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by contravariant
3035 days ago
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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: U(x) = min_i f(xi) + k|x - xi| |
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