For the LP:
max $-x_1 + x_2$
s.t.
$-x_1 \le -1$
$x_2 \le 1$
$x_1 \ge 0, x_2 \ge 0$
a) Construct and graph the central path to the optimal point by:
i) using the log-barrier function using $\mu_0 = 1000$, and $\mu_{k+1} = 0.5\mu_k$ (and use fmincon in MATLAB or
EXCEL Solver to solve each log-barrier function problem)
ii) solving the KKT conditions of the log-barrier problem which is the same as the joint KKT conditions
of the primal LP and its dual. (Hint: The dual is exactly the same as the primal. So exploit this
symmetry in simplifying the joint KKT conditions of the form:
$Ax + w = b$
$A^Ty - s = c$
$XSe = re$
$YWe = re$
Then solve for ($x_t, y_t, w_t, s_t$) as a function of $t$, and plot the locus $x_t$ as $t \to 0$.)
b) Execute two iterations of the predictor-corrector method starting from $x_1 = 3, x_2 = 0.8$.
