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

Newton's method for minimization usesFor a quantitative local bound, suppose on a convex neighbourhood containing the iterates thatwith $m>0$, and let $x^{\ast }$ be the minimizer. The integral form of the gradient and the Hessian Lipschitz bound give the quadratic convergence bound for Newton's methodIf $q=M\Vert x_0-x^{\ast }\Vert /(2m)<1$, induction yieldsSince $f(x)-f(x^{\ast })\le L\Vert x-x^{\ast }{\Vert }^2/2$,