Linear programming/seeing feasibility and unboudedness

Question

Consider the dual linear programming problem and its simplex dually feasible table: $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline -4& 0 & 1&5&16&0&4&0 \\ \hline -12& 0 & -8&-1&-7&0&-3&-1 \\ \hline -1& 1 & 1&1&1&1&1&-1 \\ \hline \end{array}$$ In the first row is our objective function. Now see the second row: can we easily determine that the dual problem is either infeasible or unbounded? What is an intuition for this observation, how can we see the unboudednes or infeasibillity for the dual problem, whichever is the acutal case? I.e. how can we modify the solution to see this?

Answer 1

In primal simplex, if for any nonbasic column that can enter the basis (positive reduced cost), all the elements are non-positive, then the problem is unbounded. In dual simplex, if there is an entering variable (negative rhs) but all the elements in front of it in the row are non-negative, then the dual problem is unbounded. You can also refer to this answer.