MATH 524 Fall 17 Chapter 4
Basic, Intermediate and CAS problems
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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).
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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).
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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.
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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 \( \mathbb{B}_3 = \{ (x,y,z) \in \mathbb{R}^3: x^2+y^2+z^2 \leq 1 \}. \)