(definition)

**Definition:**
Every *linear program* has a corresponding linear program called the dual. It is max_{y}{b· y | A^{T}y ≤ c and y ≥ 0}. For any solution x to the original linear program and any solution y to the dual we have c · x ≥ (A^{T} y)^{T} x = y^{T}(Ax) ≥ y · b. For optimal x and y, equality holds. For a problem formulated as an *integer linear program*, a solution to the dual of a *relaxation* of the program can serve as *witness*.

*Note:
From Algorithms and Theory of Computation Handbook, page 34-17, Copyright © 1999 by CRC Press LLC. Appearing in the Dictionary of Computer Science, Engineering and Technology, Copyright © 2000 CRC Press LLC.*

Author: CRC-A

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Entry modified 17 December 2004.

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Cite this as:

Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "dual linear program", in
*Dictionary of Algorithms and Data Structures* [online], Vreda Pieterse and Paul E. Black, eds. 17 December 2004. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/duallinear.html