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 minimization problem, called the primal problem, the solution to the primal problem is always greater than or equal to the solution to the dual maximization problem. Alternatively, the solution to a primal maximization 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

Consider a linear programming problem, 

where  is  and  is . The dual problem of () is

The weak duality theorem states that  for every solution  to the primal problem () and every solution  to the dual problem ().

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