Let us say that I have an 1D optimization problem. For example, I want to find the minimum of the flowing function:
y = f(x) = x^3 - 6x^2 + 4x + 12 x in [-2.1, 6.1]
The global minimum of this function is at
x = -2.1. There is a local minimum at
x =~ 3.5
If I run some local optimization algorithm starting from the point
x = 3 for example. The Algorithm will trapped in the local minimum
x= 3.5 which is expected and OK.
Is it possible after running the local optimization algorithm to extract the boundary of the
x variable that belongs to this local minimum without brute-forcing? in other words, I would like to know, analytically, what is the range of x that if I re-run the same algorithm starting from it (this range), I will got the same result. So, I can avoid running Algorithm again from this range.
P.S. the mentioned function was just an example. The real case is 6D function optimization.