Re: Minimizing the Frobenius norm


In article <1169162070.659648.204840@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"svlad" <madsvlad@xxxxxxxxx> writes:
Hi,

I need to minimize the following:

E(v1....vn) = sum_{ i=1 to L } (sum_{ j=adj(i) } || Ti - Tj ||^2 )
where || . || is the Frobenius norm.

Each of the T matrices are defined in terms of the v1....vn variables.
A subset of the v1....vn variables are known in advanced.

Each matrix T is dependent on exactly four of the n variables. adj(i)
is the subset of T that are adjacent to the current Ti. Adjacency is
defined as having two of the n variables in common.

Any help would be greatly appreciated.

Thanks.


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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