Reference no: EM132707560

Question 1: Consider the function of 2n variables

Q1.a For n = 10, n = 100 and n = 500 run the conjugate gradient method starting from x0 = 0. Experiment with different re-initialization numbers (that is, change the parameter cycmax in the code). Explain your results.

Q1.b Repeat Q1.a for the preconditioned conjugate gradient method described in the lecture and also formulated in (5.75)-(5.76) of the textbook with
V = diag{i-1, i = 1, . . . , 2n} (the inverse of the Hessian of the first sum).
Report your observations. For this question, you will need to modify the sup- plied code of the conjugate gradient method according to the preconditioning method.

Question 2: Calculate the direction of steepest descent for the function
f (x1, x2) = x1 + x2 + max 0, (x1)2 + (x2)2 - 4 at the point x = (0, -2).

Question 3: Consider the following problem
min f (x) subject to x ≥ 0, (1) where f : R3 → R is defined as follows:
f (x) = max (x1)2 + (x2 - 1)2 + (x3)2, (x1 - 1)2 + (x2 -x1)2 + 2, -x1 + 3x2 + x3 - 1 .

Q3a. Solve problem (1) using the projected subgradient method with small fixed stepsize α. Experiment with different values of α. Start from (2, 2, 2) and (1, 1, 1).

Q3b. Formulate the master problem of the cutting plane method for problem (1) as a linear optimization problem.

Q3c. Solve problem (1) by the cutting plane method starting from the same points. For this question, you will need to code the method but you can use a standard function to solve the master problem. Supply printouts from your code and its output.