Convex Optimization · Problem 9.9

Equality constrained minimization — Boyd & Vandenberghe

Mathematics — Optimization · newton-decrement

Problem

Newton decrement. Show that the Newton decrement λ(x)\lambda(x) satisfies λ(x)=sup⁡vT∇2f(x)v=1(−vT∇f(x))=sup⁡v≠0−vT∇f(x)(vT∇2f(x)v)1/2.\lambda(x) = \sup_{v^T \nabla^2 f(x) v = 1} (-v^T \nabla f(x)) = \sup_{v \neq 0} \frac{-v^T \nabla f(x)}{(v^T \nabla^2 f(x) v)^{1/2}}.

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