Basic, Intermediate and CAS problems

The following is the list of non-advanced problems for Chapter 2 of the class notes. There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please, do not post solutions.

1. Consider the function \( f(x,y) = \dfrac{x+y}{2+\cos x}. \) At what points \( (x,y) \in \mathbb{R}^2 \) is this function continuous?

2. Give an example of a \( 2 \times 2 \) symmetric matrix of each kind below:

   • positive definite,
   • positive semidefinite,
   • negative definite,
   • negative semidefinite,
   • indefinite.

3. Classify the following matrices according to whether they are positive or negative definite or semidefinite or indefinite:

   • \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix} \)
   • \( \begin{bmatrix} -1 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & -2 \end{bmatrix} \)
   • \( \begin{bmatrix} 7 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & 5 \end{bmatrix} \)
   • \( \begin{bmatrix} 3 & 1 & 2 \\ 1 & 5 & 3 \\ 2 & 3 & 7 \end{bmatrix} \)
   • \( \begin{bmatrix} -4 & 0 & 1 \\ 0 & -3 & 2 \\ 1 & 2 & -5 \end{bmatrix} \)
   • \( \begin{bmatrix} 2 & -4 & 0 \\ -4 & 8 & 0 \\ 0 & 0 & -3 \end{bmatrix}\)

4. Write the quadratic form \( \mathcal{Q}_{\boldsymbol{A}}(\boldsymbol{x}) \) associated with each of the following matrices \( \boldsymbol{A} \):

   • \( \begin{bmatrix} -1 & 2 \\ 2 & 3 \end{bmatrix} \)
   • \( \begin{bmatrix} 1 & -1 & 0 \\ -1 & -2 & 2 \\ 0 & 2 & 3 \end{bmatrix} \)
   • \( \begin{bmatrix} 2 & -3 \\ -3 & 0 \end{bmatrix} \)
   • \( \begin{bmatrix} -3 & 1 & 2 \\ 1 & 2 & -1 \\ 2 & -1 & 4 \end{bmatrix} \)

5. Write each of the quadratic forms below in the form \( \boldsymbol{x} \boldsymbol{A} \boldsymbol{x}^\intercal \) for an appropriate symmetric matrix \( \boldsymbol{A} \):

   • \( 3x^2-xy+2y^2. \)
   • \( x^2+2y^2-3z^2+2xy-4xz+6yz. \)
   • \( 2x^2-4z^2+xy-yz. \)

6. Identify which of the following real-valued functions are coercive. Explain the reason.

   • \( f(x,y) = \sqrt{x^2+y^2}. \)
   • \( f(x,y) = x^2 + 9y^2 - 6xy. \)
   • \( f(x,y) = x^4 - 3xy +y^4. \)
   • Rosenbrock functions \( \mathcal{R}_{a,b}. \)

7. Let \( C \subseteq \mathbb{R}^2 \) be a convex figure. Given a point \( P \in C \), let \( n(P) \) be the number of chords for which \( P \) is a midpoint. For instance, if \( C \) is a disk, any point \( P \in C \) satisfies \( n(P)=0 \) (if the point \( P \) is on the circle), \( n(P) = \infty \) (if the point \( P \) is the center of the disk), or \( n(P)=1 \) for any other point in the interior of \( C \).

   • Are there convex sets that contain points \( P \) with \( n(P)=2 \)? If so, sketch an example.
   • Are there convex sets that contain points \( P \) with \( n(P)=m \) for any \( m \geq 3 \)? If so, sketch and example for \( m=3 \).
8. Determine whether the given functions are convex, concave, strictly convex or strictly concave on the specified domains:
   • \( f(x) = \log(x) \) on \( (0,\infty). \)
   • \( f(x) = e^{-x} \) on \( \mathbb{R}. \)
   • \( f(x) = \lvert x \rvert \) on \( [-1,1]. \)
   • \( f(x) = \lvert x^3 \rvert \) on \( \mathbb{R}. \)
   • \( f(x,y) = 5x^2+2xy+y^2-x+2x+3 \) on \( \mathbb{R}^2. \)
   • \( f(x,y) = x^2/2+3y^2/2+\sqrt{3}xy \) on \( \mathbb{R}^2. \)
   • \( f(x,y) = 4e^{3x-y}+5e^{x^2+y^2} \) on \( \mathbb{R}^2. \)
   • \( f(x,y,z) = x^{1/2} + y^{1/3} + z^{1/5} \) on \( C = \{ (x,y,z) : x>0, y>0, z>0 \}. \)

9. Sketch the epigraph of the following functions

   • \( f(x) = e^x. \)
   • \( f(x,y)=x^2+y^2. \)

10. For the following optimization problems, state whether existence of a solution is guaranteed:

    • \( f(x) = \dfrac{1+x}{x} \) over \( [1,\infty) \)
    • \( f(x) = 1/x \) over \( [1,2) \)
    • The following piecewise function over \( [1,2] \)
    $$f(x) = \begin{cases} 1/x, &\text{if }1\leq x<2 \\\\ 1, &\text{if }x=2 \end{cases}$$

11. Use the Principal Minor Criteria to determine—if possible—the nature of the critical points of the following functions:

    • \( f(x,y) = x^3+y^3-3x-12y+20. \)
    • \( f(x,y,z) = 3x^2+2y^2+2z^2+2xy+2yz+2xz. \)
    • \( f(x,y,z) = x^2+y^2+z^2-4xy. \)
    • \( f(x,y) = x^4+y^4-x^2-y^2+1. \)
    • \( f(x,y) = 12x^3+36xy-2y^3+9y^2-72x+60y+5. \)

12. Show that the function \( f(x,y,z) = e^{x^2+y^2+z^2}-x^4-y^6-z^6 \) has a global minimum on \( \mathbb{R}^3. \)

13. Consider the function \( f(x,y) = x^3 + e^{3y} -3xe^y. \) Show that \( f \) has exactly one critical point, and that this point is a local minimum but not a global minimum.

14. Let \( f(x,y) = -\log(1-x-y)-\log x -\log y. \)

    • Find the domain \( D \) of \( f. \)
    • Prove that \( D \) is a convex set.
    • Prove that \( f \) is strictly convex on \( D. \)
    • Find the strict global minimum.
15. Find local and global minima in \( \mathbb{R}^3 \) (if they exist) for the function \( f(x,y) = e^{x+z-y}+e^{y-x-z}. \)