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.
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]
Uses
editMany primal-dual approximation algorithms are based on the principle of weak duality.[3]
Weak duality theorem
editConsider a linear programming problem,
1 |
where is and is . The dual problem of (1) is
| 2 |
The weak duality theorem states that for every solution to the primal problem (1) and every solution to the dual problem (2).
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
Generalizations
editMore 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.
See also
editReferences
edit- ^ Boyd, S. P., Vandenberghe, L. (2004). Convex optimization (PDF). Cambridge University Press. ISBN 978-0-521-83378-3.
- ^ 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.