Matrix: Dattorro/EternityII_E

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

Dattorro/EternityII_E graph
(bipartite graph drawing)


Dattorro/EternityII_E dmperm of Dattorro/EternityII_E

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  • Matrix group: Dattorro
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  • download as a MATLAB mat-file, file size: 2 MB. Use UFget(2381) or UFget('Dattorro/EternityII_E') in MATLAB.
  • download in Matrix Market format, file size: 5 MB.
  • download in Rutherford/Boeing format, file size: 3 MB.

    Matrix properties
    number of rows11,077
    number of columns262,144
    nonzeros1,503,732
    structural full rank?yes
    structural rank11,077
    # of blocks from dmperm5
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    typeinteger
    structurerectangular
    Cholesky candidate?no
    positive definite?no

    authorJ. Dattorro
    editorT. Davis
    date2011
    kindoptimization problem
    2D/3D problem?no

    Additional fieldssize and type
    bsparse 11077-by-1

    Notes:

    Dattorro Convex Optimization of Eternity II, Jon Dattoro           
                                                                       
    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 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                                              
    http://www.convexoptimization.com/wikimization/index.php           
    /Dattorro_Convex_Optimization_of_Eternity_II (url is wrapped),     
    and https://ccrma.stanford.edu/~dattorro/ .                        
    

    Ordering statistics:result
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD867,810,002
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD11,712,296

    SVD-based statistics:
    norm(A)264.319
    min(svd(A))7.18761e-14
    cond(A)3.67743e+15
    rank(A)11,076
    sprank(A)-rank(A)1
    null space dimension1
    full numerical rank?no
    singular value gap3.43781e+12

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

    Dattorro/EternityII_E 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.