## Binary Quaratic Programming For Online Tracking Of Hundreds Of People In Extremely Crowed Scene

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.

$% $ 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:

$% $ 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.