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Re: [Help-glpk] Interior point failure

From: Meketon, Marc
Subject: Re: [Help-glpk] Interior point failure
Date: Tue, 18 Dec 2012 19:16:56 -0600

Perhaps solving the dual is better.

In general, if Y is an n x 1 vector (dependent variables), X is an n x p matrix 
(independent variables), b is an p x 1 vector of regression coefficients (the 
unknowns), the L1 regression problem is:

min ||Y - Xb||1 (the L1 norm of Y-Xb)

This has dual of:

max WY, s.t. WX = 0 and -1 <= Wi <= 1, where W are the unknown dual variables 
and is a 1 x n vector.

The interior point method, when applied to this dual problem, is essentially 
performing a series of weighted least squares on the original problem.  The 
dual values in this dual problem are the "b" in the original problem.  I 
somewhat suspect that this is an easier and more stable problem for GLPK, 
because the Cholesky factorization is of a p x p matrix instead of an n x n 
matrix (p is usually much smaller than n).

For the simple case in which p=2, with a constant and slope term (the cf12a.mod 
example), the GMPL code would be:


  Curve fitting problem 12.11(a) H P Williams "Model Building in Mathematical 

  Dr. H J Mackenzie
  HARD software


  modified to solve the dual:  Dr. M. Meketon


# set of points

set I;

# independent variable

param x {i in I};

# dependent variable

param y {i in I};

# output input values

printf {i in I} "x = %.1f; y = %.1f\n", x[i], y[i];

# define equation variables

var w{i in I}, >= -1, <= 1;

# define objective function

maximize dual_obj: sum {i in I} w[i]*y[i];

# define equation constraint

s.t. constant_term : sum{i in I} w[i] = 0;
s.t. slope_term: sum{i in I} w[i]*x[i] = 0;


# should be: y = 0.6375x + 0.5812
printf "y = %.4fx + %.4f\n", slope_term.dual, constant_term.dual;

 * DATA section


param : I :   x    y :=
        1     0    1
        2   0.5  0.9
        3     1  0.7
        4   1.5  1.5
        5   1.9    2
        6   2.5  2.4
        7     3  3.2
        8   3.5    2
        9     4  2.7
       10   4.5  3.5
       11     5    1
       12   5.5    4
       13     6  3.6
       14   6.6  2.7
       15     7  5.7
       16   7.6  4.6
       17   8.5    6
       18     9  6.8
       19    10  7.3


-----Original Message-----
From: address@hidden [mailto:address@hidden On Behalf Of Reginald Beardsley
Sent: Tuesday, December 18, 2012 12:22 PM
To: glpk
Subject: [Help-glpk] Interior point failure

I've been fooling around w/ L1 fits to a line w/ random errors in X & Y to 
collect performance profiling data.  The model is cf12a.mod w/ the data 
generated by an awk script.

At slightly over 700 points, I find that the interior point method fails to 
converge sometimes.  The number of failures in a run of 10 instances w/ 714 
points varies from run to run.  This is the case whether the data points vary 
from run to run or if the points are the same but the order in which they are 
supplied varies.

command is:

glpsol --interior -m tst.mod -d tst.dat

can anyone point me to an explanation of why?  I'm guessing this is a result of 
floating point arithmetic, but would like to better understand it.


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