I have implemented a saddle point optimization problem based on the algorithm by Prof. Nesterov, primal-dual. Unfortunately, it doesn't work. It seems it is converging. But unfortunately, not to the optimal solution which I get with some out of the box solver. I am confident that out of the box solver's result is correct.
My question is, based on the arguments by paper, algorithm must work correctly. But somethings gets wrong. I want to check inequalities and conditions in proofs of the paper, But I don't know how to do this?especially because many of the inequalities involved unknown optimal solution ($x^*$). Is there any way to check inequalities in the paper, and understand what is wrong which cause my implementation doesn't work?
Note that I solved inner optimization problems in algorithm, using the same solver.
, Yurii Nesterov, Primal-dual subgradient methods for convex problems , Math. Program. 2009