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Perturbation Theorems Theorem 2[Shapiro, Dentcheva, and Ruszczyński(2009)] Under all the assumptions we have made in the previous two slides, for t ≥ 0, we have the second-order expansion of the optimal function v(t) = ¯v +tηt(¯x)+ t2 2 hf,η(¯x)+o(t 2 ), where hf,η(¯x) is the optimal value of the auxiliary problem over h, minimize 2h ⊤ ∇η(¯x)+h ⊤ ∇ 2 f(¯x)h subject to h ∈ C crit (¯x) := {d ∈ TC(¯x) : d ⊤ ∇f(¯x) = 0}, TC(x) := {c ∈ R n : dist(x +td,C) = o(t),t ≥ 0}. Moreover, if the solution to the auxiliary problem is a unique, say ¯g, we have for t ≥ 0 the first-order expansion of the optimal solution ˜x(t) = ¯x +t ¯ h+o(t). Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 26 / 26
Perturbation Theorems J.F. Bonnans and A. Shapiro. Perturbation Analysis of Optimization Problems. Springer, New York, 2000. W. Cook, A.M.H. Gerards, and A. Schrijver. Sensitivity theorems in integer linear programming. Mathematical Programming, 34:251–264, 1986. O.L. Mangasarian. Nonlinear Programming. SIAM, 1994. R.T. Rockafellar and R.J.-B. Wets. Variational Analysis. Springer, third edition, 1998. A. Shapiro, D. Dentcheva, and A. Ruszczyński. Lectures on Stochastic Programming: Modeling and Theory. SIAM, 2009. Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 26 / 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 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: More Assumptions... Perturbation Th

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