objective function and linear constraints. Its solver in matlab (a function called quadprog) uses the formulation shown below.

$\begin{matrix} \mathop{min}_\limits x \frac{1}{2} x^T H x + f^T x & s.t.\\ LB \leq x \leq UB & \\ A_{eq} x = b_{eq} & \\ Ax \leq b \end{matrix}$ where $f=\left[\begin{matrix} f_1 \\ f_2 \end{matrix}\right], A_{eq} \in \mathbb{R}^{n_{eq}\times n}, b_{eq}\in \mathbb{R}^{n_{eq}\times 1},b\in \mathbb{R}^{n_{ieq}\times 1}, A \in \mathbb{R}^{n_{ieq}\times n}$.

$n$ is the number of decision variable, $n_{eq}$ is the number of equality constraints, $n_{ieq}$ is the nubmer of inequality constrains.

Lots of function :(. It’s simple to understand. We just remember two keys: 1) the objective function is quaratic, 2) the constraints of objective function is linear.

We introduce an example of problem.

A manufactureer has done empirical modeling of their production system and has determined that their production cost is determined by the equation:

$\begin{matrix} \mathop{cost} = 0.4x^2 - 5x_1 + x_2^2 - 6x_2 + 50 & s.t. \\ 10 \le x_1 < 1000 & 10 \le x_2 < 1000 & \end{matrix}$ where $x_1$ and $x_2$ are the decision variable, in this case, the rate at which they should run Unit1 and Unit 2 respectively. Find the optimal combination of $x_1$ and $x_2$ that minimizes the cost.