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.[1]: 225 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.[2]
The weak duality theorem states that for every solution to the primal problem (1) and every solution to the dual problem (2).
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:
cTx
= xTc
≤ xTATy
≤ bTy
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.