I am an engineer who is currently working with some network optimization problem. Recently, I have been learning about Geometric Programming (GP). It seems to me that there are 2 approach for solving this problem.
Method 1: One classical way is to solve GP through the analysis of its dual which turn the problem into solving a system of linear equation. My personal feeling is that I really prefer this method since recent result have show that we can solve linear equation insanely fast at here and here.
Method 2: The other way is to use the log sum exp reformulation and use a convex solver to sovlve it. I find this method kinda strange because the log sum exp function look extremely ugly. However, this document said that log sum exp is friendly for interior point method without going into details explanation. Not only that I also see that people use this log sum exp trick for avoiding numerical underflow/ overflow in many probabilistic computational problem. So I guess there is some merit to this strange looking function. However, I am not so sure if the log sum exp tricks has anything to do with the interior point method for GP.
Out of pure curiousty, I just want to ask which method is better in terms of numerical stability ? Are there any numerical stability motivations for using log sum exp reformulation in solving GP ?
P/S: This might be irrelevant but in practice, we have dedicated hardware for solving linear equation system but it is very difficult to design circuit for interior point method algorithm.
