I learned about the simplex method, how to pivot by hand before commercial grade solvers were introduced. And I’m still a little foggy on what slack variables and objectives are actually doing.
My current understanding is that pivots are based on Gaussian elimination. This method solves systems of equations and can’t (directly) accept inequalities as inputs.
So the simplex method introduces slack variables and a tableau. At ever pivot, a new system of equations is dynamically generated and it either does (or does not) optimize the objective.
The optimal solution will always occur an intersection of variables.
Once the negative numbers are purged from the bottom row of the tableau, the system no longer generates new systems of equations and has arrived at the optima.
So far, am on I on the right track?