add feature inequality constraints (all xi>0) to an existing solver "minimize{||Ax-b||}"

Question

I have a library that can solve minimize{||Ax-b||} (Linear least squares).

• A is a matrix, while x and b is a vector.
• It is possible that A is not invertible.
• A is not a square matrix.

Now, I want to solve it with additional constraints : all xi>0.

Question

How to adapt/encapsulate the solver to support constraints all xi>0?

I prefer to think that the existing library is a blackbox function - solve(A,b,x).
I want to encapsulate it to be solvePositive(A,b,x).

For an answer, I expect rough idea, i.e. it doesn't need to contain any code at all.

Edit: .... or should I write a new algorithm instead?
If so, what are the names of applicable algorithms?
I seek for a beginner-friendly guide.

My search

• An answer in SO describes something around LSQR.

• An answer at math stackexchange describes how to do it, but with Lagrangian style. It is a wonderland for me.

• The 3rd post in Eigen library's forum offer a code, but I don't find description how the algorithm works.