Are you talking about doing it in some particularly clever way? If you simply intend to solve the problem in the original dual space (i.e. seeing $X$ as a matrix parameterized by its elements and optimizing over those variables) it is just a matter of deriving a solution and basis for $Ax=c$ where $x$ are the elements parameterizing $X$, i.e., write $x = x_0 ...
I doubt you can eliminate the fixed variables in general but given some special structure it might be possible.
In my paper I describe how to exploit fixed variables inside a primal-dual interior algorithm for conic quadratic optimization. However, I have not been able to generalize this idea to the semi-definite cone in a useful way.
solve a bundle of nonlinear convex problems simultaneously, where each
convex optimization problem is unrelated to each other.
If you are using Linux this will do exactly what you want:
for problem in $problems
ipopt $problem &
if [[ $(jobs -r -p | wc -l) -ge $max_jobs ]]; then
Almost all convex optimization problems can be formulated as a conic optimization problem using only the cone types we can handle in practice. See the Mosek modeling cookbook for details. This often leads to the best solution time and you do not have to mess with derivatives. For instance Mosek can solve such conic optimization problems.
The upcoming Mosek ...
As a partial answer, Equation 16 is a condition that leads to convergence. The replacement follows from maximization by a method that seems similar to LP rounding. I have not ran the numbers, but a factor of 3/2 seems plausible for that technique.
I suppose convergence is chosen as a criterion because it doesn't know where to optimize to. I think it related ...