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Re: Linear Programing

From: Etienne Grossmann
Subject: Re: Linear Programing
Date: Fri, 24 Nov 2000 08:05:19 +0000
User-agent: WEMI/1.13.7 (Shimada) FLIM/1.13.2 (Kasanui) Emacs/20.7 (i386-debian-linux-gnu) (with unibyte mode)


 Jeff's program does

       max f'*x     subject to   A*x == b and x >= 0

 And you want :

 LP     Linear programming.
        X=LP(f,A,b) solves the linear programming problem:

      min f'x    subject to:   Ax <= b

  I think you can transform your problem into a problem that Jeff's
program can solve. The reasoning (no warranty of correctness : I would
not at all be surprised if there were some pitfall I fell in) is :

  Your problem is equivalent to (set y = A*x - b, y should be >= 0)

      max f'*x         subject to   A*x - y == b and x >= 0 and y>=0

  which is equiv. to            (rewrite in vectorial form) 

      max [f',0]*[x;y] subject to   [A,-eye]*[x;y] == b and [x;y]>=0

  equiv. to 
  (set x = x1 - x2 with x1 >= 0 and x2 >= 0, idem for y = y1 - y2)

      max [f',-f',0,0]*[x1;x2;y1;y2] 

      subject to   [A,-A,-eye,-eye]*[x1;x2;y1;y2] == b 
      and          [x1;x2;y1;y2]>=0

  Which is the kind of problem that Jeff's function solves. 


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