Homework 1 Problems#
Note
This homework is due Friday, Jan 30, 11:59pm on Crowdmark. No credit will be given after Sunday, Feb 1, 11:59pm.
(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.).
(10 points) For each of the following matrices, determine whether they are positive/negative definite/semi-definite or indefinite. Justify your answer.
(a) \(A = \begin{pmatrix} -2 & 1 \\ 1 & -2 \end{pmatrix}\)
(b) \(B = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}\)
(c) \(C = \begin{pmatrix} 4 & 2 & 0 \\ 2 & -4 & 2 \\ 0 & 2 & 4 \end{pmatrix}\)
(d) \(D = \begin{pmatrix} 2 & 2 & 0 & 0\\ 2 & 2 & 0 & 0\\ 0 & 0 & 3 & 1\\ 0 & 0 & 1 & 3 \end{pmatrix}\)
(e) \(E = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}\)
(7 points) Consider the function \(f(x,y)= x^2 + y^2 + xy\).
a. Write the quadratic form of the function, \(f(\mathbf{x}) = \mathbf{x}^T \mathbf{A} \mathbf{x} + 2\mathbf{b}^T \mathbf{x} + c\).
b. Using this quadratic form, determine whether the function is coercive. Justify your answer.
c. Using this quadratic form, determine whether the function is convex. Justify your answer.
(10 points) Find the global minimizer and maximizer of the function \(f(x,y)= x^2+y^2-3y\) over the unit ball \(B[0,1]=\{(x,y): x^2+y^2 \leq 1 \}\).
(10 points) Find all stationary points of \(f(x,y) = 2x^{2}+3y^{2}-2xy+2x-3y+5\) and check whether they are saddle points, strict/nonstrict local/global minimizer/maximizer.
(5 bonus points) Find all stationary points of \(f(x,y)= (9x^2-4y)^2\) and check whether they are saddle points, strict/nonstrict local/global minimizer/maximizer.
Note
I have realized that problem 6 is tricker than I was aiming for. I’ve decided to turn it into a bonus question instead of a requirement. You’re welcome!