Consider the Primal-Dual problem,
(P)min $c^Tx$
s.t. Ax = b, x $\geq 0$
(D) max $b^Ty$
s.t. $A^Ty + s = c$, s $\geq 0$
The log-barrier function for (P) is :
min $c^Tx - \mu \sum_{i=1}^n ln(x_i)$
s.t. Ax = b, x > 0
How to prove that if $\mu > \mu^{\prime}$, then $c^Tx(\mu) > c^Tx(\mu^{\prime})$, where $x(\mu)$ is the optimal solution of log-barrier function with parameter $\mu$.
