How to convert the following Quadratic optimization problem to a linear one? I have an optimization problem 
$$Min : \ 3x_{11} + 5x_{12} + 4x_{21} + 3 x_{22}  - (10x_{11}x_{22} + 2x_{12}x_{21})$$
subject to the following constraints :
$$ x_{11} + x_{12} = 1$$
$$ x_{21} + x_{22} = 1$$
where $x_{11}. x_{12}, x_{21}, x_{22}$ are Integer Variables taking values either $0$ or $1$.
How do I convert the objective function to a linear one?
 A: The general linearization of a product of binary variables is overkill here.  Instead, first substitute $x_{12} = 1 - x_{11}$ and $x_{22} = 1 - x_{21}$ to obtain objective function
\begin{align}
&3x_{11} + 5x_{12} + 4x_{21} + 3 x_{22}  - 10x_{11}x_{22} - 2x_{12}x_{21}\\
&=3x_{11} + 5(1 - x_{11}) + 4x_{21} + 3 (1 - x_{21})  - 10x_{11}(1 - x_{21}) - 2(1 - x_{11})x_{21}\\
&=3x_{11} + 5 - 5 x_{11} + 4x_{21} + 3 - 3 x_{21}  - 10x_{11} +10x_{11} x_{21} - 2x_{21} +2 x_{11} x_{21}\\
&=8 - 12x_{11} - x_{21} +12x_{11} x_{21}.
\end{align}
Now introduce $y \ge 0$ to represent $x_{11} x_{21}$.  Because we are minimizing, and the objective coefficient is $12>0$, we need only enforce $y\ge x_{11} x_{21}$, which we can do with a single linear constraint $y\ge x_{11} + x_{21} - 1$.  In summary, the problem is to minimize $$8 - 12x_{11} - x_{21} +12y$$ subject to:
\begin{align}
y&\ge x_{11} + x_{21} - 1\\
x_{11} &\in \{0,1\}\\
x_{12} &\in \{0,1\}\\
y &\ge 0
\end{align}
The optimal solution is $(x_{11},x_{21},y)=(1,0,0)$, which yields objective value $-4$ and corresponds to $(x_{11},x_{12},x_{21},x_{22})=(1,0,0,1)$ in the original space.
