🚀 go-pugleaf

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