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2 1 0 Introduction and Notation x 2 −1 2 0 x 1 4 1 0 −1 2 0 x 1 6 1 0 −1 0 1 2 1 0 x 2 − 0.5x −1 0 1 2 1 0 x 4 − 0.5x −1 0 1 2 1 0 x 6 − 0.5x −1 0 1 Figure: Minimizing x 2m over the domain C = [0,1], for m = 1,2,3. If the objective is perturbed by −0.5x (tilt perturbation), the optimal solution is less “stable” with higher values of m. The ɛ-optimal solution sets are shown for ɛ = 0.05. Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 4 / 26
Questions Introduction and Notation Suppose that we have an objective function fθ and a set Cθ that depend on some parameter θ ∈ Θ (See Figure 2 on the next slide). We would ask questions including: • What are conditions on Θ ensuring that minCθ fθ is finite for all θ ∈ Θ? • Given Θ, what is the range of the optimal value mapping θ ↦→ minCθ fθ? Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 5 / 26
Page 1 and 2: Stochastic Optimization Seminar Lec
Page 3: Motivation Introduction and Notatio
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 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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