• SJSU Singular Matrix Database
  • Matrix group: LPnetlib
  • Click here for a description of the LPnetlib group.
  • Click here for a list of all matrices
  • Click here for a list of all matrix groups


  • Matrix: LPnetlib/lp_dfl001
  • Description: Netlib LP problem dfl001: minimize c'*x, where Ax=b, lo<=x<=hi
  • download as a MATLAB mat-file, file size: 192 KB. Use SJget(512) or SJget('LPnetlib/lp_dfl001') in MATLAB.
  • download in Matrix Market format, file size: 182 KB.
  • download in Rutherford/Boeing format, file size: 154 KB.

    LPnetlib/lp_dfl001

    Routine svd from Matlab 7.6.0.324 (R2008a) used to calculate the singular values.

    LPnetlib/lp_dfl001

    Matrix properties (click for a legend)  
    number of rows6,071
    number of columns12,230
    structural full rank?yes
    structural rank6,071
    numerical rank 6,058
    dimension of the numerical null space6,172
    numerical rank / min(size(A))0.99786
    Euclidean norm of A 15.917
    calculated singular value # 60580.045648
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    2.1725e-011
    calculated singular value # 60595.322e-015
    gap in the singular values at the numerical rank:
    singular value # 6058 / singular value # 6059
    8.5771e+012
    calculated condition number4.162e+015
    condest-2
    nonzeros35,632
    # of blocks from dmperm1
    # strongly connected comp.1
    entries not in dmperm blocks0
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    typereal
    structurerectangular
    Cholesky candidate?no
    positive definite?no

    authorM. Meketon
    editorD. Gay
    date1990
    kindlinear programming problem
    2D/3D problem?no
    SJid512
    UFid619

    Additional fieldssize and type
    bfull 6071-by-1
    cfull 12230-by-1
    lofull 12230-by-1
    hifull 12230-by-1
    z0full 1-by-1

    Notes:

    A Netlib LP problem, in lp/data.  For more information                    
    send email to netlib@ornl.gov with the message:                           
                                                                              
    	 send index from lp                                                      
    	 send readme from lp/data                                                
                                                                              
    The following are relevant excerpts from lp/data/readme (by David M. Gay):
                                                                              
    The column and nonzero counts in the PROBLEM SUMMARY TABLE below exclude  
    slack and surplus columns and the right-hand side vector, but include     
    the cost row.  We have omitted other free rows and all but the first      
    right-hand side vector, as noted below.  The byte count is for the        
    MPS compressed file; it includes a newline character at the end of each   
    line.  These files start with a blank initial line intended to prevent    
    mail programs from discarding any of the data.  The BR column indicates   
    whether a problem has bounds or ranges:  B stands for "has bounds", R     
    for "has ranges".  The BOUND-TYPE TABLE below shows the bound types       
    present in those problems that have bounds.                               
                                                                              
                           PROBLEM SUMMARY TABLE                              
                                                                              
    Name       Rows   Cols   Nonzeros    Bytes  BR      Optimal Value         
    DFL001     6072  12230    41873     353192  B     1.12664E+07 **          
                                                                              
            BOUND-TYPE TABLE                                                  
    DFL001     UP                                                             
                                                                              
    Submitted by Marc Meketon.                                                
                                                                              
    DFL001, says Marc Meketon, "is a 'real-world' airline schedule planning   
    (fleet assignment) problem.  This LP was preprocessed by a modified       
    version of the KORBX(r) System preprocessor.  The problem reduced in      
    size (rows, columns, non-zeros) significantly.  The row and columns were  
    randomly sorted and renamed, and a fixed adjustment to the objective      
    function was eliminated.  The name of the problem is derived from the     
    initials of the person who created it."                                   
                                                                              
