I have a solution space of an under-determined linear system Ax = b with n x m matrix A:
$$x= x0 + V2 * c (1)$$
with [U, S, V] = svd(A); V2 = V(:,r+1:end); $x0 = A^+ b; $ r = rank(A);
I need only non-negative solutions for x, so I want to intersect K-dim manifold defined in (1) with the positive m-space, $R^m_+$. As an intersection of convex sets - if it exists - it is convex too and is a finite convex K-Dim polyhedron in m-space. Currently I derive the vertices of this polyhedron by solving
Loop over all Idx = combinations 1:k out of 1:m
c=V2(Idx,:)\-x0(Idx);
vertex=x0+V2*c
accept vertex if ~any(vertex<0)
This loop is slow as there are the number of combinations is large and I need a faster method for finding this intersection and was wondering if there are more direct approaches for interection the solution manifold with $R^m_+$?