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.

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]

^ 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.

[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.

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