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Remark Growth Conditions Ideally, we are interested in the largest c > 0 that can satisfies the inequality for a fixed γ. Compared to Hölder-type conditions, here only x needs to vary, and the absolute value is not taken on the left-hand side. Hence in practice, S stands for a set of local minimizers. Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 14 / 26
Growth Conditions Example(strongly convex function) Let f(x) be a twice differentiable, strongly convex function over a convex set C comprising its minimum: there exists a constant m > 0 such that ∇ 2 f(x) mI for all x ∈ C. Since for any x,y ∈ C, there exists a point z = tx +(1−t)y with t ∈ [0,1] such that f(y)−f(x) = ∇f(x) ⊤ (y −x)+ 1 2 (y −x)⊤ ∇ 2 f(z)(y −x) , we have, by the strong convexity assumption, f(y)−f(x) ≥ ∇f(x) ⊤ (y −x)+ m 2 ||y −x||2 . Consider S = argmin Cf, s ∈ S, and write fS = minC f. We have f(y)−fS ≥ ∇f(s) ⊤ (y −s)+ m 2 ||y −s||2 ≥ m 2 ||y −s||2 , using the fact that ∇f(s) ⊤ (y −s) ≥ 0 for all y ∈ C and s ∈ S, by optimality of S. This proves that S is in fact reduced to {s}, and that the second-order growth condition holds with modulus c = m/2: f(x)−fS ≥ c[x −dist(x,S)] 2 for all x ∈ C. Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 15 / 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: Growth Conditions Definition 6(γ-o
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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