How to find the smallest position of an element of an ascending list {1,4,10,12,20} such that the element is not less than 11?

FirstPosition[{1, 4, 10, 12, 20}, _?(# >= 11 &)]

(* Out: {4} *)

If you want just the number $4$, then use First@FirstPosition[...].

This question is closely related to the question Best way to insert element into an ordered list at the correct position? , where it was noted that Leonid Shifrin's bsearchMax from this answer is a fast way to solve the problem. Like flinty's answer it uses a binary search but it appears to be faster.

bsearchMax[{1, 4, 10, 12, 20}, 11]

4

For a list of decent size, bsearchMax can be orders of magnitude faster than FirstPosition.

A couple of slower alternatives to FirstPosition:

data = {1, 4, 10, 12, 20};

Position[data, SelectFirst[data, # >= 11 &]][[1, 1]]

Needs["Combinatorica`"]; Ceiling@BinarySearch[data, 11]