Matrix: Dattorro/EternityII_Etilde

Description: Dattorro Convex Optimization of Eternity II (second reduction)

Dattorro/EternityII_Etilde graph
(bipartite graph drawing)

Dattorro/EternityII_Etilde dmperm of Dattorro/EternityII_Etilde

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  • download as a MATLAB mat-file, file size: 2 MB. Use UFget(2382) or UFget('Dattorro/EternityII_Etilde') in MATLAB.
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    Matrix properties
    number of rows10,054
    number of columns204,304
    structural full rank?yes
    structural rank10,054
    # of blocks from dmperm5
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    Cholesky candidate?no
    positive definite?no

    authorJ. Dattorro
    editorT. Davis
    kindoptimization problem
    2D/3D problem?no

    Additional fieldssize and type
    bsparse 10054-by-1


    Dattorro Convex Optimization of Eternity II, Jon Dattoro           
    An Eternity II puzzle ( problem         
    formulation A*x=b is discussed thoroughly in section 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 and very sparse.                              
       * 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 and very sparse.                         
       * 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.                                                   
    For more details, see                                                    
    /Dattorro_Convex_Optimization_of_Eternity_II (url is wrapped),     
    and .                        

    Ordering statistics:result
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD683,079,081
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD12,141,269

    SVD-based statistics:
    null space dimension0
    full numerical rank?yes

    singular values (MAT file):click here
    SVD method used:s = svd (full (R)) ; where [~,R,E] = spqr (A') with droptol of zero

    Dattorro/EternityII_Etilde svd

    For a description of the statistics displayed above, click here.

    Maintained by Tim Davis, last updated 12-Mar-2014.
    Matrix pictures by cspy, a MATLAB function in the CSparse package.
    Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.