I had the idea to use an additional term in an numerical optimization problem for a box constraint. It is somewhat a mix between penalty and barrier function and I am wondering what the drawbacks are and would be really happy for alll opinions on it!
For \begin{align} \min & \; f(x) \\ s.t. & \; a<x<b\end{align}
I had in mind to use: \begin{align} \\min & \; f(x)+(\frac{2}{b-a}(x-\frac{b+a}{2}))^{2n} \\ & \; x \in \mathbb{R} \end{align} To me it seems that the solution should approach the solution of the bounded problem if $n$ approaches infinity. (Okay, at $x=a$ and $x=b$ there is a difference)
The advantages I see, compared to classic barrier functions like $-log(x-a)$, is that it is defined on $\mathbb{R}$ and compared to other penalty functions e.g. $(min(0,x-a))^2$, that it is smooth.
Am I missing something big and do you have any ideas why this is not used (or was I too stupid to find it)?
This penalty function is not self-concordant for $n>1$, which means that Newton's method will struggle in practice, see wikipedia. To put it another way, there are plenty of possible barrier/penalty functions, which have all kinds of properties (e.g. exact, or smooth, or something else), but not all of them are useful for solving problems numerically.