Minimization question (lin. system subject to inequality constraints)

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3 total messages Started by dynaflow Thu, 13 Aug 1992 16:32

#430

Author: dynaflow
Date: Thu, 13 Aug 1992 16:32

34 lines
1452 bytes

I have the following problem, and not being familiar with 
minimization techniques, am appealing to net gurus for help

I need to solve the linear system 

              A x = b

where A is an NxN matrix 
      x is a N vector,
  and b another N vector,

subject to the constraint that each of the x_i , i=1,..,N are
positive.

The  thing to do seems to be to minimize 

|| Ax -b|| subject to  x_i > 0.


Now the question is how. I cannot formulate the above inequality as an
over-det system for example and use some standard SVD code. 

Having no idea of minimizing with inequality constraints (except some
hazy recollection of the simplex method from my undergraduate classes)
I  am at a loss. 
Any ideas, references, pointers or code would
be appreciated. 

Some more information:
The maximum dimension N is smaller than 40.
The ratio of the max and min. singular values of A is about 10^7.
The matrix is symmetric and guaranteed to be positive definite.

 Thanks in advance

Looking for CFD pointers

#431

Author: Charles Holden Winstead
Date: Thu, 13 Aug 1992 18:21

11 lines
901 bytes

Hey All,

I'm interested in learning how to numerically solve the incompressible 
Navier-Stokes equations, 2D.  My goal is to program my own solver.  I've
looked into formulating this myself using Finite Elements, but I was wondering
if there's a standard, accepted method for solving these.  I'm interested in
any references or suggestions.

Thanks,

-Chuck

<None>

#448

Author: dsimon
Date: Mon, 17 Aug 1992 22:54

27 lines
1129 bytes

> I have the following problem, and not being familiar with 
> minimization techniques, am appealing to net gurus for help
> 
> I need to solve the linear system 
> 
>               A x = b
> 
> where A is an NxN matrix 
>       x is a N vector,
>   and b another N vector,
> 
> subject to the constraint that each of the x_i , i=1,..,N are
> positive.
> 
> The  thing to do seems to be to minimize 
> 
> || Ax -b|| subject to  x_i > 0.
> 
> 
> Now the question is how. I cannot formulate the above inequality as an
> over-det system for example and use some standard SVD code. 

One method I've used is called Hildreth's method.  The reference is
Luenberger, "Optimization by Vector Space Methods," Wiley, 1968,
pages 299-300.  It's a pretty straight-forward iterative method.
----------
Dan Simon

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