Re: Minimizing the Frobenius norm

Peter,

So the thing that I am stumbling on is the fact that both Ti and Tj are
defined in terms of the unknown v:

Considering a single term in the inner summation, don't I really have
the following:

|| Ti * v - Tj * v||^2

I am not seeing how this turns into the familiar Ax - b = 0 type of
problem. If I knew either Ti or Tj then it easily fits into that form.
Treating (Ti - Tj) as the A matrix and b as a zero-filled vector
doesn't seem correct as the optimal value for x would always end up
being a zero-filled vector: Ax = 0

Am I missing the obvious?

P.S.

The v are linearly related to the T so it is indeed a linear
optimization problem.


the frobenius norm squared is the sum of the squares of the elements of the
matrix. hence you have here a typical "least squares" problem.
now it depends on the kind how your variables v_i enter the matrices T_i
what to do: should each of the elements of these matrices be an affine linear
function of the v_i, then you have a simple linear least squares problem and
can use one of the well established linear least squares solvers, for example
the one from LAPACK. (DGELSS)
Otherwise you have a nonlinear least squares problem and need a corresponding
code. there exist several good codes for this , for example ELSUNC is such ,
or , should the optimal sum of differences not be small,
port/n2f or port/n2g
anything in
http://www.netlib.org
or look here
http://plato.asu.edu/sub/nonlsq.html#lsqnres

hth
peter

.

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