MATH 309 Spring 2017 - Homework 2 The due date is 11:59 pm, February 3 (Friday). Please submit your your source codes and the required results via email to Xiaorui Li at xla97@sfu.ca. No late submission will be accepted. Partial points will be deducted for those who copy or duplicate the others' homework or work. Problem 1: Implement the steepest descent method with the exact step length for solving 1 min f (x) = xT Ax bT x, x 2 where b = (1, 1, . . . , 1)T and A is the nn Hilbert matrix, whose elements are Ai,j = 1/(i+j 1). Set the initial point to x0 = 0. Try n = 20 and terminate the algorithm when kf (xk )k 102 . Report the objective function value f , the step length and the norm of f of the last 10 iterations. Problem 2: Implement the steepest descent method with the inexact step length satisfying the strong Wolfe conditions with c1 = 104 and c2 = 0.1 for solving the same optimization problem as given in Problem 1. Set the initial step length 0 = 1 and the initial point x0 = 0. Try n = 20 and terminate the algorithm when kf (xk )k 102 . Report the objective function value f , the step length and the norm of f of the last 10 iterations. Compared with the algorithm implemented in Problem 1, which one is faster? (The codes for finding the step length satisfying the strong Wolfe conditions with c1 = 104 and c2 = 0.1 are posted on the Canvas.) Problem 3: Implement the steepest descent method with the backtrack line search (that is, Algorithm 3.1) to minimize the Rosenbrock function f (x) = 100(x2 x21 )2 + (1 x1 )2 . Set = 1, = 0.9 and c = 104 and the initial point x0 = (1.2, 1)T . Terminate the algorithm once kf (xk )k 104 . Report the objective function value f , the step length and the norm of f of the last 10 iterations. Problem 4: Implement the steepest descent method with the inexact step length satisfying the strong Wolfe conditions with c1 = 104 and c2 = 0.1 to minimize the above Rosenbrock function. Set the initial step length 0 = 1 and the initial point x0 = (1.2, 1)T . Terminate the algorithm once kf (xk )k 104 . Report the objective function value f , the step length and the norm of f of the last 10 iterations. Compared with the algorithm implemented in Problem 3, which one is faster? 1