# Duality theorem

**DOI:**https://doi.org/10.1007/1-4020-0611-X_261

A theorem concerning the relationship between the solutions of primal and dual linear-programming problems. One form of the theorem is as follows: If either the primal or the dual has a finite optimal solution, then the other problem has a finite optimal solution, and the optimal values of their objective functions are equal. From this it can be shown that for any pair of primal and dual linear programs, the objective value of any feasible solution to the minimization problem is greater than or equal to the objective value of any feasible solution to the dual maximization problem. This implies that if one of the problems is feasible and unbounded, then the other problem is infeasible. Examples exist for which the primal and its dual are both infeasible. Another form of the theorem states: if both problems have feasible solutions, then both have finite optimal solutions, with the optimal values of their objective functions equal. Strong duality theorem.