lecture. - CASTLE Lab - Princeton University

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