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.
Many primal-dual approximation algorithms are based on the principle of weak duality.
Weak duality theorem
The primal problem:
- Maximize cTx subject to A x ≤ b, x ≥ 0;
The dual problem,
- Minimize bTy subject to ATy ≥ c, y ≥ 0.
The weak duality theorem states cTx ≤ bTy.
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: 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 where and are the objective functions for the primal and dual problems respectively.
- 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.