Homework 8

1. Book 10.9 and 10.15 (related problems)
2. Consider the equality constrained problem in (10.37) page 553 of the textbook with $X \in \mathbf{S}^3$ and $p=2$ equality constraints using the quantities

$$A_1 = \begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 0\end{bmatrix}, \quad b_1 = 1,\\ A_2=\begin{bmatrix}0 & 0 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}, \quad b_2 = 1.$$

Compute the solution of the problem using the following methods.

(a) Infeasible start Newton method. Use the initial point $X^{(0)} = I$.

(b) Dual Newton method, i.e., the standard Newton method applied to the dual problem.

Verify that the two methods compute the same optimal point (and Lagrange multiplier). Compare the computational effort per step for the two methods.

The optimal point is

$$X^\ast = \frac{1}{18} \begin{bmatrix} 7 & -2 & 1 \\ -2 & 4 & -2 \\ 1 & -2 & 7\end{bmatrix}.$$