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Hölder and Lipschitz Continuity for Functions Definition 1 (distance from a point to a set) The distance from a point x ∈ X to a set S ⊂ X is written dist(x,S) = infs∈S x −s. Definition 2 (neighborhood of a set) A neighborhood of a set S ⊂ X, written NS, is a set that is a neighborhood of each point of S. Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 8 / 26
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
Page 1 and 2: Stochastic Optimization Seminar Lec
Page 3 and 4: Motivation Introduction and Notatio
Page 5 and 6: Questions Introduction and Notation
Page 7: Hölder and Lipschitz Continuity fo
Page 11 and 12: 1 Introduction and Notation Growth
Page 13 and 14: Growth Conditions Definition 6(γ-o
Page 15 and 16: Growth Conditions Example(strongly
Page 17 and 18: Directionally Differentiable First-
Page 19 and 20: 1 Introduction and Notation Perturb
Page 21 and 22: Perturbed Optimization Problem Pert
Page 23 and 24: Proof of Theorem 1 Perturbation The
Page 25 and 26: More Assumptions... Perturbation Th
Page 27: Perturbation Theorems J.F. Bonnans

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