If you are given an optimal primal solution x to an LP and y

If you are given an optimal primal solution x to an LP and you wish to deduce an optimal dual solution y.

Determine y using Complementary Slackness of y with x.

Solution

To detremine the optimal dual solution y* , first thing we need to ascertain is that y* is unique or not , that is we need to find the condition in which, of the many possible solutions of the dual problem, which one gives optimal solution.

Consider two conditions:

1]Complementary Slackness of y with x.

2]y* is a feasible solution of the dual problem i.e. contraints of the dual are satisfied .

Every optimal dual solution y is feasible and hence satisfies the second condition, and also by the other direction of the Complementary Slackness Theorem, since x*and y*are optimal to their respective LP\'s, We must have first condition satisfied.

Hope this helps!!