Dattorro Convex Optimization of Eternity II
From Wikimization
An Eternity II puzzle problem formulation is discussed thoroughly in section 4.6.0.0.15 of online book Convex Optimization & Euclidean Distance Geometry. This matrix is obtained by presolving a sparse 864,593 1,048,576 system. This Matlab binary contains three successive reductions, each equivalent to this larger system:
- and is the million column Eternity II matrix having redundant rows and columns removed analytically,
- and has columns removed corresponding to known zero variables (removal produces dependent rows),
- and has columns removed not in smallest face (containing ) of polyhedral cone ,
- zero_varbs: row-vector identifying 57,840 columns removed from to make
- zero_varbs_cone: row-vector identifying 53,666 more columns removed from to make ;
i.e., those columns not belonging to smallest face of containing .
The following linear program is a very difficult problem that remains unsolved:
Matrix is sparse having only 782,087 nonzeros. All entries of are integers from the set .
Vector has only 358 nonzeros.
Vector is left unspecified because it is varied later in a Convex Iteration to find a minimal cardinality solution . Constraints bound the variable from above by . Any minimal cardinality solution is binary and solves the Eternity II puzzle. Minimal cardinality of this Eternity II problem is equal to number of puzzle pieces, 256.
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.