Dattorro Convex Optimization of Eternity II

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An Eternity II puzzle problem formulation LaTeX: A_{}x\!=\!b\, is discussed thoroughly in section 4.6.0.0.15 of book Convex Optimization & Euclidean Distance Geometry.  This LaTeX: A\, matrix is obtained by presolving a 864,593 LaTeX: \!\times\! 1,048,576 system. This Matlab binary contains three successive reductions, each equivalent to this larger system:

  • LaTeX: \tau\!\in\!\mathbb{R}^{11077} and LaTeX: \,E\!\in\!\mathbb{R}^{11077\times262144} is the million column Eternity II matrix having redundant columns removed,
  • LaTeX: \tilde{\tau}\!\in\!\mathbb{R}^{10054} and LaTeX: \,\tilde{E}\!\in\!\mathbb{R}^{10054\times204304} has columns removed corresponding to known zero variables,
  • LaTeX: b\!\in\!\mathbb{R}^{7362}   and LaTeX: A\!\in\!\mathbb{R}^{7362\times150638}  has columns removed not in smallest face (containing LaTeX: \tilde{\tau}) of polyhedral cone LaTeX: \mathcal{K}\triangleq\{\tilde{E}^{}x~|~x\!\succeq\!0\},
  • zero_varbs: row-vector identifying 57,840 columns removed from LaTeX: \,E to make LaTeX: \tilde{E}
  • zero_varbs_cone: row-vector identifying 53,666 more columns removed from LaTeX: \,E to make LaTeX: \,A;
    i.e., those columns not belonging to smallest face of LaTeX: \mathcal{K} containing LaTeX: \tilde{\tau}.

The following linear program is a very difficult problem:

LaTeX: \begin{array}{cl}\mbox{minimize}_x&c^{\rm T}x\\ \mbox{subject to}&A\,x=b\\ &x\succeq_{}\mathbf{0}\end{array}

Matrix LaTeX: A\!\in\!\mathbb{R}^{7362\times150638} is sparse having only 782,087 nonzeros. All entries of LaTeX: A\, are integers from the set LaTeX: \{{-1},0,1\}\,.

Vector LaTeX: b_{\!}\in\!\{0,1\}^{7362} has only 358 nonzeros.

Vector LaTeX: c\, is left unspecified because it is varied later in a Convex Iteration to find a minimal cardinality solution LaTeX: x\,.  Minimal cardinality of this Eternity II problem is equal to number of puzzle pieces, 256. Any minimal cardinality solution is binary and solves the Eternity II puzzle. Constraints LaTeX: A_{}x\!=\!b\, bound the variable from above by LaTeX: \mathbf{1}.

The technique, convex iteration, requires no modification (and works very well) when applied instead to mixed integer programming (MIP, not discussed in book). There is no modification to the linear program statement except 256 variables are declared binary.

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