Self-dual embedding of cone LPs

[AlexR]

Having looked at the pdf document explaining how CVXOPT works (http://www.seas.ucla.edu/~vandenbe/publications/coneprog.pdf), I was hoping to have some theoretical questions cleared up since I'm struggling to understand with amazingly interesting topic. In particular, I'm a bit stumped by section 6 which covers self-dual embedding of cone LPs. I'm willing to accept equations (27) and (28) (although any sources to their derivations would be greatly appreciated), however I'm struggling to see how we get to equations (30a) and (30b). Just above equation (30a) it states "therefore, at optimum the solution must satisfy a complementarity condition with itself, and we can write the optimality conditions for (27) as.", and similarly under equation (30b) it states "combined with (29), this implies that at the optimum $\theta = 0$ and the extended embedding reduces to the homogeneous embedding."

My problem is that I can't seem to figure out where the objective function from (27), $(m+1)\theta$ comes into the optimality conditions of (30a) and (30b). Is this because of equation (29), where since $(m+1)\theta = s^T z + \kappa \tau$ and optimality occurs at $\theta = 0$, this would imply that the equation given by (30b), $z^T s + \kappa \tau = 0$, incorporates the objective function into the optimality conditions?