# 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. That means the solution to the primal (minimization) problem is *always* greater than or equal to the solution to an associated dual problem. This is opposed to strong duality which only holds in certain cases.[1]

## Uses

Many primal-dual approximation algorithms are based on the principle of weak duality.[2]

## Weak duality theorem

The *primal* problem:

- Maximize
**c**^{T}**x**subject to*A***x**≤**b**,**x**≥ 0;

The *dual* problem,

- Minimize
**b**^{T}**y**subject to*A*^{T}**y**≥**c**,**y**≥ 0.

The weak duality theorem states **c**^{T}**x** ≤ **b**^{T}**y**.

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:
**c**^{T}**x**
= **x**^{T}**c**
≤ **x**^{T}*A*^{T}**y**
≤ **b**^{T}**y**

### 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

## References

- Boţ, Radu Ioan; Grad, Sorin-Mihai; Wanka, Gert (2009),
*Duality in Vector Optimization*, Berlin: Springer-Verlag, p. 1, doi:10.1007/978-3-642-02886-1, ISBN 978-3-642-02885-4, MR 2542013. - Gonzalez, Teofilo F. (2007),
*Handbook of Approximation Algorithms and Metaheuristics*, CRC Press, p. 2-12, ISBN 9781420010749.