User talk:Wotao.yin

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I regard the following as a very difficult problem, having spent considerable time with it.

LaTeX: \begin{array}{cl}\mbox{minimize}_X&c^{\rm T}\mbox{vec}\,X\\
\mbox{subject to}&A\,\mbox{vec}\,X=b\\
&X^{\rm T\!}X=I\\
&X\geq_{}\mathbf{0}\end{array}

Nonnegative rectangular submatrix LaTeX: \,X\!\in\mathbb{R}^{1024\times256}\, comes directly from a permutation matrix LaTeX: \,\Xi\!\in\!\mathbb{R}^{1024\times1024}\, having three out of every four consecutive columns discarded.   This discard occurs because of structural redundancy in LaTeX: \Xi\,.

Notation LaTeX: \mbox{vec}\,X\!\in\mathbb{R}^{262144} denotes vectorization; it means, the columns of LaTeX: \,X are stacked with column 1 on top and column 256 on the bottom.

Matrix LaTeX: A\!\in\!\mathbb{R}^{10565\times262144} is sparse having only 979,444 nonzeros.   All its entries are integers from the set LaTeX: \{{-1},0,1,2\}\,.  The 2 appears only in the fifth row from the bottom of LaTeX: A\,.

Vector LaTeX: b\, is quite sparse having only a single nonzero entry: LaTeX: 1\,.

A Matlab binary contains matrices LaTeX: \,A and LaTeX: \,b.   Vector LaTeX: c\, is left unspecified because I want to vary it later as part of a Convex Iteration.   Vector LaTeX: c\, may arbitrarily be set to LaTeX: \mathbf{0} or LaTeX: \mathbf{1}, for your purposes, but leave a hook for it in case you require another value.

A good presolver can eliminate about 50,000 columns of LaTeX: \,A because one of the constraints (fifth row from the bottom of LaTeX: \,A\,) has only nonnegative entries.   This means that about 50,000 entries in permutation submatrix LaTeX: X\, can be set to zero before numerical solution begins.   The Matlab binary possesses all 262,144 columns of LaTeX: A\,;   none of its columns have yet been discarded by a presolve.

--Dattorro 03:31, 5 November 2010 (PDT)

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