I was wondering why don't we select a very large number to a penalty function $c$ of the augmented function $f(x) + cP(x)$ instead of doing algorithms to increase c slowly? I know that as $c$ is large enough, the sharpness of the curve decreases.
More information from my reply on a comment after editing:
It's a general question for penalty method in solving optimization problem. You know, penalty method uses an augmented function θ(c,x)=f(x)+cP(x) to optimize the function f(x) under some constraints g(x). P(x) is the penalty function. so if we draw cP(x) with respect to x, it appears as we increase c, more accuracy and less sharpness of this curve. In numerical analysis, as I see, they tend to use algorithms using SUMT technique by increasing c as steps. The question is why? Why don't we set c for example 10000 directly without needing to iteration