lecture. - CASTLE Lab - Princeton University
lecture. - CASTLE Lab - Princeton University
lecture. - CASTLE Lab - Princeton University
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Second-Order Expansions<br />
First- and Second-Order Expansions<br />
A directionally differentiable function g(x) is second order directionally<br />
differentiable at x in the direction h if the (parabolic) second order directional<br />
derivatives<br />
g ′′ (x,h,w) = lim<br />
t→0 +<br />
g(x +th+ t2<br />
2w)−[g(x)+tg′ (x,h)]<br />
t<br />
exist for all (parabolic) directions w. In that case we can expand<br />
g(x +th+ t2<br />
2 w) for t > 0 as<br />
g(x +th+ t2<br />
2 w) = g(x)+tg′ (x,h)+ t2<br />
2 g′′ (x,h,w)+o(t 2 ).<br />
Boris Defourny (ORFE) Lecture 6: Perturbation Analysis in Optimization March 15,2012 18 / 26
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