Given the optimal solution of the dual problem, how to get the optimal solution of the primal problem?

Question

For linear programming, can we get the optimal solution of the primal problem based on the optimal solution of the dual problem? It seems that for dual problems, people are more concerned about the objective function value rather than the optimal solution.

Answer 1

Yes, solving the dual problem will give you a solution to the primal problem as well. Remember that each slack variable associated with a constraint in the primal problem corresponds to a decision variable in the dual problem. In fact, the dual decision variable is nonzero if and only if the corresponding primal constraint is satisfied with equality (i.e. the primal slack variable is zero). This is the notion of complimentary slackness. Once you know which constraints must hold with equality in the primal, you can obtain the values of the primal decision variables with a linear system solve.

It's been a while since I took linear programming, so double check what I said before using it in your own work.