Basic, Intermediate and CAS problems

The following is the list of non-advanced problems for Chapter 3 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 following problem: Find the global minimum of the function \( f(x,y) = 6(x-10)^2+4(y-12.5)^2 \) on the set \( S = \{ (x,y) \in \mathbb{R}^2 : x^2+(y-5)^2 \leq 50, x^2+3y^2\leq 200, (x-6)^2+y^2 \leq 37 \}.\)

   • Write the statement of this problem as a program with the notation from equation (22). Label the objective function, as well as the inequality constraints accordingly.
   • Is the objective function \( f \) pseudo-convex? Why or why not?
   • Are the inequality constraints quasi-convex? Why or why not?
   • Sketch the feasibility region. Label all relevant objects involved.
   • Is the point \( (7,6) \) feasible? Why or why not?
   • Employ Theorem 4.4 to write a necessary condition for optimality and verify that is satisfied by the point \( (7,6). \)
   • Employ Theorem 4.5 to decide whether this point is an optimal solution of (P).

2. Let \( f(x,y)=(x-4)^2+(y-6)^2.\) Consider the program (P) to find the global minimum of \( f \) on the set \( S = \{ (x,y) \in \mathbb{R}^2 : y-x^2\geq 0, y\leq 4 \}. \)

   • Write the statement of this problem as a program with the notation from equation 22. Label the objective function, as well as the inequality constraints accordingly.
   • Is the objective function \( f \) pseudo-convex? Why or why not?
   • Are the inequality constraints quasi-convex? Why or why not?
   • Sketch the feasibility region. Label all relevant objects involved.
   • Is the point \( (2,4) \) feasible? Why or why not?
   • Employ Theorem 4.4 to write a necessary condition for optimality and verify that is satisfied by the point \( (2,4) \).
   • Employ Theorem 4.5 to decide whether this point is an optimal solution of (P).

3. Let \( f(x,y)=(x-9/4)^2+(y-2)^2. \) Consider the program (P) to find the global minimum of \( f \) on the set \( S = \{ (x,y) \in \mathbb{R}^2 : y-x^2 \geq 0, x+y\leq 6, x\geq 0, y \geq 0 \}. \)

   • Write down the KKT optimality conditions and verify that these conditions are satisfied at the point \( (3/2, 9/4) \).
   • Present a graphical interpretation of the KKT conditions at \( (3/2, 9/4) \).
   • Show that this point is the optimal solution to the program.

4. Find examples of non-diagonal \( 3\times 3 \) symmetric square matrices with integer-valued eigenvalues of each type below:

   • \(\boldsymbol{A}_1\) positive definite,
   • \(\boldsymbol{A}_2\) positive semi-definite,
   • \(\boldsymbol{A}_3\) negative definite,
   • \(\boldsymbol{A}_4\) negative semi-definite, and
   • \(\boldsymbol{A}_5\) indefinite.

   For each of these matrices, find the maximum of their corresponding quadratic form \( \mathcal{Q}_{\boldsymbol{A}_k}(x,y,z) \) over the unit ball \( \mathbb{B}_3 = \{ (x,y,z) \in \mathbb{R}^3: x^2+y^2+z^2 \leq 1 \}. \)