A Lipschitz function is one that doesn't change to quickly, such that the difference between two values |f(x) - f(y)| is at most as large as the distance between x and y times a constant k i.e. |f(x) - f(y)| <= k|x-y|. In particular this implies that:
f(x) - f(y) <= k|x-y|
hence
f(x) <= f(y) + k|x-y|.
Therefore the function:
u(x) = f(y) + k|x-y|
is an upper bound for the function f(x). Repeating this for multiple points x1, x2, x3,... you can construct a stricter upper bound by taking the minimum:
f(x) - f(y) <= k|x-y|
hence
f(x) <= f(y) + k|x-y|.
Therefore the function:
u(x) = f(y) + k|x-y|
is an upper bound for the function f(x). Repeating this for multiple points x1, x2, x3,... you can construct a stricter upper bound by taking the minimum:
U(x) = min_i f(xi) + k|x - xi|