Homework 2 Problems#
Note
This homework is due Thurs, Feb 12, 11:59pm on Crowdmark. No credit will be given after Sunday, Feb 15, 11:59pm. Note this is one day longer than the usual homework deadline due to Rememberance day on Feb 13.
(3 points) Include information on Crowdmark related to resources used to complete this homework (e.g., textbook, lecture notes, online resources, generative AI, study groups, etc.).
Consider the following function:
\[ \min_{x_1,x_2} \Big\{ f(x)= \frac{1}{4}x_1^4+ \frac{1}{4}x_2^4-x_1x_2+4 \Big\} \]a. (10 points) Find the three stationary points of the function analytically and classify them as being local minimizers, local maximizers, or saddle points. Evaluate the function at these points.
b. (5 points) What is the global minimum of the function and at what point(s) is it attained? Note that the function \(f(x)\) is coercive, so it must have a global minimizer.
c. (10 points) Use the functions developed in Python during the course coding assignments (or feel free to write your own new versions for this part) to find the minimizer of \(f(x)\) using two methods:
gradient method with backtracking with \((s=1, \alpha= \beta= 0.5)\)
pure Newton’s method
Choose the four initial points:
\((4,4)^\top\)
\((-4,-4)^\top\)
\((-0.25,0.25)^\top\)
\((30,30)^\top\)
For the stopping condition in the methods, choose a reasonable \(\varepsilon\), e.g., \(10^{-5}\).
For this question, post the code you used for this part to Crowdmark showing the solution found for each initial point and method. Unless you are changing the functions from class, you do not need to post the code for the methods themselves, but you should post the code showing how you called the methods with printouts showing what solution was found and how many iterations it took in each case.
d. (5 points) Did all of your methods converge to the same point for each initial point? If not, explain why.
e. (5 points) Did all of your methods take the same number of iterations to converge for each initial point? If not, explain why.