In linear programming we sometimes use artificial variables for the simplex method when constraints are expressed as equalities. After a certain number of iterations of the simplex method we reach a point in which all coefficients are non negative. What does a nonzero value of the artificial variable indicate at that point? Infeasible solution? Error in the computation? Would you be able to illustrate with an example?
Edit: This is an example. Initial simplex tableau on the right. I start the simplex method by choosing entering-leaving variable and I stop if there are no negative coefficients in the last row. What does a nonzero value for the artificial variable mean at this point?
My understanding is that you make an initial optimization pass with an objective function that maximizes the negative of the artificial variables. If these artificial variables are set to zero from that pass, then the problem is feasible, and you can then eliminate them and continue on with a second pass using the real objective function. If the artificial variables are not zero (after that first pass) then the original problem was not feasible. (I believe some texts describe this as Phase I and Phase II.)
(Note that in CLRS (the textbook), instead of artificial variables, it prescribes replacing an equality constraint $ax=b$ with two counteracting inequality constraints $ax\le b$ and $-ax \le -b$)