Here I am giving a heuristic scheme based on an engineering point of view. We have a very complicated function P(X,Y,Z,U,...) involving any type of expression such as log, polynomial, square root, inverse, etc. Now we want to minimize this w.r.t some constraints (again very complex) over the variables, but some variables may be independent (indeed we may have $X=(x_1,x_2)$ such that $x_1,x_2$ are dependent but independent of $Y,Z$) . Now the heuristic is as follows;
$$\min_{X,Y,Z} P(X,Y,Z) \approx \min_X P(X,Y,Z)$$ where X is considered to be variable and Y,Z constant (w.r.t X, and assuming that they are independent of X).
We then get something like this;
$$\min P(X,Y,Z) \approx \min_X P(X,Y,Z) = f(Y,Z)$$
We again do the same thing with $f$, thus
$$\min P(X,Y,Z) \approx \min_Y f(Y,Z) = g(Z)$$ $$\min P(X,Y,Z) \approx \min_Z g(Z)$$
I am wondering if this way of optimization is true? is the final approximation good enough (maybe even can be equal with min)?
