lecture. - CASTLE Lab - Princeton University

Hölder and Lipschitz Continuity for Functions
Definition 3 (Hölder condition of degree α)
A function f(x) satisfies the Hölder condition of degree α on a set S ⊂ X (for
some α ≥ 0) if there exists a constant c ≥ 0 such that
|f(x)−f(y)| ≤ cx −y α for all x,y ∈ S. (1)
When α = 1, it corresponds to Lipschitz condition.
Definition 4 (modulus)
The smallest possible constant c > 0 satisfies Hölder condition of function f(x)
and the degree α. Note here we implicitly assume f(x) is of real value. A
similar definition could be easily obtained for vector-valued function where
|f(x)−f(y)| is replaced by f(x)−f(y).
Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 9 / 26