For any linear program,  we have one of 3 alternatives: it has an optimal solution (opt), it is unbounded (nub), it is infeasible (inf).

The theorem on 4 alternatives gives the following 4 joint alternatives for the program and its dual:



opt

inf

unb

opt

√ min = max

-

-

inf

-

unb

-

-


where √ means possible and - means impossible.


In the  (opt, opt) case,  the optimal values for the primal and dual programs are the same.

This outcome  (the duality theorem) can be stated as follows: if both programs are feasible, they have optimal solutions and the optimal values are the same.

Another version is: if a linear program has an optimal solution, then the dual program has an optimal solution and the optimal values art the same.