Geometric Presolver example

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Assume that the following problem is massive:
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Assume that the following optimization problem is massive:
<center>
<center>
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<math>\begin{array}{rl}\mbox{find}&x\\
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<math>\begin{array}{rl}\mbox{find}&x\in\mathbb{R}^n\\
\mbox{subject to}&E\,x=t\\
\mbox{subject to}&E\,x=t\\
&x\succeq_{}\mathbf{0}\end{array}</math>
&x\succeq_{}\mathbf{0}\end{array}</math>
</center>
</center>
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The problem is presumed solvable but not computable by any contemporary means.
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The problem is presumed solvable but not computable by any contemporary means.<br>
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The most logical strategy is to make the problem smaller.
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The most logical strategy is to somehow make the problem smaller.<br>
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Finding a smaller but equivalent problem is called ''presolving.''
[http://www.convexoptimization.com/TOOLS/EAndy.mat This Matlab workspace file]
[http://www.convexoptimization.com/TOOLS/EAndy.mat This Matlab workspace file]
contains a real <math>E</math> matrix having dimension <math>533\times 2704</math> and compatible <math>t</math> vector. There exists a cardinality <math>36</math> binary solution <math>x</math>. Before attempting to find it, we presume to have no choice but to reduce dimension of the <math>E</math> matrix prior to computing a solution.
contains a real <math>E</math> matrix having dimension <math>533\times 2704</math> and compatible <math>t</math> vector. There exists a cardinality <math>36</math> binary solution <math>x</math>. Before attempting to find it, we presume to have no choice but to reduce dimension of the <math>E</math> matrix prior to computing a solution.
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A lower bound on the number of rows of <math>\,E\in\mathbb{R}^{533\times 2704}\,</math> retained is <math>217</math>.<br>
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A lower bound on number of rows of <math>\,E\in\mathbb{R}^{533\times 2704}\,</math> retained is <math>217</math>.<br>
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A lower bound on the number of columns retained is <math>1104</math>.
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A lower bound on number of columns retained is <math>1104</math>.
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The present exercise is to determine those rows and columns using any contemporary presolver.
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The present exercise is to determine whether any contemporary presolver can meet this lower bound.

Revision as of 17:15, 11 April 2013

Assume that the following optimization problem is massive:

LaTeX: \begin{array}{rl}\mbox{find}&x\in\mathbb{R}^n\\ \mbox{subject to}&E\,x=t\\ &x\succeq_{}\mathbf{0}\end{array}

The problem is presumed solvable but not computable by any contemporary means.
The most logical strategy is to somehow make the problem smaller.
Finding a smaller but equivalent problem is called presolving.

This Matlab workspace file contains a real LaTeX: E matrix having dimension LaTeX: 533\times 2704 and compatible LaTeX: t vector. There exists a cardinality LaTeX: 36 binary solution LaTeX: x. Before attempting to find it, we presume to have no choice but to reduce dimension of the LaTeX: E matrix prior to computing a solution.

A lower bound on number of rows of LaTeX: \,E\in\mathbb{R}^{533\times 2704}\, retained is LaTeX: 217.
A lower bound on number of columns retained is LaTeX: 1104.

The present exercise is to determine whether any contemporary presolver can meet this lower bound.

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