Could you provide values of step size, decision variables, objective function, and etc. that are needed to fill in the tables?
Question 2 (Gradient and Newton's methods)
The function below is called the Rosenbrock function, also known as the banana function or De Jong's function:
f(x) = 100(x2 - x)2 + (1 - x)2
In the module folder (the folder for coursework 2 on the BB page), you'll find 8 M-functions (for steepest descent search), including the M-function for Armijo rule (as below), a special case of the inexact line search scheme.
Armijo Rule:
1. Choose > 0, e.g. = 1, set m = 1, = 1.
2. While f(x + αVx) > f(x) + αt∇g0pxs1, α = α/2, e.g. α = 1/2.
3. Set α = α, set m = 1, k = k + 1, go to 2.
Write your own MATLAB code/functions to implement Newton's method using the Armijo rule for inexact backtracking line search Armijo rule. Provide code/script of all the following required functions in your report:
(1) rosenbrock_hessian.m
(2) newton_armijo.m
(3) newton_armjo_main.m
Use your code/functions to minimize the Rosenbrock function. You should consider the following requirements:
a. Use Newton's method without Armijo rule; use Newton's method with Armijo rule.
b. Use the following initial parameter values: Initial step size 1. = 0.5, = 0.4, stopping criterion x0.01
c. Record the step size at each iteration. Test your algorithms with the initial value xi = 5.
In your report, you should provide the following information:
2.1 The values of step size, decision variables, and the objective function obtained using Newton's method with and without using Armijo rule Table 2-1
The values of the following parameters for the first three iterations:
α, x, f(x), k = 1, 2, 3.
Without using Armijo rule
With Armijo rule
Iteration 1
Iteration 2
Iteration 3
Table 2-2
The values of the following parameters for the last three iterations:
α, x, f(x), k = N-2, N-1, N.
Without using Armijo rule
With Armijo rule
Iteration N-2
Iteration N-1
Iteration N
2.2 How does Newton's method with Armijo rule differ from without using Armijo rule?
