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least-squares solutions of linear systems

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minibela
minibela 2011-10-7
回答： Matt J 2017-10-12
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.

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Matt J
Matt J 2017-10-12
Minimize
-h(6) %or h(7)
subject to the optimality constraint,
A.'*(A*x-b)=0

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