I encountered the following problem in studying convex optimization: In order to solve the convex optimization problem $$\begin{cases} \mathrm{min}\ f(x)\\\mathrm {s.t.}\ g_i(x)\leq0\ (i=1,...,m)\ \end{cases} (1)$$ One needs to solve the KKT condition equations: $$\begin{cases} \nabla f(x)+\sum_{i=1}^m \lambda_i \nabla g_i(x)+\mu ^T(Ax-b)=0\\ g_i(x)\leq0\\ \lambda_i\geq0\\ \lambda_i g_i(x) = 0 \end{cases} (2)$$ However, in the primal-dual interior point method, the pertubed KKT condition equations are solved (in which the parameter $t$ is updated during iterations): $$\begin{cases} \nabla f(x)+\sum_{i=1}^m \lambda_i \nabla g_i(x)+\mu ^T(Ax-b)=0\\ g_i(x)\leq0\\ \lambda_i\geq0\\ \lambda_i g_i(x) = -1/t \end{cases} (3)$$ I can understand that the pertubed KKT condition is constructed under the inspiration of barrier interior point method (in which the problem $\mathrm{min}\ f(x)-1/t\sum_{i=1}^m\log {(-g_i(x))}$ is solved). However, my problem is,
What is the numerical advantage of solving problem (1) by solving Eq. (3) rather than Eq. (2)?
I see no difference between solving Eq. (2) and Eq. (3), while Eq. (2) is definitely easier to solve (at least no need to update $t$)...