Suppose we have the following optimization problem to solve:
\begin{equation} \begin{split} \textbf{(P1)}~~~~~~~~ \underset{x_1, \, x_2, \,x_3}{\text{minimize}}\, \left\{ \, f\left(x_1, \, x_2, \, x_3 \right) + \lambda \, \Big(g\left(x_1\right) + g\left(x_2\right) + g\left(x_3\right) \Big) \, \right\} \\ s.t. \,\,\,\,\,\,\, x_1\geq 0 \\ x_1 + x_2^2 - 5 \geq 0 \\ x_3 - 25 \geq 0 \,. \end{split} \end{equation} where both $f$ and $g$ are convex functions.
How can I apply the logarithmic barrier function in order to transform problem (P1) into an unconstrained problem? BTW I am doing that as follows:
\begin{equation} \small \textbf{(P2)} ~~~~~~~~\underset{x_1, \, x_2, \, x_3}{\text{minimize}} \, \left\{\, f\left(x_1, \, x_2, \, x_3 \right) + \lambda \, \Big(g\left(x_1\right) + g\left(x_2\right) + g\left(x_3\right) \Big) - \mu \, \sum_{i=1}^3 \, ln\Big( c_i(x) \Big)\, \right\} \end{equation} with $c_1(x) = x_1$, $~~c_2(x) = x_1 + x_2^2 - 5$, $~~$ and $~~c_3(x) = x_3 - 25$.
Is my use of the logarithmic barrier function right to transform problem (P1) into (P2)? If not, can someone provide me details about how can I do it in the right way?
Any help will be very appreciated!