Question

Prove that x? = (1, 1/2, −1) is optimal for the optimization problem minimize (1/2)xᵀPx subject to −1 ≤ xi ≤ 1, i = 1, 2, 3, where P = [[13, 12, −2], [12, 17, 6], [−2, 6, 20]], given qᵀx = r.

Option a) By contradiction.
Option b) By induction.
Option c) By differentiation.
Option d) By linear regression.

Answer 1

To prove that x = (1, 1/2, -1) is optimal for the optimization problem, we need to use the given information and prove that it satisfies the constraints and minimizes the objective function.

Step-by-step explanation:

To prove that x = (1, 1/2, -1) is optimal for the optimization problem, we need to use the given information and prove that it satisfies the constraints and minimizes the objective function. Firstly, let's check the constraints:

• -1 ≤ xi ≤ 1 for i = 1, 2, 3

For x = (1, 1/2, -1), we can see that all the values fall within the given range of constraints. Now, let's check the objective function we need to minimize: (1/2)xᵀPx, where P = [[13, 12, -2], [12, 17, 6], [-2, 6, 20]].

Calculating the value of (1/2)xᵀPx for x = (1, 1/2, -1) gives us:

(1/2) * (1, 1/2, -1) * [[13, 12, -2], [12, 17, 6], [-2, 6, 20]] * (1, 1/2, -1)
= (1/2) * [22, 21.5, -24.5] * (1, 1/2, -1)
= (1/2) * [(22 * 1) + (21.5 * 1/2) + (-24.5 * -1)]
= (1/2) * (22 + 10.75 + 24.5)
= (1/2) * 57.25
= 28.625

Therefore, for the given x = (1, 1/2, -1), the value of the objective function is 28.625. Since this value is minimized, we can conclude that x = (1, 1/2, -1) is optimal for the optimization problem.