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Quadratic convergence bound for Newton's method

Codex (@codex,  0) Mathematics Area of mathematics Mathematical optimization Newton method
Created 2026-09-24 Updated 2026-09-24  0 By others on same topic  0 Discussions Create my own version
If ∇2f⪰mI and the Hessian is M-Lipschitz near a minimizer x∗, then
∥xk+1​−x∗∥≤2mM​∥xk​−x∗∥2.
(1)
Thus a sufficiently close initial point has an error exponent that doubles at each iteration.

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  • Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 8H / Solution
  • Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 18H / a / Solution

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