The following is a nonlinear programming problem:

Problem 2 (1.4.3, 6 points): Consider the iteration #+1 = pk +afd where ok is chosen by the Armijo rule with initial stepsize s = 1, 0 (0.1/2), and dk is equal to d' = -(02f(x"))-f(x) if V2 f() is nonsingular and the following two inequalities hold: Gi|| Vf(r)||PS-Vf(ryd, || C P 0,c2 > 0, p > 2, P2 > 1. Show that the sequence {d*} is gradient related. Furthermore, every limit point of {z*} is stationary, and if {zk} converges to a nonsingular local minimum r*, the rate of convergence of {* - * ||} is superlinear. Problem 2 (1.4.3, 6 points): Consider the iteration #+1 = pk +afd where ok is chosen by the Armijo rule with initial stepsize s = 1, 0 (0.1/2), and dk is equal to d' = -(02f(x"))-f(x) if V2 f() is nonsingular and the following two inequalities hold: Gi|| Vf(r)||PS-Vf(ryd, || C P 0,c2 > 0, p > 2, P2 > 1. Show that the sequence {d*} is gradient related. Furthermore, every limit point of {z*} is stationary, and if {zk} converges to a nonsingular local minimum r*, the rate of convergence of {* - * ||} is superlinear