# 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 cTx subject to A xb, x ≥ 0;

The dual problem,

Minimize bTy subject to ATyc, y ≥ 0.

The weak duality theorem states cTxbTy.

Namely, if ${\displaystyle (x_{1},x_{2},....,x_{n})}$ is a feasible solution for the primal maximization linear program and ${\displaystyle (y_{1},y_{2},....,y_{m})}$ is a feasible solution for the dual minimization linear program, then the weak duality theorem can be stated as ${\displaystyle \sum _{j=1}^{n}c_{j}x_{j}\leq \sum _{i=1}^{m}b_{i}y_{i}}$, where ${\displaystyle c_{j}}$ and ${\displaystyle b_{i}}$ are the coefficients of the respective objective functions.

Proof: cTx = xTcxTATybTy

### Generalizations

More generally, if ${\displaystyle x}$ is a feasible solution for the primal maximization problem and ${\displaystyle y}$ is a feasible solution for the dual minimization problem, then weak duality implies ${\displaystyle f(x)\leq g(y)}$ where ${\displaystyle f}$ and ${\displaystyle g}$ are the objective functions for the primal and dual problems respectively.