## Group Dattorro

Group Description ```Dattorro Convex Optimization of Eternity II, Jon Dattoro http://www.convexoptimization.com/wikimization/index.php/Dattorro_Convex_Optimization_of_Eternity_II An Eternity II puzzle (http://www.eternityii.com/) problem formulation A*x=b is discussed thoroughly in section 4.6.0.0.15 of the book Convex Optimization & Euclidean Distance Geometry which is freely available. That A matrix is obtained by presolving a sparse 864,593-by-1,048,576 system. The 3 problems in this set contains three successive reductions, each equivalent to that larger system: * EternityII_E: a 11077-by-262144 system E*x=tau, where tau is 11077-by-1. This is the million column Eternity II matrix having redundant rows and columns removed analytically. In the UF Collection, E is the Problem.A matrix, and tau is Problem.b. All entries in E are from the set {-1,0,1,2}. tau is binary. * EternityII_Etilde: a 10054-by-204304 system Etilde*x=tautilde with tautilde of size 10054-by-1. The system has columns removed corresponding to some known zero variables (removal produced dependent rows). In the UF Collection, Etilde is the Problem.A matrix, and tautilde is Problem.b. All entries in Etilde are from the set {-1,0,1}. tautilde is binary. * EternityII_A: a 7362-by-150638 system A*x=b, where b is 7362-by-1. This system has columns removed not in smallest face (containing tautilde) of polyhedral cone K = { Etilde*x | x >= 0 }. The following linear program is a very difficult problem that remains unsolved: maximize_x z'*x, subject to A*x=b and x >= 0 The matrix A in the EternityII_A problem is sparse, having only 782,087 nonzeros. All entries of A are integers from the set {-1,0,1}. The vector b is binary, with only 358 nonzeros. Direction vector z is determined by Convex Iteration: maximize_z z'*x^{star} subject to 0 <= z <= 1 and z'*1 = 256 (for a vector x, x >= 0 means all(x>=0) in MATLAB notation). These two problems are iterated to find a minimal cardinality solution x. Constraint A*x=b bounds the variable from above by 1. Any minimal cardinality solution is binary and solves the Eternity II puzzle. The Eternity II puzzle is solved when z^{star}'*x^{star} = 256. Minimal cardinality of this Eternity II problem is equal to number of puzzle pieces, 256. Comment: 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 here except 256 variables, corresponding to the largest entries of iterate x^{star}, are declared binary.```
Displaying all 3 collection matrices
Id Name Group Rows Cols Nonzeros
2381 EternityII_E Dattorro 11,077 262,144 1,503,732
2382 EternityII_Etilde Dattorro 10,054 204,304 1,170,516
2383 EternityII_A Dattorro 7,362 150,638 782,087