Convex Optimization · Problem 9.18

Equality constrained minimization — Boyd & Vandenberghe

Mathematics — Optimization · quadratic-convergence-self-concordant

Problem

Quadratic convergence. Let

f : \mathbf{R}^n \to \mathbf{R}

be a strictly convex self-concordant function. Suppose

\lambda(x) < 1

, and define

x^+ = x - \nabla^2 f(x)^{-1} \nabla f(x)

. Prove that

\lambda(x^+) \leq \lambda(x)^2 / (1 - \lambda(x))^2

. Hint. Use the inequalities in exercise 9.17, part (c).