Now I am new to the world of primal-dual algorithms and I want to understand the SOCP-Code of Lobo/Vandenberghe/Boyd (primal dual interior point method). Currently I am working through Goldfarb and Idnani (1981):
A Numerically Stable Dual Method for Solving Strictly Convex Quadratic Programs
and I was wondering what they do with equality constraints. In the introduction they say that $x$ may also be subject to equality constraints $\hat{C}^Tx-\hat{b}=0$ but they will ignore these to simplify the presentation. Sadly though I cant find a reference to these later in the paper.
What to do with these type of constraints? Should I split them up into two sets of inequality constraints like this?
$$\hat{C}^Tx\leq\hat{b},\\-\hat{C}^Tx\leq-\hat{b}.$$
Does this method work for the Goldfarb/Idnani (QP) case? Does it work for the Lobo/Vandenberghe/Boyd (SOCP) case? Does it matter that the solution is not in the interior of the set of feasible points when equality constraints are in place? If this does not work, how to deal with these constraints?
In general an equality constraint can be handled by using it to eliminate one of the variables (using ordinary linear algebra) before any of the clever optimization algorithms is let loose on the problem.
This works well for a general optimization algorithm, but if one wants to allow your method to be specialized to a particular sparse shape of the goal functions and/or constraints by providing optimized solvers for subproblems, one may have to handle equality constraints more explicitly during the algorithm proper.