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Second-Order Expansions First- and Second-Order Expansions A directionally differentiable function g(x) is second order directionally differentiable at x in the direction h if the (parabolic) second order directional derivatives g ′′ (x,h,w) = lim t→0 + g(x +th+ t2 2w)−[g(x)+tg′ (x,h)] t exist for all (parabolic) directions w. In that case we can expand g(x +th+ t2 2 w) for t > 0 as g(x +th+ t2 2 w) = g(x)+tg′ (x,h)+ t2 2 g′′ (x,h,w)+o(t 2 ). Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 18 / 26
1 Introduction and Notation Perturbation Theorems 2 Hölder and Lipschitz Continuity for Functions 3 Growth Conditions 4 First- and Second-Order Expansions 5 Perturbation Theorems Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 19 / 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 and 8: Hölder and Lipschitz Continuity fo
Page 9 and 10: 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: Directionally Differentiable First-
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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