# Solving for primary variables of a linear program after already having solved for the dual

I was wondering if there is a general procedure of solving for the primary variables of a linear or quadratic or, in general, a convex program after already having solved the dual program.

The problem I am working on is a least squares problem with l1-norm regularization terms and these regularization terms can be changed into constraints on a quadratic program for the dual variables. I can solve this for the dual variables. One of these ends up being the residual between the data and the model. So I can easy arrive at the model. However the model is a sum of various terms and I would like to know them individually, not just their sum (the model).

The only way I know how to get them is to do a second minimization problem of minimizing the l1-norms with the least squares term now held constant (so negligible). This is even more involved than the first problem but still reducible to a linear program and so is solvable.

However I suspect this is not the simplest route and that there should be a way of writing the primary variable solution in terms of the dual variable solutions.