Consider the following optimization problem:

.

- Find the optimal solution and the optimal value of .
- Write down the dual function and find the closed form.
- Write down the dual problem.
- Find the optimal solution to the dual problem.

Consider the following optimization problem:

- Sketch the feasible set of the problem, and the level sets of the objective function.
- Find the optimal solution and the optimal value .
- Give the KKT conditions. Do there exist Lagrange multipliers and that satisfy KKT conditions?
- Write down the Lagrange dual function and the dual problem.
- Solve the dual problem.
- Does strong duality hold? Does Slater's condition hold?