Consider the LP
min f(x) = x_1 + x_3
s.t. x_1 + x_2 + x_3 = 1
x \geq 0
x \in \mathbb{R}^3
A. Determine a parametric representation of the central path. That is, find expressions for each coordinate as functions of $\mu$. (See Exercise 9.11).
B. Show that as $\mu \to 0$, ($x_1(\mu)$, $x_2(\mu)$, $x_3(\mu)$) converges to the unique optimal solution.
C. Find the analytic center of the feasible region by carefully determining $\lim_{\mu \to \infty} (x_1(\mu), x_2(\mu), x_3(\mu))$.
Exercise 9.11 (Computation of the central path) Consider the problem
minimize $x_1 + x_2$
subject to $x_1 + x_2 + x_3 = 1$
$x_1, x_2, x_3 \geq 0$.
(a) Show that the central path is given as follows:
$x_1(\mu) = \frac{1 + 3\mu - \sqrt{1 + 9\mu^2 - 2\mu}}{4}$,
$x_2(\mu) = x_1(\mu)$,
$x_3(\mu) = 1 - 2x_1(\mu)$.
(b) Show that as $\mu$ decreases to zero, ($x_1(\mu)$, $x_2(\mu)$, $x_3(\mu)$) converges to the
unique optimal solution.
