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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?

dykw
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1 Answers1

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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).

abiessu
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  • yes, I agree with you abiessu. What happen if we can make some assumptions on f. (other than assuming f is strictly increasing or decreasing.) – dykw Feb 11 '14 at 22:06
  • What assumptions are we allowed to make? Do we assume that the function is a polynomial in `n` variables? Do we assume that the function is a linear combination of `n` variables? If the latter is assumed, what assumptions can we make about the linear factors? Such factors will cause the same bounding issue as the problem with the constant `C` above. – abiessu Feb 11 '14 at 22:10
  • What happen if we assume the function is a polynomial in n variables? – dykw Feb 11 '14 at 22:12
  • Then we must account for the exponents and coefficients which have the same issue as the constant `C` in my first example. – abiessu Feb 11 '14 at 22:13
  • The only way to limit this function in a boundable way is to claim that only the inputs to the function and mathematical operators are present in the function definition. Even then, if you have division involved it is possible to get an unbounded function like `f(x1,x2,x3)=x1/(x2-x3)`. – abiessu Feb 11 '14 at 22:15
  • I see. So, there is not general method to compute the exact upper bound if we don't know f. Are there any formal statistical way to estimate the upper bound? – dykw Feb 11 '14 at 22:17
  • I do not know of any, but you could cross-post to math.stackexchange.com and see what happens. – abiessu Feb 11 '14 at 22:22