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

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Many primal-dual approximation algorithms are based on the principle of weak duality.[3]

Weak duality theorem

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Consider 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 = xTcxTATybTy

Generalizations

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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.

See also

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References

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  1. ^ Boyd, S. P., Vandenberghe, L. (2004). Convex optimization (PDF). Cambridge University Press. ISBN 978-0-521-83378-3.
  2. ^ 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.
  3. ^ Gonzalez, Teofilo F. (2007), Handbook of Approximation Algorithms and Metaheuristics, CRC Press, p. 2-12, ISBN 9781420010749.