MATH 524 Fall 2017 Chapter 4
Basic, Intermediate and CAS problems

Consider the following problem: Find the global minimum of the function \( f(x,y) = 6(x10)^2+4(y12.5)^2 \) on the set \( S = \{ (x,y) \in \mathbb{R}^2 : x^2+(y5)^2 \leq 50, x^2+3y^2\leq 200, (x6)^2+y^2 \leq 37 \}.\)
 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 \( f \) pseudoconvex? Why or why not?
 Are the inequality constraints quasiconvex? 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).

Let \( f(x,y)=(x4)^2+(y6)^2.\) Consider the program (P) to find the global minimum of \( f \) on the set \( S = \{ (x,y) \in \mathbb{R}^2 : yx^2\geq 0, y\leq 4 \}. \)
 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 \( f \) pseudoconvex? Why or why not?
 Are the inequality constraints quasiconvex? 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).

Let \( f(x,y)=(x9/4)^2+(y2)^2. \) Consider the program (P) to find the global minimum of \( f \) on the set \( S = \{ (x,y) \in \mathbb{R}^2 : yx^2 \geq 0, x+y\leq 6, x\geq 0, y \geq 0 \}. \)
 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.

Find examples of nondiagonal \( 3\times 3 \) symmetric square matrices with integervalued eigenvalues of each type below:
 \(\boldsymbol{A}_1\) positive definite,
 \(\boldsymbol{A}_2\) positive semidefinite,
 \(\boldsymbol{A}_3\) negative definite,
 \(\boldsymbol{A}_4\) negative semidefinite, 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 \( \mathbb{B}_3 = \{ (x,y,z) \in \mathbb{R}^3: x^2+y^2+z^2 \leq 1 \}. \)