Netlib LP problem dfl001: minimize c'*x, where Ax=b, lo<=x<=hi
Name lp_dfl001
Group LPnetlib
Matrix ID 619
Num Rows 6,071
Num Cols 12,230
Nonzeros 35,632
Pattern Entries 35,632
Kind Linear Programming Problem
Symmetric No
Date 1990
Author M. Meketon
Editor D. Gay
Structural Rank 6,071
Structural Rank Full true
Num Dmperm Blocks 1
Strongly Connect Components 1
Num Explicit Zeros 0
Pattern Symmetry 0%
Numeric Symmetry 0%
Cholesky Candidate no
Positive Definite no
Type real
SVD Statistics
Matrix Norm 1.591695e+01
Minimum Singular Value 4.036579e-15
Condition Number 3.943179e+15
Rank 6,058
sprank(A)-rank(A) 13
Null Space Dimension 13
Full Numerical Rank? no
Download Singular Values MATLAB
Download MATLAB Rutherford Boeing Matrix Market
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.