least-squares solutions of linear systems

Hello, I'm trying to find a way to determine what elements of a least-squares solution may come out as negative numbers. My linear system A*x = b has h = [h1, h2,..., h8] as the minimun-norm l.s.s. and h6 <0, h7 <0. I would like to know what is the maximum value of h6 (and h7) such that h is still l.s.s but not min.norm. If I transform the system into A1*y = b1 with y = x - eps, and use lsqnonneg(A1,b1), then I get g = [g1, g2,..., g8] having g6 always equal to that small non-negative eps. That makes me think the constraint is artificial and no l.s.s. of A*x = b will ever have all elements > 0. Could I prove this mathematically? Could I say without solving the system "least-squares solution has exactly 2 negative elements"? Thanks in advance.

Answer 1

[Matt J]

Minimize

-h(6)  %or h(7)

subject to the optimality constraint,

A.'*(A*x-b)=0