Consider the following optimization problem:

,

where , with , , with and .

- Write down the KKT system.
- Find the closed form solution for the optimal solution and the corresponding Lagrange multiplier .

Consider the following optimization problem:

- Write down the Lagrangian function and the Lagrange multiplier condition.
- Solve the problem by the Lagrange multiplier method.
- Apply Newton's method to solve it from the the initial point : write down the KKT system; find the closed form expression for the Newton direction; and then run an iteration.

Consider the following optimization problem:

.

- Sketch the feasible set and label it properly.
- Write down the KKT conditions.
- Find the optimal solution and corresponding multipliers.

Consider the following optimization problem:

.

Suppose that is a continuously differentiable function, and .

Let be the optimal solution and we further assume that is a regular point. Apply KKT condition to determine:

- Whether or ?
- Whether , , or ?