This post deals with the minimization of the log barrier function that is:

where defined as

We shall attempt an intuitive explanation on the solution of the following unconstrained minimization problem

First off, let’s say our set of inequalities defined by is closed, the optimal solution of problem can not lie outside the closed area defined by since function would not be defined. So we have no other choice than to force to lie inside . If is close to the hyperplane defined by , say the one, i.e. , then the term would explode to infinity, hence the minimization process fails. The best compromise would be to place as far as possible from all hyperplanes , which is then termed the analytic center of the inequalities for all . See the above video for a nice illustration.