# Weak duality

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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.

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

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