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On linear least-squares problems with diagonally dominant weight
matrices

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Anders Forsgren

The solution of the unconstrained weighted linear least-squares
problem is known to be a convex combination of the basic solutions
formed by the nonsingular subsystems if the weight matrix is diagonal
and positive definite. In particular, this implies that the norm of
this solution is uniformly bounded for any diagonal and positive
definite weight matrix. In addition, the solution set is known to be
the relative interior of a finite set of polytopes if the weight
matrix varies over the set of positive definite diagonal matrices. In
this paper, these results are reviewed, and generalized to the set of
weight matrices that are symmetric, positive semidefinite, diagonally
dominant and give unique solution to the least-squares problem. This
is done by means of a particular symmetric diagonal decomposition of
the weight matrix, giving a finite number of diagonally weighted
problems, but in a space of higher dimension. Extensions to
equality-constrained weighted linear least-squares problems are
given. A discussion of why the boundedness properties do not hold for
general symmetric positive definite weight matrices is given. The
motivation for this research is from interior methods for
optimization.
SIAM Journal on Matrix Analysis and Applications 17 (1996), 763-788.

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