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<H1>dual linear program</H1>
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(definition)

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<strong>Definition:</strong>
Every <a href="linearprogrm.html" tppabs="http://hissa.nist.gov/dads/HTML/linearprogrm.html"><em>linear program</em></a> has a corresponding linear program called the dual.  It is  max<sub>y</sub> {b &middot; y | A<sup>T</sup>y  <img src="leq.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/leq.gif" border=0 height=7  width=14 alt="less than or equal to"> c  and y  <img src="geq.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/geq.gif" border=0 height=7  width=14 alt="greater than or equal to">  0 }.  For any solution x to the original linear program and any solution y to the dual we have c  &middot; x  <img src="geq.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/geq.gif" border=0 height=7  width=14 alt="greater than or equal to"> (A<sup>T</sup> y)<sup>T</sup> x = y<sup>T</sup>(Ax)  <img src="geq.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/geq.gif" border=0 height=7  width=14 alt="greater than or equal to"> y  &middot; b.  For optimal x and y, equality holds.  For a problem formulated as an <a href="integerliner.html" tppabs="http://hissa.nist.gov/dads/HTML/integerliner.html"><em>integer linear program</em></a>, a solution to the dual of a <a href="relaxation.html" tppabs="http://hissa.nist.gov/dads/HTML/relaxation.html"><em>relaxation</em></a> of the program can serve as <a href="witness.html" tppabs="http://hissa.nist.gov/dads/HTML/witness.html"><em>witness</em></a>.
<P><em>Note:
From Algorithms and Theory of Computation Handbook, page 34-17, Copyright &copy; 1999 by CRC Press LLC.</em>
<P>Author: <a href="terms.html#authorCRC-A" tppabs="http://hissa.nist.gov/dads/terms.html#authorCRC-A">CRC-A</a>

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Entry modified Thu Jun 17 09:56:40 1999.<BR>
HTML page formatted Wed Dec 22 09:35:12 1999.

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