# How to prove that $c^Tx(\mu)$ is strictly decreasing with $\mu$ in interior point method for LP

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 $$\mu$$.

• can you prove it by contradiction? – LinAlg Nov 10 '19 at 22:40
• I think constradiction is a little difficult. – Icy Nov 12 '19 at 6:31