EXERCISE 3 [15pt] (a) Implement the dogleg trust region method for strictly convex functions (with s.p.d. Hessian). Your implementation should return the Cauchy point whenever the gradient and Newton steps are collinear. Include your implementation into the report. Highlight the part where you solve for the intersection point between the trust region and the dogleg path and provide a short narrative explanation. [6pt] (b) Apply the dogleg trust region method to minimise the Rosenbrock function f : R- - R f(z,y) = 100(y - 23)3 + (1 -x)2 with two different starting points ro = (0.5, 1) and ro = (-1.5, 1) . Plot the trajectories traced by the iterates over the function contours. State your choice of the stopping condition and any relevant parameters. What do you observe? [2pt] (c) Investigate convergence of the dogleg iterates in (b) a posteriori and include one relevant error plot. What are the empirical convergence rates and how did you obtain them? Do they agree with the theoretical predictions? Paraphrase the relevant theoretical results. [4pt] (d) Can global convergence be expected or not, and why? Paraphrase the relevant theoretical results. [3pt]