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.

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.

So, in short: weak duality states that any solution feasible for the dual problem is an upper bound to the solution of the primal 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] Many primal-dual approximation algorithms are based on the principle of weak duality.

[3] Consider a linear programming problem, where

The dual problem of (1) is

minimize

subject to

{\displaystyle {\begin{aligned}{\underset {y\in \mathbb {R} ^{m}}{\text{minimize}}}\quad &b^{\top }y\\{\text{subject to}}\quad &A^{\top }y\geq c,\\&y\geq 0.\end{aligned}}}

The weak duality theorem states that

for every solution

to the primal problem (1) and every solution

to the dual problem (2).

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

are the coefficients of the respective objective functions.

Proof:

cTx = xTc ≤ xTATy ≤ bTy

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

are the objective functions for the primal and dual problems respectively.