Let's say we apply dual simplex method in the process of doing a gomory cut in integer programming. Is the "algorithm" the same when working with the dual problem?
I ask this because we sometimes get as exercise to first take the dual of the original problem, then to continue with the exercise (applying the simplex method) and find an integer solution for the original problem (where dual simplex comes into play). The exercise asks for the dual problem to help us reduce the number of variables (for example to be able to visualize the different steps of the tableau in a 2D plot). I know that to get the final result from a dual problem for the original problem, I have to look at the coefficients in the tableau and not at the value of the variables, but I am not sure whether the dual simplex method for gomory cut stays the same when applying it to the dual problem.
Example exercise:
constraints
objective
(a) Formulate the dual form.
(b) Convert the system to second primal form.
(c) Simulate the simplex method.
(d) If the solution is not an integer, calculate the Gomory cut.
When working with linear programming (and real-valued variables), we are always free to switch to the dual instead, because strong duality holds. The primal and dual have the same optimal value, and if we solve the dual, we will even be able to recover an optimal primal solution.
With integer linear programming, this is no longer true. All you know is that the LP relaxation of the primal agrees in optimal value with the dual. It is not even meaningful to consider the dual as an integer program, and even if you did, solving that integer program (using fractional cuts or otherwise) would not help us with the primal.
Each stage of the cutting plane algorithm can be solved by using the dual instead of the primal. However:
In order to add a Gomory cut, you need an inequality with fractional coefficients that's true for all primal feasible solutions and tight at the optimal solution to the LP relaxation. This is straightforward to do from the simplex method if you're solving the primal: just take a row of the tableau corresponding to a variable with a fractional value. It's not straightforward if you're solving the dual.
When we add a cutting plane, we add a new constraint to the primal. This corresponds to adding a new variable to the dual. This eliminates the primary advantage of using the dual (to reduce the number of variables).
Unless you're using the revised simplex method, it is trickier to add a new variable to a tableau than it is to add a new constraint.
I do not recommend it.