Let f(x,y)=0 which is an implicit function of x and y. Let, "y" be such a variable that I can control. I mean, I can give any value for "y" and then solve f(x,y). So "y" is my control varible, and "x" is uncontrlled. Let "x*" denote a solution for a a value of "y*". I have another function g(x*) which depends the solution of f. I want to minimize(local minimum is sufficient) g by changing y. How can I do that?
The main problem is that g is not known. It is known that its value depends on "x*". Just think it like, a computer can calculate its value, but does not show you the closed form of the function. One very kwown property of "g" is that, it has all properties of a differantiable function.
"y" can take any nonnegative real value. So, initially, I think as a starting point to give "0" to "y". Then calculate "x*". And then using this "x*", learn g(x*). But at the second step, what will I do? How much will I increase y? I will be very glad for any help. Thanks lot.