How to estimate the upper bound of y in this situation? [closed]

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How to estimate the upper bound of y in this situation?

Given 1. a function y=f(x_1,x_2,x_3,x_4,x_5) with 5 parameters (y=f(...) can be any function). 2. for each x_i there are k_i possible values.

I want to estimate the upper bound of y but I don't want to try all combinations of input, which has size k_1*k_2*k_3*k_4*k_5.

Any idea or direction?

Answer 1

There is no practical upper bound possible without bounding f(...) to a particular type of function. Suppose y=f(x1,x2,x3,...)=C for some constant C. There is then no way to put any bound on f(...) based on inputs x1,x2,... since C has no connection to any of them.

Alternatively, suppose f(x1,x2,...)=x1^(x2^(...)) and compare with f(x1,x2,...)=x1+x2+... Both of these functions are possible to place a bound on, although the bound on the first function may be exceptionally large (in absolute value).