# 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 is a feasible solution for the primal minimization linear program and is a feasible solution for the dual maximization linear program, then the weak duality theorem can be stated as , where and are the coefficients of the respective objective functions.

More generally, if is a feasible solution for the primal minimization problem and is a feasible solution for the dual maximization problem, then weak duality implies where and are the objective functions for the primal and dual problems respectively.