    Bob Bixby reports that the CPLEX solver (running on a Sparc station)      
    finds slightly different optimal values for some of the problems.         
    On a MIPS processor, MINOS version 5.3 (with crash and scaling of         
    December 1989) also finds different optimal values for some of the        
    problems.  The following table shows the values that differ from those    
    shown above.  (Whether CPLEX finds different values on the recently       
    added problems remains to be seen.)                                       
                                                                              
    Problem        CPLEX(Sparc)          MINOS(MIPS)                          
    DFL001       1.1266396047E+07            **                               
                                                                              
    David Gay reports:                                                        
    ** On an IEEE-arithmetic machine (an SGI 4D/380S), I (dmg) succeeded in   
    getting MINOS 5.3 to report optimal objective values, 1.1261702419E+07    
    and 1.1249281428E+07, for DFL001 only by starting with LOAD files         
    derived from the solution obtained on the same machine by Bob             
    Vanderbei's ALPO (an interior-point code); starting from one of the       
    resulting "optimal" bases, MINOS ran 23914 iterations on a VAX before     
    reporting an optimal value of 1.1253287141E+07.  When started from the    
    same LOAD file used on the SGI machine, MINOS on the VAX reported an      
    optimal value of 1.1255107696E+07.  Changing the FEASIBILITY TOLERANCE    
    to 1.E-10 (from its default of 1.E-6) led MINOS on the SGI machine to     
    report "optimal" values of 1.1266408461E+07 and 1.1266402835E+07.  This   
    clearly is a problem where the FEASIBILITY TOLERANCE, initial basis, and  
    floating-point arithmetic strongly affect the "optimal" solution that     
    MINOS reports.  On the SGI machine, ALPO with SPLIT 3 found               
     primal:  obj value =  1.126639607e+07      FEASIBLE   ( 2.79e-09 )       
     dual:    obj value =  1.126639604e+07      FEASIBLE   ( 1.39e-16 )       
                                                                              
    Bob Bixby reports the following about his experience solving DFL001       
    with CPLEX:                                                               
      First, the value for the objective function that I get running          
      defaults is 1.1266396047e+07, with the following residuals:             
                                                                              
      Max. unscaled (scaled) bound        infeas.: 4.61853e-14 (2.30926e-14)  
      Max. unscaled (scaled) reduced-cost infeas.: 6.40748e-08 (6.40748e-08)  
      Max. unscaled (scaled) Ax-b          resid.: 4.28546e-14 (4.28546e-14)  
      Max. unscaled (scaled) c_B-B'pi      resid.: 8.00937e-08 (8.00937e-08)  
                                                                              
      The L_infinity condition number of the (scaled) optimal basis is        
      213737.  I got exactly the same objective value solving the problem in  
      several different ways.  I played a bit trying to get a better          
      reduced-cost infeasibility, but that seems hopeless (if not pointless)  
      given the c-Bpi residuals.                                              
                                                                              
      Just as an aside, this problem exhibits very interesting behavior when  
      solved using a simplex method.  I ran reduced-cost pricing on it in     
      phase I, with the result that it took 465810 iterations to get          
      feasible.  Running the default CPLEX pricing scheme, the entire         
      problem solved in 94337 iterations (33059 in phase I) on a              
      Sparcstation.  Steepest-edge pricing (and a different scaling) took     
      25803 iterations.  This is a nasty problem.                             
                                                                              
                                                                              
    Added to Netlib on  11 Oct. 1990                                          
    9 Jan. 1991: Bixby's remarks about DFL001 added to lp/data/readme.        
                                                                              
    

    Ordering statistics:AMD METIS
    nnz(V) for QR, upper bound nnz(L) for LU9,529,914 7,249,767
    nnz(R) for QR, upper bound nnz(U) for LU1,544,706 1,383,665

    Maintained by Leslie Foster, last updated 24-Apr-2009.

    Entries 5 through 14 in the table of matrix properties and the singular
    value plot were created using SJsingular code. The other plots
    and statistics are produced using utilities from the SuiteSparse package.
    Matrix color plot pictures by cspy, a MATLAB function in the CSparse package.