lecture. - CASTLE Lab - Princeton University

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Perturbation Theorems
For t > 0, consider the perturbed problems
P(t) : minimize f(x)+tηt(x) subject to x ∈ X,
where the perturbations ηt(x) satisfy the following assumptions:
• B1’. The functions ηt are Lipschitz continuous on U.
• B2. ηt converges to a function which is Lipschitz continuous as t ↓ 0.
• B3. The function η is differentiable at ¯x. In particular we can expand
η(x(t)) where x(t) = x +th+ t2
s w +o(t2 ) as follows:
η(x(t)) = η(¯x)+t∇η(¯x)h+o(t).
Let gt(x) = f(x)+tηt(x), and consider v(t) = minCg and ¯x(t) = argmin C gt.
Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 25 / 26