The quadratic penalty function is commonly written as follows
$$\color{red}{\frac{1}{2}}\mu \sum_{i=1}^n c_i^2 = \color{red}{\frac{1}{2}}\mu c^\top c$$
One could just as easily drop this factor and use $\mu' = \frac 1 2\mu$, so I'm assuming there is a reason why it's included, but what? Is it just a matter of style to halve quadratic terms?