Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / U / U2 / b / i / Solution

For scalar minimization, apply Newton's root-finding method to the stationarity equation $f^{\prime }(x)=0$. Linearize $f^{\prime }$ at the current point to obtainThe iteration requires $f^{\prime \prime }(x_k)\ne 0$. Near a nondegenerate local minimum, where $f^{\prime }(x_{\ast })=0$, $f^{\prime \prime }(x_{\ast })>0$, suitable smoothness and a sufficiently close starting value give local convergence. A stationary limit must still be checked to be a minimum. This undamped method is not globally convergent and need not decrease $f$ at every step.