Weak duality
In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0. This means that for any maximization problem, called the primal problem, the solution to the primal problem is always less than or equal to the solution to the dual (minimization) problem. Weak duality is in contrast to strong duality, which states that the primal optimal objective and the dual optimal objective are equal. Strong duality only holds in certain cases. Uses
Many primal-dual approximation algorithms are based on the principle of weak duality. Weak duality theorem
The primal problem:
 Maximize   subject to ;
The dual problem,
 Minimize    subject to . The weak duality theorem states . Namely, if  is a feasible solution for the primal maximization linear program and  is a feasible solution for the dual minimization linear program, then the weak duality theorem can be stated as 
, where  and  are the coefficients of the respective objective functions. Proof:

Generalizations
More generally, if  is a feasible solution for the primal maximization problem and  is a feasible solution for the dual minimization problem, then weak duality implies  where  and  are the objective functions for the primal and dual problems respectively. See also
Convex optimization
Max–min inequality

References

Linear programming
Convex optimization