For the following minimization problem:
\begin{array}{*{20}{c}} {\min }&{f\left( x \right)}\\ {}&{s\left( x \right) \le 0}\\ {}&{r\left( x \right) = 0} \end{array}
A common way to turn this into unconstraint optimization problem is to penalize it using the penalty $p$ as:
$\begin{array}{*{20}{c}} {\min }&{f\left( x \right) + p{{\left[ {r\left( x \right)} \right]}^2}} \end{array} + p{\left[ {\max \left( {s\left( x \right),0} \right)} \right]^2}$
My question is that:
If I have $m$ constraint ${s_1}\left( x \right),{s_2}\left( x \right),...,{s_m}\left( x \right)$ and $n$ constraint ${r_1}\left( x \right),{r_2}\left( x \right),...,{r_n}\left( x \right)$, is it possible to use just one penalization constant $p$ for penalizing all of them ?
Thank you for your enthusiasm !
What about this ?$$\min f(x) + p\left( \sum_{i=1}^nr_i^2(x) \right)+p\left( \sum_{i=1}^m \left(\max \{s_i (x),0\}\right)^2 \right)$$